List of papers of V. S. Popov

Copied from INSPEC database 1/20/2004

Ovid: Bibliographic Records

Author: Zeldovich BYa. Perelomov AM. Popov VS.
Title: Relaxation of a quantum oscillator.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.55, no.2, 1968, pp.589-606. USSR.
Abstract: Relaxation of a quantum harmonic oscillator to the thermodynamic equilibrium state is considered on basis of a kinetic equation. The method of generating functions is employed for solving the equation. A number of exact solutions for various initial conditions is considered. For large quantum numbers it can be transformed to the Fokker-Planck equation. The Green function for the latter is determined. It is shown how the results obtained can be applied to the problem of statistics of photocounts. The given kinetic equation is extended to the case when an external classical force f(t) which arbitrarily depends on time acts on the oscillator. Methods for describing the evolution of the oscillator (characteristic functions, distributions of quasiprobabilities on the alpha plane) are briefly considered. The methods are valid in the presence of a force f(t). The equations of motion of the corresponding quantities are derived. The partial solutions for the density matrix rho are presented in a table. The so-called 'harmonic oscillator paradox' is discussed.

Author: Kotova LP. Perelomov AM. Popov VS.
Title: Quasiclassical approximation in ionization problems.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.54, no.4, April 1968, pp.1151-61. USSR.
Abstract: The paper is dedicated to the development of the complex trajectory technique (description of the tunnel effect in the quasiclassical case with the help of trajectories corresponding to imaginary 'times'). The following problems are considered: ionization of a bound level in collisions with a charged particle; ionization of atoms by a constant electric field in the presence of a magnetic field; evaluation of adiabatic corrections to the probability for tunnelling through a potential barrier of arbitrary shape.

Author: Belavin AA. Zel'dovich BYa. Perelomov AM. Popov VS.
Title: Relaxation of quantum systems with equidistant spectra.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.56, no.1, 1969, pp.264-74. USSR.
Abstract: The structure of a kinetic equation which describes the relaxation of quantum systems with equidistant spectra is discussed. Examples of such systems are harmonic oscillators (with linear or nonlinear damping), a spin in a magnetic field, an ensemble of two-level atoms interacting with a radiation field (Dicke problem). It is demonstrated that equations of this type retain positive determinacy of the density matrix. A solution of the kinetic equation is obtained for the Dicke problem and an exact quantum expression is found for the shape of the radiation line. Interference line narrowing due to the equidistant nature of the spectrum is discussed.

Author: Popov VS. Perelomov AM.
Title: Parametric excitation of a quantum oscillator.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.56, no.4, 1969, pp.1375-90. USSR.
Abstract: An exact solution is obtained for the problem of a quantum oscillator with a time dependent frequency omega (t) varying in an arbitrary manner. The quantum transition probabilities w/sub mn/ between the discrete spectrum states which are stationary for t to +or- infinity are calculated. The variation of the adiabatic invariant I=E/ omega under the action of a variable frequency omega (t) is calculated for an arbitrary initial state.

Author: Zel'dovich BYa. Perelomov AM. Popov VS.
Title: Relaxation of quantum oscillator in the presence of an external force.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.57, no.1, 1969, pp.196-206. USSR.
Abstract: A solution of the kinetic equation for a quantum oscillator with damping on which an external classical force f(t) is acting has been obtained. A number of exact solutions for various initial conditions are considered and the spectrum of the radiation emitted by the oscillator is determined.

Author: Popov VS. Perelomov AM.
Title: Parametric excitation of a quantum oscillator. II.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.57, no.5, Nov. 1969, pp.1684-90. USSR.
Abstract: For Pt.I see ibid. 56, 1375, 1969. An exact solution of the Schroedinger equation is obtained for a quantum oscillator with a varying frequency omega (t) which is acted upon by an external force f(t). The time dependence of omega (t) and f(t) may be arbitrary. The population distribution at the end of the processes is calculated for the case when at t to - infinity the oscillator was in the ground state mod 0, omega /sub -/). The quasi-energy spectrum for an oscillator with periodically varying parameters is determined.

Author: Popov VS.
Title: On the mean diffusion escape time.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.58, no.4, 1970, pp.1400-5. USSR.
Abstract: The problem of determining the mean time for escape from an arbitrary region by particles performing Brownian movement or diffusion is considered. The dependence of the diffusion coefficient on coordinates is assumed to be arbitrary. In the case of spherical symmetry the expressions for the mean escape time tau and its dispersion Delta tau /sup 2/ can be reduced to quadratures.

Author: Gribov VN. Kobsarev IYu. Mur VD. Okun LB. Popov VS.
Institution: Acad. Scis., Moscow, USSR.
Title: On properties of amplitudes not satisfying the Pomeranchuk theorem conditions.
Source: Physics Letters B, vol.32b, no.2, 8 June 1970, pp.129-32. Netherlands.
Abstract: Properties of the scattering amplitudes as functions of the momentum transfer (t) and of the impact parameter ( rho ) are investigated in the case when the total cross sections for particle (p) and antiparticle (a) are symptotically (s to infinity ) constant and are not equal to each other. It is shown that if the equality Imf/sup p/(s,t)=Imf/sup a/(s,t) is violated, the weaker integral equality integral ImF/sup p/(s,t)s/sup -2/ds approximately= integral ImF/sup a/(s,t)s/sup -2/ds, t[left angle bracket]0 must hold. The equality provides for the absence of singularities at t=0 in the amplitude. It follows from this equality that t-(and s-) dependence of ImF/sup p/ and ImF/sup a/ must be different.

Author: Popov VS.
Institution: USSR Acad. Sci., USSR.
Title: Electron energy levels at Z>137.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.11, no.5, 5 March 1970, pp.254-6. USSR.
Abstract: As is well known, the solution of the Dirac equation in the field of a point charge Ze is mathematically correct only if Z[left angle bracket]137. The energy of the lower level of the discrete spectrum (n=1, j=1/2) is epsilon /sub 0/= square root (1-a/sup 2/) (a=Ze/sup 2/=Z/137, h(cross)=c=m=1). (1) epsilon /sub 0/ reaches zero when alpha =1, and the continuation of formula (1) to the region alpha [right angle bracket]1 leads to imaginary values of epsilon /sub 0/. When alpha [right angle bracket]1 it is therefore necessary to take into account the finite dimensions of the nucleus. A critical charge is reached which determines the 'electrodynamic boundary' of the periodic system of elements and is also the pair production threshold.

Author: Popov VS.
Title: 'Fall down to the center' at Z>137 and critical nuclear charge.
Source: Yadernaya Fizika, vol.12, no.2, 1970, pp.429-47. USSR.
Abstract: Solution is considered of the Dirac equation for electron in the Coulomb field with Z[right angle bracket]137, when for the point nuclear charge an 'incidence to the center' occurs. Account of a finite size of the nucleus resulting in regularization of the Coulomb potential at r to 0 allows one to deal with charges Z[right angle bracket]137 up to a 'critical' value Z/sub cr/, such that the ground state energy reaches the boundary of the lower continuum, E=-mc/sup 2/. An equation for Z/sub cr/ is obtained and dependence of Z/sub cr/ on the nuclear radius R and on the charge distribution inside the nucleus is studied. Mean radius of the ground state is calculated and it is shown that at Z to Z/sub cr/ the electron state remains localized, though Sigma =E/mc/sup 2/ to -1 and lambda = square root (1- epsilon /sup 2/) to 0. Relativistic Coulomb problem is also discussed for particles with spins 0 and 1.

Author: Popov VS.
Title: Positron production in a coulomb field with Z>137.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.59, no.3, 1970, pp.965-84. USSR.
Abstract: Spontaneous production of positrons in a strong coulomb field with Z[right angle bracket]Z/sub cr/ is considered, where Z/sub cr/ is the critical value of the nuclear charge (for Z=Z/sub cr/ the energy of /sup 1/S/sub 1/2/ ground state reaches the limit of the lower continuum epsilon =-mc/sup 2/; the value of Z/sub cr/ always exceeds 137 and depends on the cut-off radius R and, to a less degree, on the nature of the electric charge distribution over the nuclear volume). It is shown that at the threshold (i.e. for (Z-Z/sub cr/)[left angle bracket][left angle bracket]Z/sub cr/) the probability for spontaneous positron creation vanishes exponentially. It is important for the analysis that for epsilon =-mc/sup 2/ (m is the electron mass). In this respect this case sharply differs from the usual nonrelativistic situation with epsilon to +mc/sup 2/.

Author: Gribov VN. Kobsarev IYu. Mur VD. Okun LB. Popov VS.
Title: On properties of the amplitudes not satisfying the conditions of the Pomeranchuk theorem.
Source: Yadernaya Fizika, vol.12, no.6, pp.1271-85. USSR.
Abstract: Properties of the scattering amplitudes as functions of the momentum transfer t and of the impact parameter rho are considered in case when the total cross sections for a particle and the antiparticle are asymptotically (s to infinity ) constant and not equal. A detailed consideration is performed for a so called scale-invariant case when the partial amplitudes ( up to a common factor) depend only on one variable x= rho /1n s.

Author: Popov VS.
Title: On the properties of a discrete spectrum for values of Z close to 137.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.60, no.4, 1971, pp.1228-44. USSR.
Abstract: The nature of the discrete spectrum for an electron in the Coulomb field of a bare nucleus of charge close to 137 is investigated. In this region of Z values the level spectrum significantly depends on cut-off of the Coulomb potential at small distances. An equation is derived for the level energies epsilon /sub n/. In the logarithmic approximation (i.e. on the condition that L=In (h(cross)/mcR)[right angle bracket][right angle bracket]1, where m is the electron mass and R the nuclear radius) the equation can be solved explicitly and the simple expressions can be obtained for epsilon /sub n/.

Author: Popov VS.
Institution: Acad. Sci., USSR.
Title: Production of e/sup +/e/sup -/ pairs in an alternating external field.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.13, no.5, 5 March 1971, pp.261-3. USSR.
Abstract: Interaction of an electromagnetic field with a vacuum of charged particles leads to the appearance of nonlinear additions to the Lagrangian L of the electromagnetic field. In particular, there appears an imaginary part Im L, which determines the probability w of pair production by the external field. In the case of fields E and H constant in time (but alternating in space) it is possible to solve this problem exactly. The author using Dirac's theory in a quasiclassical approach obtains probabilities for pair production in time-varying fields E(t).

Author: Popov VS.
Title: On the problem of the critical charge of nucleus.
Source: Yadernaya Fizika, vol.14, no.2, 1971, pp.458-68. USSR.
Abstract: The critical charge Z/sub cr/ is by definition the value of the charge of nucleus (or of the system of nuclei) at which the energy of the ground state of the discrete spectrum decreases down to the boundary of the lower continuum E=-mc/sup 2/. To calculate Z/sub cr/, the method of the effective potential is developed and the variational principle is formulated. With their help one can obtain Z/sub cr/ for the system of two colliding nuclei.

Author: Popov VS.
Title: Pair production in a varying external field (quasiclassical approximation).
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.61, no.4, 1971, pp.1334-51. USSR.
Abstract: A quasiclassical method is developed for calculating the probability for pair production in a varying electromagnetic field. The conditions of applicability of the method are: E[left angle bracket][left angle bracket]m/sup 2/c/sup 3//eh(cross), h(cross) omega [left angle bracket][left angle bracket]mc/sup 2/, where omega is the effective field variation frequency, E is the field amplitude and m mass of the particles produced by the field. Underlying the method is the concept of sub-barrier particle motion taking place along a classical trajectory but for an imaginary value of 'time'. The exact Green function for the Dirac equation in an external field is not required and the method can be applied to a very broad class of fields.

Author: Migdal AB. Perelomov AM. Popov VS.
Title: Analytical properties of wave functions at low energy.
Source: Yadernaya Fizika, vol.14, no.4, 1971, pp.874-85. USSR.
Abstract: Analytical properties of the continuum wave functions chi /sub k/(r) are investigated for the case when there is a level with energy near zero (the level may be real, virtual, or quasistationary). It is shown that at small distances, r[left angle bracket][left angle bracket]k/sup -1/, (k is the wave vector) the wave functions became factorizable: chi /sub k/(r) approximately=( Delta (k))/sup 1/2/ chi /sub 0/(r) while the factor Delta (k) changes sharply within the resonance width. Delta (k) is expressed in terms of the scattering amplitude at small k and is determined by the analyticity.

Author: Popov VS. Rozhdestvenskaya TI.
Title: Critical charge in collisions of nuclei.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.14, no.4, 20 Aug. 1971, pp.267-70. USSR.
Abstract: In order to verify the theory of supercritical atoms a calculation is made for the collision of the two bare uranium nuclei and the probability of positron emission as a function of internuclear distance.

Author: Perelomov AM. Popov VS.
Title: Perturbation of a continuous spectrum by a near-lying level.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.61, no.5, 1971, pp.1743-55. USSR.
Abstract: The nature of the perturbation of continuous spectrum functions chi /sub k/(r) and the analytic properties of the S matrix in the region k to 0 in the presence of a level close to zero are considered. Formulas are obtained which express parameters of the low energy expansion of the S matrix in terms of the wave function chi /sub 0/(r) at the moment of formation of the level. Exact solutions of the problem are obtained for a number of potentials.

Author: Popov VS.
Title: Method for a calculation of the nuclear critical charge.
Source: Yadernaya Fizika, vol.15, no.5, May 1972, pp.1069-81. USSR.
Abstract: A simple formula is given for the nuclear critical charge. The formula is asymptotically exact at the condition R[left angle bracket][left angle bracket]h(cross)/(m/sub e/c), where R is the nuclear radius. The dependence of the critical charge Z/sub c/ on R, on to character of the charge distribution inside the nucleus, etc., is investigated by the use of this formula. Perturbation theory for Z/sub c/ is considered.

Author: Marinov MS. Popov VS.
Title: Pair production in electromagnetic field (case of arbitrary spin).
Source: Yadernaya Fizika, vol.15, no.6, 1972, pp.1271-85. USSR.
Abstract: The quasiclassical method of 'imaginary time' is generalized for the case of a particle with spin. The production probability of pairs in strong electromagnetic field for particles with an arbitrary spin and magnetic moment is calculated.

Author: Migdal AB. Perelomov AM. Popov VS.
Title: Notes on analytic properties of wave functions at very low energy.
Source: Yadernaya Fizika, vol.16, no.1, 1972, pp.222-4. USSR.
Abstract: Proposes some additions and refinements to results obtained in studies of the behaviour of continuous spectral wave functions in the case where there is an almost-zero energy level in the system, and in the so-called 'narrow-pit' example. In regard to wave function factorization, a refinement is introduced for barrier permeability. Also an interpretation of the 'narrow pit' leads to expansion of the region of applicability of factorization.

Author: Zel'dovich YaB. Popov VS.
Title: Electronic structure of superheavy atoms.
Source: Uspekhi Fizicheskikh Nauk, USSR.
Abstract: The authors describe the status of the problem of the electron structure of superheavy atoms with nuclear charge Z[right angle bracket]Z/sub c/; here Z/sub c/ approximately=170 is the critical value of the nuclear charge, at which the energy of the ground state of the 1S/sub 1/2/ electron reaches the limit of the lower continuum of the solutions of the Dirac equation ( epsilon =-m/sub e/c/sup 2/). They discuss the dependence of Z/sub c/ on the nuclear radius R and on the character of the distribution of the electric charge inside the nucleus, and also the form of the wave functions at Z close to Z/sub c/. Owing to the Coulomb barrier, the state of the electron remains localized at Z[right angle bracket]Z/sub c/, in spite of the fact that its energy approaches the continuum. An analysis of the polarization of the vacuum in a strong Coulomb field shows that a bare nucleus with supercritical charge Z[right angle bracket]Z/sub c/ produces spontaneously two positrons and, in addition a charge density with a total of two units of negative charge in the vacuum.

Author: Popov VS. Marinov MS.
Title: e/sup +/e/sup -/-pair production in variable electric field.
Source: Yadernaya Fizika, vol.16, no.4, 1972, pp.809-22. USSR.
Abstract: The calculation of probability w of production of fermion pairs in vacuum under the influence of variable electric field E(t) comes to a solution of the problem on parametric excitation of quantum oscillator. The cases, when external field E(t) preserves constant direction in space or is situated in a definite plane, are treated in detail. By the 'imaginary time' method a closed formula has been obtained for probability w in the case of the field with elliptical polarization.

Author: Popov VS.
Title: Critical charge in the two-center problem.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.16, no.6, 20 Sept. 1972, pp.355-8. USSR.

Author: Popov VS.
Institution: Inst. Theoretical & Experimental Phys., USSR.
Title: Pair production in a variable and homogeneous electric field as an oscillator problem.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.62, no.4, April 1972, pp.1248-62. USSR.
Abstract: The calculation of the probability w of pair production in a strong electric field E(t) which is homogeneous in space reduces to the problem of the parametric excitation of a quantum oscillator (or, what is equivalent, to the calculation of the reflection coefficient for a plane wave incident on a one-dimensional potential barrier). Because of this relation, the calculation of the pair-production probability does not require the determination of the exact solutions of the relativistic wave equations and the Green's function. In the particular case when E(t)=E(cosh omega t)/sup -2/ the probability w coincides with the coefficient of reflection from the Eckart potential. The relation between the exact formulas for w and the quasiclassical approximation (the 'imaginary time' method) is investigated in detail, and the limits of applicability of the quasiclassical approximation are indicated.

Author: Popov VS.
Title: Critical charge for colliding nuclei.
Source: Yadernaya Fizika, vol.17, no.3, 1973, pp.621-33. USSR.
Abstract: Using the approximation (Z/sub i/+Z/sub 2/-Z/sub c/)[left angle bracket][left angle bracket]Z/sub c/, formulae are obtained for the critical charge in relativistic two-center problem. It allows one to find a critical distance R/sub c/ (Z/sub 1/, Z/sub 2/) between colliding nuclei at which the ground state energy of quasimolecule (Z/sub 1/, Z/sub 2/, e) reaches the boundary of the lower continuum and spontaneous production of positrons begins.

Author: Gershtein SS. Popov VS.
Institution: Inst. High-Energy Phys., Serpukhov, USSR.
Title: Spontaneous production of positrons in collisions of heavy nuclei.
Source: Lettere Al Nuovo Cimento, vol.6, ser.2, no.14, 7 April 1973, pp.593-6. Italy.
Abstract: The authors show that spontaneous production of positrons takes place even when only one of the nuclei is bare (and the second one has an ordinary electron shell structure). This fact provides an opportunity to carry out the experiment with a heavy target.

Author: Marinov MS. Popov VS.
Title: Effect of screening on the critical charge of a nucleus.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.17, no.9, 5 May 1973, pp.511-4. USSR.
Abstract: In view of the difficulty of calculating Z/sub c/ in the two-center problem the authors consider, in order to determine the scale of the screening correction, a model problem with a spherical superheavy nucleus.

Author: Popov VS. Mur VD.
Title: Quasistationary levels in lower continuum.
Source: Yadernaya Fizika, vol.18, no.3, 1973, pp.684-98. USSR.
Abstract: Quasistationary levels in the lower continuum of solutions of the Dirac equation are considered. Depth of the potential V(r) is assumed to be sufficient ( mod V mod [right angle bracket]2mc/sup 2/) for the bound states to cross the boundary of the lower continuum epsilon =-mc/sup 2/ at V=V/sub c/. In so doing the bound state becomes a quasistationary state whose width gamma determines probability of spontaneous positron production in the static field. Behaviour of gamma at the threshold V to V/sub c/ and at V[right angle bracket][right angle bracket]V/sub c/ is investigated. It is shown that gamma is limited even at mod epsilon mod [right angle bracket][right angle bracket]mc/sup 2/, i.e. for a level sunken deeply into the lower continuum. For the Coulomb field gamma /mc/sup 2/ approximately exp(-2 pi Z alpha )[left angle bracket][left angle bracket]1. This result shows that the description of the K shell of a supercritical atom with the nuclear charge Z[right angle bracket]Z/sub c/, (see abstr. A67385 of 1972), is valid not only at epsilon to -mc/sup 2/, but also for deep levels. The angular momentum (j) dependence of gamma and analytical properties of the S matrix near V=V/sub c/ are studied in detail. In particular, the case of the p/sub 1/2/ level, having some peculiar properties, is considered, as well as the s levels of a scalar particle.

Author: Popov VS.
Institution: Inst. Theoretical & Experimental Phys., Acad. Sci., USSR.
Title: Spontaneous positron production by the Coulomb field of a heavy nucleus.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.18, no.1, 5 July 1973, pp.53-8. USSR.
Abstract: The spontaneous production of positrons in a Coulomb field with Z[right angle bracket]Z/sub c/ (where Z/sub c/ approximately=170 is the critical charge of the nucleus) is of interest as an example of e/sup +/e/sup -/ pair production via an unusual static mechanism. The most realistic way of searching for this phenomenon is in collisions of heavy nuclei with Z/sub 1/+Z/sub 2/[right angle bracket]Z/sub c/, for example uranium or californium nuclei. It is not necessary that both colliding nuclei be 'bare' (i.e., have no electron shells). When a beam of bare nuclei Z/sub 1/ is incident on an ordinary heavy target Z/sub 2/, vacant places are produced in the K shell of the combined atom if Z/sub 1/[right angle bracket]Z/sub 2/. The cross section sigma for the production of e/sup +/ has in this case practically the same value as for bare nuclei.

Author: Popov VS.
Institution: Inst. Theoretical & Experimental Phys., USSR.
Title: The 'imaginary-time' method in problems concerning the ionisation of atoms and pair production.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.63, no.5, Nov. 1972, pp.1586-98. USSR.
Abstract: The 'imaginary-time' method was originally proposed in order to solve the problem of the ionisation of non-relativistic bound systems of atoms and ions in the field of an intense light wave. The method is a generalisation of the quasiclassical WKB approximation to the case of time-dependent fields. It describes the tunnelling transition of an electron from a bound state to the continuum by using the classical equations of motion, but with an imaginary time. Such an approach has also been found useful in connection with investigations of ionisation in Coulomb collisions. It has also been established that the method is successful in calculating the probability for pair production from vacuum due to an external field, although the relation between these two problems has remained nuclear. It is shown that the ionisation of atoms and the production of e/sup +/e/sup -/ pairs from vacuum correspond to two limiting cases of the general relativistic problem of the ionisation of a level with arbitrary energy E due to the influence of an internal electric field. Advantages associated with such a generalisation are discussed. In particular, by going to the non-relativistic limit a closed expression can be obtained for the atomic ionisation probability omega in an arbitrary electric field-a new result. The findings are of interest in laser technology.

Author: Popov VS.
Title: Spontaneous production of positrons at collisions of heavy nuclei (production cross section and the energy spectrum of e/sup +/).
Source: Yadernaya Fizika, vol.19, no.1, 1974, pp.155-68. USSR.
Abstract: The analytical solution of the two-centre Coulomb problem for the Dirac equation is obtained in the approximation (Z/sub 1/+Z/sub 2/-Z/sub c/)/Z/sub c/[left angle bracket][left angle bracket]1. Using the solution, the total cross section sigma of positron production in the quasistatic field of the colliding nuclei, the cross section d sigma /d Omega corresponding to the scattering of the nucleus under a fixed angle theta , and the energy spectrum of the produced positrons are calculated. Dependence of these quantities on the incident nucleus energy E is considered. This enables one to suggest the optimal energy region for the experiment. The similar problem of e/sup +/e/sup -/-pair production from vacuum in a variable electric field E(t) is discussed briefly.

Author: Popov VS.
Institution: Inst. Theoretical & Experimental Phys., USSR.
Title: Resonant pair production in a strong electric field.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.18, no.7, 5 Oct. 1973, pp.435-9. USSR.
Abstract: Pair production anomalies connected with the periodicity of the field are investigated. An analogy is found between the problems of fermion pair production from vacuum and resonant excitation of atomic levels in a strong optical wave field. The analogy no longer holds if excitation of multilevel atomic systems is considered.

Author: Popov VS.
Title: Method of imaginary time for periodical fields.
Source: Yadernaya Fizika, vol.19, no.5, 1974, pp.1140-56. USSR.
Abstract: The method of imaginary times is formulated for periodical fields, T(t+T)=F(t), where F(t) is an external electric field satisfying the conditions necessary for the quasiclassical approach to be consistent. Ionization of atoms by a strong light wave and pair creation from vacuum are considered as applications. Explicit expressions for the pair production probability w, applicable for any periodical field, are obtained in case of the linear polarization of the field F(t). The probability w has different forms for fermions and for bosons. Effect of the statistics of the created particles is most essential at p/sub ///=0, i.e. for particles produced in a direction transversal to the electric field. In this case boson pairs are created only if an even number of quanta is absorbed, n=2K, and fermion pairs are created only if an odd number is absorbed, n=2K+1. Properties of the exact solutions of the Klein-Gordan and Dirac's equations with a periodical field and the relation to the quasiclassical approach are discussed. Pair creation by an electric field with elliptic and circular polarizations is also considered by means of the method of imaginary times.

Author: Perelomov AM. Popov VS.
Title: Variational principle for critical charge.
Source: Teoreticheskaya i Matematicheskaya Fizika, vol.14, no.1, Jan. 1973, pp.18-26. USSR.
Abstract: In order to find the critical charge Z/sub V/, the effective potential method is developed and a variational principle is formulated.

Accession Number: 000676438
Author: Marinov MS. Popov VS. Stolin VL.
Title: Variational calculation of the critical distance between nuclei.
Source: Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.19, no.1, 5 Jan. 1974, pp.76-80. USSR.
Abstract: A variational calculation is made of the critical distance R/sub c/ between colliding nuclei, at which the fundamental term of the quasimolecule (Z,Z,e) drops to the limit of the lower continuum and spontaneous positron generation begins.

Author: Marinov MS. Popov VS.
Title: A variant of the Wentzel-Kramers-Brillouin method and calculation of the critical nuclear charge.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.67, no.4, Oct. 1974, pp.1250-62. USSR.
Abstract: A modification of the WKB method is proposed which consists in exclusion of centrifugal energy from the momentum p(r) and in a corresponding alteration of the constant gamma in the quantization condition gamma = gamma (l,a), where l is the orbital moment and V(r) approximately r/sup - alpha / for r to 0. The accuracy of the approximation and application of the WKB method for a strong coulomb field with a charge Z[right angle bracket]137 are considered in detail.

Author: Marinov MS. Popov VS.
Institution: Inst. Theoretical & Experimental Phys., USSR.
Title: On the production of e/sup +/e/sup -/ pairs from the vacuum under the influence of an electric field.
Source: Teoreticheskaya i Matematicheskaya Fizika, vol.17, no.1, Oct. 1973, pp.34-46. USSR.
Abstract: A study is made of the problem of pair production of particles with spin 1/2 in an external homogeneous electric field E(t). The Heisenberg equations of motion are found for the operators of creation and annihilation, a solution for which has the form of a linear canonical transformation. A system of differential equations that determine the coefficients U(t) and V(t) of this transformation are obtained. The probabilities of pair creation are expressed in terms of the eigenvalues of the matrix VV/sup +/. The case of particles with anomalous magnetic moment is also considered.

Author: Marinov MS. Popov VS.
Title: On energy spectrum of positrons produced in nuclear interactions at R
Source: Yadernaya Fizika, vol.20, no.6, 1974, pp.1223-8. USSR.
Abstract: The energy spectrum positrons produced in the collision of heavy nuclei with charges Z/sub 1/+Z/sub 2/[right angle bracket]Z/sub c/ is calculated when the ground level of the discrete spectrum dives into the lower continuum. The spectrum positron has a sharp peak near the upper boundary T=T/sub max/.

Author: Marinov MS. Popov VS.
Title: Critical distance in heavy ion collisions.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.62, no.2, Feb. 1975, pp.421-31. USSR.
Abstract: A variational calculation is carried out for the critical distance R/sub cr/ between colliding ions. Values of R/sub cr/ are calculated in the 90[left angle bracket]or=Z[left angle bracket]or=100 region with allowance for finite size of the nuclei. In this case R/sub cr/ greatly exceeds the nuclear radius r/sub N/ (for example R/sub cr/ approximately=50F for uranium nuclei). The results can be employed for checking the predictions regarding the cross section for spontaneous, positron production in Coulomb collisions of heavy nuclei.

Author: Mur VD. Popov VS.
Title: Pole model of the scattering amplitude with a highest possible growth of the cross section.
Source: Yadernaya Fizika, vol.21, no.4, 1975, pp.868-77. USSR.
Abstract: According to the known Froissart inequality, the total cross section sigma /sub t/ at s to infinity cannot grow more rapidly than ln/sup 2/ s. The pole models of partial amplitudes of the cross-channel leading to the highest possible growth sigma /sub t/ approximately ln/sup 2/ s and consistent with the unitarity condition in the s channel are considered. It is shown that in the region of the parameter rho approximately ln s' an automodel regime is being settled, i.e. the partial amplitudes of the s channel asymptotically depend only upon the ratio rho /ln s. General features of the amplitudes as well as the cases sigma /sub t/ approximately ln s are briefly discussed.

Author: Popov VS.
Title: Spontaneous positron production in collisions between heavy nuclei.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.65, no.1, July 1973, pp.35-53. USSR.
Abstract: It is shown that in collisions between a completely stripped (bare) nucleus (Z/sub 2/) and the atoms of a heavy target (Z/sub 1/), vacancies are formed in the K-shell of the united atom, providing Z/sub 2/[right angle bracket]or=Z/sub 1/. Therefore such collisions, as well as those between two bare nuclei, will result in spontaneous production of electron-positron pairs in a static Coulomb field of the colliding nuclei if the nuclei approach each other at a distance R[left angle bracket]R/sub c/ (R/sub c/ is a critical distance at which the ground level of the discrete spectrum becomes as low as the boundary of the lower continuum). In this case one or two electrons fill the vacancies in the K-shell of the united atom and the positrons escape to infinity. This circumstance should make feasible the experimental observation of spontaneous production of positrons in an ordinary heavy target by a bare nuclear beam. The kinematics of the process is considered and the threshold behavior of the positron production cross section and positron energy spectrum are considered.

Author: Marinov MS. Popov VS.
Institution: Inst. for Theoretical & Experimental Phys., Moscow, USSR.
Title: A modification of the Bohr-Sommerfeld quantization condition.
Source: Journal of Physics A-Mathematical & General, vol.8, no.10, Oct. 1975, pp.1575-85. UK.
Abstract: The Bohr-Sommerfeld quasiclassical quantization condition for a central potential is modified in such a way that the constant gamma becomes dependent on the angular momentum l and on the potential behaviour at small distances. This form of the quantization condition is especially simple in the case l=0. When applied to some particular potentials, taken as examples, the present approximation provides more accurate values of the level energies than the conventional form of the quantization condition. Application of the quasiclassical approach to singular potentials with 'collapse' and to the superstrong Coulomb field induced by a charge Z[right angle bracket]137 is also considered.

Author: Marinov MS. Popov VS.
Title: Phase shift technique for the Dirac equation and the calculation of the critical nuclear charge.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.65, no.6, Dec. 1973, pp.2141-54. USSR.
Abstract: A simple method for solving the Dirac and Schroedinger equations with an arbitrary potential V(r) is developed which is convenient for qualitative analysis of the problem and for numerical calculations. The idea of the method is that the equation under consideration is replaced by first order nonlinear equation for the phase function theta (r). Application of the method to the problem of appearance of a bound level and its application for calculation of the critical charge Z/sub c/ are considered. The effect of screening of the nuclear Coulomb field by the electron shell on Z/sub c/ is estimated. Variational calculations of Z/sub c/ for a system of colliding nuclei (relativistic two-center problem) are performed.

Author: Marinov MS. Popov VS. Stolin VL.
Institution: Inst. for Theoretical & Experimental Phys., Moscow, USSR.
Title: Variational approach to the relativistic two-center problem: Critical internuclear distance.
Source: Journal of Computational Physics, vol.19, no.3, Nov. 1975, pp.241-56. USA.
Abstract: As shown recently, if the distance between two heavy nuclei becomes less than its critical value, R/sub cr/, their electric potential is strong enough to create electron-positron pairs from the vacuum. This process may occur when the ground level of the quasimolecule (Z,Z,e) is lower than -m/sub e/c/sup 2/. In order to find R/sub cr/ the Dirac equation with two Coulomb centres is considered. By means of a proper choice of coordinates and using the variational principle the problem is reduced to a set of ordinary linear differential equations with singular coefficients. To solve the boundary problem the Riccati equation for the logarithmic derivative matrix was considered. The computations are carried out and R/sub cr/ is found as a function of the nuclear charges. By virtue of the variational principle the obtained quantity is a sure lower limit for R/sub cr/.

Author: Marinov MS. Popov VS.
Title: Two-center problem for the Dirac equation.
Source: Yadernaya Fizika, vol.23, no.2, 1976, pp.479-87. USSR.
Abstract: The two-center problem for the Dirac equation is considered in view of an experiment on the spontaneous production of positrons in collisions of heavy nuclei. The positrons are created when the internuclear distance is less than its critical value R/sub cr/. The critical distance R/sub cr/ is calculated by means of the variational approach for all the nuclear charges Z[left angle bracket]137. It is shown that at Z to 137 the function R/sub cr/(Z) has a finite limit. Formulae necessary for numerical calculations of the critical distance by means of the variational approach are presented.

Accession Number: 000977788
Author: Migdal AB. Voskresenskii DN. Popov VS.
Title: On the distribution of vacuum charge near supercharged nuclei.
Source: Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.24, no.3, 5 Aug. 1976, pp.186-8. USSR.
Abstract: A supercritical nucleus, surrounded by a vacuum shell, which remains around the nucleus after the departure of the positrons to infinity, is investigated. The behaviour of the potential and of the density of the electron cloud, near the edge of the supercharged nucleus, at Ze/sup 3/[right angle bracket][right angle bracket]1, is determined. The analysis is performed in the Thomas-Fermi approximation.

Accession Number: 000992915
Author: Eletskii VL. Popov VS.
Title: Critical charge for abnormal nuclei.
Source: Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.24, no.4, 20 Aug. 1976, pp.253-6. USSR.
Abstract: Results are presented of analysis and calculation of the critical charge Z/sub c/ for abnormal nuclei, specifically, a relationship is obtained for Z/sub c/ as a function of density of the nuclear substance and Z/A ratio. The results indicate that Z/sub c/ falls with rising n/sub p/ and, in the range n/sub p/[right angle bracket][right angle bracket]n/sub p//sup (o)/, approaches 137 value (critical charge for a point nucleus).

Author: Popov VS. Eletskii VL. Mur VD.
Title: Properties of deep-lying levels in a strong electrostatic field.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.71, no.3, Sept. 1976, pp.856-71. USSR.
Abstract: Properties of deep levels (real, virtual and quasistationary) lying near the boundary of the lower continuum are considered. The effective radius expansion is extended to the case of the Dirac equation. It is employed for investigating the level motion near the boundary epsilon =-mc/sup 2/ for various values of the moments j and l. In the limiting cases of R[left angle bracket][left angle bracket]h(cross)/mc and R[right angle bracket][right angle bracket]h(cross)/mc (R is the range of forces) the formulas can be simplified. The case of a wide potential well R[right angle bracket][right angle bracket]h(cross)/mc, in particular, can be reduced to the determination of the spectrum of bound and quasistationary states in the Schrodinger equation with a power potential. The asymptotic behavior of the critical nuclear charge Z/sub cr/ in the region R/sub N/[right angle bracket][right angle bracket]h(cross)/mc (R/sub N/ is the nuclear radius) is found and Z/sub cr/ is calculated for the muon for various distributions of electric charge in the nucleus. Differences in the behavior of the levels near epsilon =-mc/sup 2/ for scalar and spinor particles and the inapplicability of the Klein-Gordon single particle equation for Z[right angle bracket]or=Z/sub cr/ are discussed.

Author: Migdal AB. Popov VS. Voskresenskii DN.
Title: Distribution of a vacuum charge near super-charged nuclei.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.72, no.3, March 1977, pp.834-50. USSR.
Abstract: The distribution of a vacuum charge near super-critical nuclei (Ze/sup 2/[right angle bracket][right angle bracket]1) is found. The calculation is performed in the Thomas-Fermi approximation extended to the relativistic case. For Ze/sup 3/[right angle bracket]or approximately=1 the electron shell penetrates the nucleus and almost completely screens its charge. For Ze/sup 3/[right angle bracket][right angle bracket]1 the supercharged nucleus is an electrically neutral plasma with equal concentrations of electrons, protons and neutrons (when N=Z). The potential within the nucleus is constant and equals -V/sub 0/=-(3 pi /sup 2/n/sub p/)/sup 1/3/ approximately=-1.94 m/sub pi /c/sup 2/. Near the boundary of the nucleus there is a transition region with a width which is independent of Z. The electric field and surface charge are located in this region. The Coulomb energy of the nucleus, which impedes its stability, is reduced as a result of screening and the E/sub Q/ varies as Z/sup 5/3/ dependence is replaced by a Z/sup 2/3/ dependence. The Thomas-Fermi equation for a neutral atom is also considered.

Author: Mur VD. Popov VS.
Institution: Moscow Engng. Phys. Inst., Moscow, USSR.
Title: Bound states near the limit of the lower continuum (fermion case).
Source: Teoreticheskaya i Matematicheskaya Fizika, vol.27, no.2, May 1976, pp.204-16. USSR.
Abstract: The effective range approximation is generalized to the case of the Dirac equation. Formulas are obtained that express the parameters in the expansion of the S-matrix as epsilon to -mc/sup 2/ in terms of the wave function of the level at the critical point. These formulas are used to investigate the entry of levels of the discrete spectrum into the lower continuum. The critical charge of the nucleus for muons is calculated (the case R[right angle bracket][right angle bracket]h(cross)/mc).

Author: Mur VD. Popov VS.
Institution: Moscow Engng. Phys. Inst., Moscow, USSR.
Title: Bound states near the limit of the lower continuum (boson case).
Source: Teoreticheskaya i Matematicheskaya Fizika, vol.27, no.1, April 1976, pp.81-93. USSR.
Abstract: The well-known effective range approximation in nonrelativistic scattering theory is generalized to the case of scalar particles satisfying the Klein-Gordon equation. Exact expressions are obtained for the parameters of the expansion of the S-matrix and the energy of levels near the limit of the lower continuum in terms of the wave function at the point at which a bound state arises for antiparticles (i.e., at epsilon =-mc/sup 2/). These expressions are used to investigate the motion of the levels near the limit epsilon =-mc/sup 2/ for different values of the angular momentum l. It is shown that the p-level curve can 'bend' for potentials with sharp edge; this was known previously only for s-levels. A number of exactly solvable examples is considered. In particular, the exact solution of the Klein-Gordon equation for the s levels in the Hulthen potential is investigated in detail, together with the passage to the limit of unscreened Coulomb potential.

Author: Eletskii VL. Popov VS.
Title: Critical nuclear charge for higher levels. (Numerical calculations and the quasiclassical approach).
Source: Yadernaya Fizika, vol.25, no.5, 1977, pp.1107-12. USSR.
Abstract: Numerical results are presented for the critical nuclear charges corresponding to all the electronic levels crossing the boundary of the lower continuum at Z alpha [left angle bracket]9. Quasi-classical formulae for the number of states in the lower continuum are obtained and compared with the numerical calculation. Integral representations are obtained for the Macdonald function with an imaginary index and its derivative, that are appropriate for numerical computations at any values of the argument and the index.

Author: Lisin VI. Marinov MS. Popov VS.
Institution: Inst. for Theoretical & Experimental Phys., Moscow, USSR.
Title: Critical distance for the electron two-center problem.
Source: Physics Letters B, vol.69B, no.2, 1 Aug. 1977, pp.141-2. Netherlands.
Abstract: Critical distance R/sub cr/ for the spontaneous positron creation in heavy-ion collisions is calculated precisely for point-like nuclei with 85[left angle bracket]or=Z[left angle bracket]or=126. For example, in case of U-U collision R/sub cr/=38.4 fm. Rapid convergence of the method is clearly demonstrated.

Author: Popov VS. Eletskii VL. Turbiner AV.
Title: Summing asymptotic series in quantum field theory.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.26, no.3, 5 Aug. 1977, pp.193-7. USSR.
Abstract: Examines a method of summing series in the perturbation theory with factorially increasing coefficients a/sub k/. A connection is established between the asymptote of a/sub k/ when k to infinity and the nature of the special characteristics of the sum of the series. The boundaries of the range of applicability of the improved perturbation theory are found. The results obtained are tested on a number of physics problems for which exact solutions are known.

Author: Popov VS.
Title: New expressions for the invariant operators of the unitary groups.
Source: Teoreticheskaya i Matematicheskaya Fizika, vol.29, no.3, Dec. 1976, pp.357-69. USSR.
Abstract: The invariant operators or (Casimir operators) for the unitary groups U(n) and SU(n) are considered. The eigenvalues of these operators for arbitrary irreducible representation are expanded with respect to standard power sums S/sub k/. Expansion are obtained for the coefficients ( beta )/sub nu /( nu ); they hold for arbitrary rank n-1 of the group and arbitrary order p of the invariant operator. These expressions considerably simplify the calculation of the eigenvalues of the invariant operators (especially for large p), which is demonstrated by a number of examples.

Author: Popov VS. Eletskii VL. Marinov MS.
Title: Critical charge for anomalous nuclei.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.73, no.4, Oct. 1977, pp.1241-57. USSR.
Abstract: The dependence of the critical charge of nuclei on the proton density is found. The values of Z/sub cr/ for anomalous nuclei are obtained. The calculations of Z/sub CR/ are carried out by taking into account the diffusion of the nuclear boundary and also the effect of screening of the Coulomb nuclear potential by the electron shell. The Thomas-Fermi statistical method is employed for describing the electron density in the shell. Two cases are considered: 1) screening by an ordinary electron shell formed by electrons occupying levels of the discrete spectrum (-m[left angle bracket] epsilon [left angle bracket]m); 2) screening by a vacuum shell formed by electrons located at levels with energies of the atom, q=(Z-N)/Z is also found. The properties of electron states at the critical point are considered in detail. Asymptotic formulas for solution of the Thomas-Fermi equation are derived for two extreme cases: q to O (weakly ionized atom) and q to 1.

Author: Popov VS. Eletsky VL. Turbiner AV.
Institution: Inst. of Theoretical & Experimental Phys., Moscow, USSR.
Title: Borel summation of perturbation series in quantum mechanics and field theory.
Source: Physics Letters B, vol.72B, no.1, 5 Dec. 1977, pp.99-102. Netherlands.
Abstract: The Borel summation of factorial divergent perturbation series is considered. The relation between the asymptotics of the coefficients in the series and the character of the exact function singularity is established. The applicability limits for the improved perturbation theory are obtained. The results are tested on a number of physical problems for which the exact solutions are available.

Author: Eletsky VL. Voskresensky DN. Popov VS.
Title: Quasiclassical approximation and the Thomas-Fermi method for Z>137.
Source: Yadernaya Fizika, vol.26, no.5, 1977, pp.994-1004. USSR.
Abstract: The quasiclassical approximation is applied to solve the problem of the electron in the Coulomb field of the nucleus with the charge Z when Ze/sup 2/[right angle bracket]1 and to describe the multielectronic shell of the supercritical (Ze/sup 2/[right angle bracket][right angle bracket]1) atom. An approximate method based on the condition R[left angle bracket][left angle bracket]r (the short nuclear range approximation) is developed to calculate the physical characteristics of the electron states at the energy close to the boundary of the lower continuum. The accuracy of these methods is discussed. The 'fall to the centre' in the Thomas-Fermi relativistic equation for the electronic shell of the supercritical atom is studied.

Author: Eletskii VL. Popov VS.
Title: The Thomas-Fermi method for Z>137.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.73, no.6, Dec. 1977, pp.2046-59. USSR.
Abstract: A solution is obtained for the relativistic Thomas-Fermi equation for the electron cloud density near supercritical (Ze/sup 2/[right angle bracket][right angle bracket]1) nuclei. The cases of a vacuum ion (supercritical nucleus surrounded by a vacuum shell), neutral atom and positive ion are considered. The angular momentum distribution of the vacuum shell electrons is found. For Ze/sup 3/[left angle bracket][left angle bracket]1 (weak screening) the electron shell of the supercritical atom consists of two parts which almost do not overlap. These are the vacuum shell with a radius r/sub a/ approximately Ze/sup 2//2 and the outer shells with a radius r/sub e/ approximately (Z/sup 1/3/e/sup 2/)/sup -1/. Here r/sub a//r/sub e/ approximately (Ze/sup 3/)/sup 4/3/[left angle bracket][left angle bracket]1. 'Falling to the center' is investigated in the relativistic Thomas-Fermi equation. The nature of the singularities of the equation solutions at r to 0 is determined by the renormalization group technique. A differential equation for the Gell-Mann-Low function beta ( mu ) is obtained and its asymptotic values for mu to 0 and mu to infinity are found. A numerical solution is obtained for all values of the effective coupling constant mu .

Author: Popov VS. Yeletskii VL. Turbiner AV.
Title: Higher orders in perturbation theory and the summation of series in quantum mechanics and field theory.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.74, no.2, Feb. 1978, pp.445-65. USSR.
Abstract: The Borel method for summation of the perturbation theory series with factorially increasing coefficients is considered. The relation between the asymptotic value of the coefficients of the series a/sub k/, for k to infinity , and the singularity of the sum is established. An improved perturbation theory is developed and the limits of its applicability are found. The results are verified for a number of physical problems for which the coefficients of the perturbation theory series exhibit factorial growth and the exact solutions are known. By applying the improved perturbation theory to the scalar field theory g phi /sup 4//4] it is possible to determine the behavior of the Gell-Mann-Low function psi (g) for 0[left angle bracket]g/16 pi /sup 2/[left angle bracket]or approximately=0.3. In this interval psi (g) is a monotonously growing function of g.

Author: Mur VD. Popov VS. Voskresenskii DN.
Title: On levels in the lower continuum.
Source: Yadernaya Fizika, vol.27, no.2, 1978, pp.529-41. USSR.
Abstract: Quasistationary levels belonging to the lower (negative) continuum of solutions of the Dirac equation are considered. Application of the WKB approximation enables one to solve the problem of the real part of the level energy completely. Explicit formulae are obtained for the level spectrum with the energy epsilon [left angle bracket]-mc/sup 2/, their numbers and the angular momentum distribution are found. Comparison with the numerical calculation shows that the WKB method has a reasonable accuracy even at zeta =Ze/sup 2//h(cross)c[right angle bracket]or approximately=2. The change of the asymptotics (r to infinity ) of the solutions of the Dirac equation at crossing the boundary of the lower continuum is explicated.

Author: Mur VD. Popov VS. Voskresenskii DN.
Title: The WKB method for Z>137.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.28, no.3, 5 Aug. 1978, pp.140-4. USSR.
Abstract: WKB approximation formulae are obtained for Dirac's equations in a strong ([right angle bracket]2 M/sub e/C/sup 2/ electron binding energy) external field, and used to calculate the energies and widths of the quasistationary states in the lower continuum. A preexponential factor for the probability of spontaneous positron production is also found.

Author: Eletskii VL. Popov VS.
Title: Calculation of the Gell-Mann-Loy function in scalar field theory.
Source: Yadernaya Fizika, vol.28, no.1, 1978, pp.268-70. USSR.
Abstract: Lipatov's work (1976, 1977) suggests a quasi-classical method of calculating functional integrals in perturbation theory, and analyses the coefficients of the Gell-Mann-Loy function in scalar field theory. It is proposed that Lipatov's method of calculating the asymptotic coefficients should be used in conjunction with the Pade-Borel approximation for accurate and economical computation of the Gell-Mann-Loy function. An example is presented where the function is calculated for a scalar field g phi /sup 4//4].

Author: Eletsky VL. Popov VS.
Institution: Inst. of Theoretical & Experimental Phys., Moscow, USSR.
Title: On the Pade-Borel summation of perturbation series.
Source: Physics Letters B, vol.77B, no.4-5, 28 Aug. 1978, pp.411-14. Netherlands.
Abstract: Pade-Borel summation of perturbation series with factorially growing coefficients is considered. Application of this method to scalar field theory g phi /sup 4//4] allows one to compute the Gell-Mann-Low function psi (g) at 0[left angle bracket]g/16 pi /sup 2/[left angle bracket]0.6. In this interval it grows monotonously with the coupling constant g and has no zeros.

Author: Mur VD. Popov VS.
Title: Semiclassical approximation for the Dirac equation in strong fields.
Source: Yadernaya Fizika, vol.28, no.3, 1978, pp.837-49. USSR.
Abstract: A consistent scheme of the semiclassical approximation for the Dirac equation in strong external field is developed. The approximate wave function is found both in classically allowed region and in the forbidden region and the joining conditions in the turning point are obtained. Equations are obtained for the energies of quasistationary levels lying in the lower continuum of solutions of the Dirac equation. The method is applied to relativistic corrections to the Bohr-Sommerfeld quantization rule, to calculation of the preexponential factor in probability of the position spontaneous production by the Coulomb field of a nucleus with charge Z[right angle bracket]Z/sub cr/. Semiclassical formulae are obtained for the number of levels dropped into the lower continuum.

Author: Eletskii VL. Popov VS.
Title: Summing of perturbation theory series by Pade and Pade-Borel methods.
Source: Yadernaya Fizika, vol.28, no.4, 1978, pp.1109-22. USSR.
Abstract: The series in the perturbation theory and quantum mechanics often have coefficients which increase as factorials a/sub k/ approximately const.(k alpha )]a/sup k/k/sup beta /, k to infinity . Therefore the convergence radius of these series is zero. The Pade and Pade-Borel methods allow to find the sum of the perturbation theory series psi (g)= Sigma a/sub k/(-g)/sup k/ (in a certain g interval) using the first coefficients a/sub k/ and their asymptotic behaviour at k to infinity . These methods (especially the Pade-Borel method) have essential advantages as compared to the ordinary and improved perturbation theory. The results are illustrated by a number of physical problems for which the coefficients a/sub k/ increase as factorials and at the same time the exact solutions are known. Application of the Pade-Borel method to the scalar field theory with the interaction H/sub int/=(g/4]) integral phi /sup 4/d/sup 4/x allows to find the Gell-Mann-Low function psi (g) at 0[left angle bracket]g/16 pi /sup 2/[left angle bracket]0.6. In this range psi (g) rises monotonously with increase of the coupling constant g and has no zeroes.

Author: Popov VS.
Institution: Inst. of Theoretical & Experimental Phys., USSR.
Title: Superheavy atoms.
Source: Izvestiya Akademii Nauk SSSR Seriya Fizicheskaya, vol.41, no.12, 1977, pp.2577-84. USSR.
Conference Information: Proceedings of the All-Union Conference 'Modern Theory of Atoms and Atomic Spectra. Leningrad, USSR. 1-4 Feb. 1977.
Abstract: A review is given of the present state of the theory of superheavy (atomic number greater than 137) atoms. Problems reviewed include the critical charge of the nucleus, entry of discrete levels into the lower (negative) continuum, properties of the electron shells of supercritical atoms and spontaneous creation of positrons in slow collisions between heavy ions.

Author: Dolgov AD. Popov VS.
Institution: Inst. of Theoretical & Experimental Phys., Moscow, USSR.
Title: Modified perturbation theories for an anharmonic oscillator.
Source: Physics Letters B, vol.79B, no.4-5, 4 Dec. 1978, pp.403-5. Netherlands.
Abstract: A simple method of higher-order calculations in perturbation theory (PT) for an arbitrary gv(r) anharmonic oscillator is presented. The method is based on using the Riccati equation instead of the Schrodinger one. The structure of the perturbation series for the energy eigenvalues is investigated. If v(r) to exp(br/sup v/) as r to infinity , then the perturbation series is not Borel summable for 0[left angle bracket]v[left angle bracket]2. If v[right angle bracket]or=2 the PT expansion in powers of g does not exist. A new convergent version of PT valid for any g, 0[left angle bracket]g[left angle bracket] infinity , is proposed.

Author: Popov VS. Eletsky VL. Mur VD. Voskresensky DN.
Institution: Inst. of Theoretical & Experimental Phys., Moscow, USSR.
Title: WKB approximation for the Dirac equation at Z>137.
Source: Physics Letters B, vol.80B, no.1-2, 18 Dec. 1978, pp.68-72. Netherlands.
Abstract: The WKB formulae are obtained for the Dirac equation in a strong external field, i.e. a field in which the electron binding energy exceeds 2m/sub e/c/sup 2/ and the energy levels cross the lower continuum boundary. The energy and the width of the quasistationary states in the lower continuum are calculated as well as the preexponential factor in the probability of spontaneous positron production by the Coulomb field. A simple quasiclassical expression for the number of levels immersed into the lower continuum is obtained. Other applications of the WKB method to the theory of supercritical atoms are briefly discussed.

Author: Popov VS.
Institution: Inst. of Theoretical & Experimental Phys., USSR.
Title: Invariant operators for the classical groups.
Source: Teoreticheskaya i Matematicheskaya Fizika, vol.32, no.3, Sept. 1977, pp.344-7. USSR.
Abstract: The eigenvalues of invariant operators of arbitrary order for all classical groups are calculated. The eigenvalues are represented as polynomials of the power sums S/sub k/.

Author: Dolgov AD. Popov VS.
Title: Higher orders and the structure of perturbation theory for an anharmonic oscillator.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.75, no.6, Dec. 1978, pp.2010-26. USSR.
Abstract: A simple method is developed for the calculation of the higher orders in perturbation theory (PT) in powers of g for a D-dimensional isotropic oscillator with arbitrary anharmonicity gv(r). The method is based on a transition from the Schrodinger equation to the Riccati equation. In the important particular case of power law anharmonicity when v(r)=r/sup 2N/, N=2, 3, 4, ..., all the xi /sub n/(r) terms of the PT series become polynomials and hence calculation of the higher orders of perturbation theory is considerably simplified. A new variant of PT is proposed which converges for all values of the coupling constant g, 0[left angle bracket]or=g[left angle bracket] infinity . The structure of the PT series for the energy levels is investigated for potentials with power (v(r) approximately r/sup v/) and exponential (v(r) approximately exp(ar/sup 2//sup v/)) growth at infinity.

Author: Kudryavtsev AE. Popov VS.
Title: On the theory of the nuclear shift of proton-antiprotonic atom levels.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.29, no.5, 5 March 1979, pp.311-16. USSR.
Abstract: A formula is obtained, expressing the shift in coulomb levels of hadronic atoms in terms of nuclear potential scattering length (without a small shift presumption). Experimental data on the pp-atom 1s-level shift point to a possible bound state with a binding energy of about 0.9 MeV and width [left angle bracket]200 keV in the pp system.

Author: Popov VS. Voskresenskii DN. Eletskii VL. Mur VD.
Title: The WKB method for Z>137 and its application to the theory of supercritical atoms.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.76, no.2, Feb. 1979, pp.431-59. USSR.
Abstract: Formulas for the Dirac equation in a strong external field are obtained in the WKB approximation. A field is considered strong if the electron coupling energy exceeds 2m/sub e/c/sup 2/ and the discrete spectrum levels may be lowered into the continuum. The wave functions in the classically allowed and forbidden regions are found and the conditions for matching them on transition through the turning point are obtained. The WKB method is applied to the following problems: 1) generalization of the Bohr-Sommerfeld quantization conditions with allowance for relativistic effects and the spin; 2) calculation of the pre-exponential factor in the probability for spontaneous positron production; 3) the energy and width of the levels in the relativistic two-center problem; 4) effect of screening on the critical distance R/sub cr/ between the colliding nuclei; 5) tunneling through a nonspherical barrier and the positron angular distribution; 6) allowance for the finite velocity of nuclei. Some other problems which lie outside the limits of applicability of the WKB method are also discussed e.g. the electron spectrum Ze/sup 2/=1 and the deep levels in the lower continuum (Ze/sup 2/[right angle bracket][right angle bracket]1, j[left angle bracket]or approximately=(Ze/sup 2/)/sup 1/2/).

Author: Dolgov AD. Popov VS.
Institution: Inst. of Theoretical & Experimental Phys., Moscow, USSR.
Title: The anharmonic oscillator and its dependence on space dimensions.
Source: Physics Letters B, vol.86B, no.2, 24 Sept. 1979, pp.185-8. Netherlands.
Abstract: The perturbation series (PS) in powers of g for the D-dimensional gr/sup 4/ anharmonic oscillators is considered. High-order terms of the ground-state-energy E(g,D) expansion are explicitly calculated with the help of recurrence relations among integers. The rate of approach of the PS coefficients to the asymptotic value as a function of D is investigated. For the case of D=0 explicit solutions are obtained and analytic properties of E(g,0) are briefly discussed. For D[right angle bracket][right angle bracket]1 the first terms in the expansion of E(g,D) in powers of D/sup -1/ are found.

Author: Alliluev SP. Eletsky WL. Popov VS.
Institution: Moscow Phys. Tech. Inst., Moscow, USSR.
Title: The high orders of perturbation theory for the Stark effect in hydrogen.
Source: Physics Letters A, vol.73A, no.2, 3 Sept. 1979, pp.103-6. Netherlands.
Abstract: An effective method is developed to calculate the high orders of the perturbation series (PS) for the Stark shift of the energy levels in the hydrogen atom. The method is based on the use of the Riccati equation instead of the Schrodinger equation. The PS coefficients alpha /sub k/ (hyperpolarizabilities) are given by simple recurrence relations among integers. For the ground state alpha /sub k/ are calculated exactly for k[left angle bracket]or=14 and numerically up to k=160. The asymptotic behaviour of alpha /sub k/ when k to infinity is discussed for the energy eigenvalues E/sub n/, 1[left angle bracket]or=n[left angle bracket]or=5. With the results obtained the energy and the width of the ground state are determined for external fields E[left angle bracket]or approximately=0.1 au.

Author: Popov VS. Kudryavtsev AE. Mur VD.
Title: Nuclear level shift and radiative transitions in a proton-antiproton atom.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.77, no.5, 1979, pp.1727-50. USSR.
Abstract: A formula for the Coulomb level shifts in a proton-antiproton atom is derived in terms of the scattering length for a strong potential V/sub s/. The rearrangement of the atomic spectrum when a bound state in the V/sub s/ potential appears is discussed on the basis of the formula. The relation between the level shift and effective radius expansion is pointed out. The mean radius of the s-state for an arbitrary value of the shift and the probabilities for radiative E1 transitions between p- and s-levels are calculated. From the experimental data on the 1s-level shift in the pp-atom it is concluded that a bound Qs-state (quasinuclear meson) with a binding energy epsilon approximately=1 MeV and width Gamma [left angle bracket]or approximately=200 keV may exist. Calculations show that the probabilities for 2p to 1s and 2p to Qs radiative transitions are approximately equal. This indicates that it should be feasible to observe experimentally the Qs-level by investigating the gamma -ray spectrum produced in transitions between pp atomic levels.

Author: Mur VD. Popov VS.
Title: The theory of a nuclear shift of levels in heavy atoms.
Source: Yadernaya Fizika, vol.31, no.3, 1980, pp.617-25. USSR.
Abstract: The results of Kudryavtsev et al. (1979) are generalized to the case of the relativistic Coulomb problem when the parameter zeta =Za approximately=Z/137 is not supposed to be small. The method of matching derivatives is used, the Klein-Gordon and Dirac equations being solved exactly outside the nucleus (r[right angle bracket]r/sub N/). For mr/sub N/[left angle bracket][left angle bracket]1 the nuclear shift of the levels is written in an explicit analytical form. The derived formulae are applied to calculate the critical distance in heavy ion collisions (taking into account finite size of the colliding nuclei).

Author: Lisin VI. Marinov MS. Popov VS.
Institution: Inst. for Theoretical & Experimental Phys., Moscow, USSR.
Title: Critical electron state in heavy-ion collisions.
Source: Physics Letters B, vol.91B, no.1, 24 March 1980, pp.20-2. Netherlands.
Abstract: The results of a calculations of the electron wave function for the critical state epsilon =-m/sub e/c/sup 2/) and of some related quantities are presented. The effect of the finite nuclear size on the critical internuclear distance R/sub cr/ is taken into account. The calculations are performed by means of the variational method for nuclei with charges 85[left angle bracket]or=Z[left angle bracket]or=100. The error of the WKB method for strong Coulomb fields is found to be a few percent.

Author: Mur VD. Kudryavtsev AE. Popov VS.
Title: On the influence of the nn-channel on the properties of a proton-antiproton atom.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.31, no.7, 5 April 1980, pp.429-33. USSR.
Abstract: The influence of the second channel (nn) on nuclear level shifts and the probabilities of radiation transitions in a pp atom are studied. The obtained formulae can also be used on other hadronic atoms ( Sigma /sup -/p, K/sup -/p etc.)

Author: Eletskii VL. Mur VD. Popov VS.
Title: On the WKB method for Z>137.
Source: Doklady Akademii Nauk SSSR, vol.249, no.1-3, Nov. 1979, pp.329-33. USSR.
Abstract: The authors study the WKB method for a strong external field (in the Coulomb problem this corresponds to the region Z[right angle bracket]137). For the relativistic wavefunctions they obtain approximations that can be used in the neighborhood of a turning point and also consider the question of the accuracy of the WKB method.

Author: Alliluev SP. Eletsky VL. Popov VS. Weinberg VM.
Institution: Moscow Phys. Tech. Inst., Moscow, USSR.
Title: Stark effect in hydrogen: high orders of perturbation theory for excited states.
Source: Physics Letters A, vol.78A, no.1, 7 July 1980, pp.43-6. Netherlands.
Abstract: The method proposed previously by the authors (1978-79) for computation of high order terms of perturbation series (PS) is generalized to excited states with wave-functions having nodes. The computation of the PS coefficients E/sub k/ for the Stark shift in hydrogen is reduced to solving a system of recurrence relations. With these relations exact values of E/sub k/ are calculated up to k=12 for all states with the principal quantum number n=2 as well as for the states with n/sub 1/=n/sub 2/=0, mod m mod =n-1 and n[left angle bracket]or=5. Numerical values of E/sub k/ are obtained for k[left angle bracket]or=100. This makes it possible to investigate the approach of E/sub k/ to their asymptotics E/sub k/ and to determine power corrections to E/sub k/.

Author: Eletsky VL. Popov VS.
Institution: Inst. of Theoretical & Experimental Phys., Moscow, USSR.
Title: On perturbation theory for the gr/sup 2N/ anharmonic oscillator.
Source: Physics Letters B, vol.94B, no.1, July 1980, pp.65-8. Netherlands.
Abstract: Perturbation series (PS) in powers of the coupling constant g for D-dimensional anharmonic oscillator with power anharmonicity gr/sup 2N/ is considered. The high order PS coefficients epsilon /sub k/ for the ground state energy are calculated explicitly with the help of recurrence relations among integers. The rate of approach of epsilon /sub k/ to their asymptotics epsilon /sub k/ as a function of space dimension D is discussed.

Author: Eletskii VL. Popov VS.
Title: The divergence of the perturbation theory series for the Stark effect and summation of the series.
Source: Doklady Akademii Nauk SSSR, vol.250, no.1-3, Jan. 1980, pp.74-7. USSR.
Abstract: Effective methods for calculating the higher orders of perturbation theory have been found in recent years in quantum mechanics and field theory. A number of problems of quantum mechanics are known for which the perturbation-theory coefficients possess a factorial increase and at the same time one can obtain (analytically or numerically) an exact solution of the Schrodinger equation. The anharmonic oscillator and the Stark effect are classical examples. The results for the Stark effect (a hydrogen atom in a uniform electric field) are described.

Author: Dolgov AD. Eletskii VL. Popov VS.
Title: A new approach to perturbation theory for a discrete spectrum (anharmonic oscillator).
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.79, no.5, Nov. 1980, pp.1704-18. USSR.
Abstract: An effective method is developed for calculating the higher orders of perturbation theory (PT) for discrete spectrum. The method is based on the transition from the Schrodinger equation to the Riccati equation for the logarithmic derivative of the wave function. For the polynomial potentials (and in particular for the case of an anharmonic oscillator with a gr/sup 2N/ nonlinearity, N=2, 3, 4,...) the calculation of higher orders of PT is reduced to simple recurrence relations. Using these relations the coefficients in the PT series for the energy levels are found exactly for k[left angle bracket]or=10 and numerically up to k approximately 200 (k is the PT order). A rapidly converging PT with the behavior of the potential at singular points taken into account is developed on the basis of the Riccati equation.

Author: Popov VS. Kudryavtsev AE. Lisin VI. Mur VD.
Title: On the theory of nuclear level shifts in hadronic atoms.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.80, no.4, 1981, pp.1271-87. USSR.
Abstract: Due to strong short range interaction the Coulomb levels of hadronic atoms experience shifts and broadening. The presence in the problem of a small parameter r/sub e//a/sub B/ makes it possible to develop an analytic theory of the phenomenon (r/sub e/ is an effective range of the strong potential and a/sub B/=h(cross)/sup 2//m/sup e2/ is the Bohr radius). Terms of the order of r/sub e/, the proton form factor and also the effect of the second channel (which is close in energy) are taken into account in the equation for the level shifts. The shifts and broadening of the s-levels and the restructuring of the atomic spectrum are investigated in the presence of absorption, i.e. for a complex scattering length. The region of applicability of perturbation theory in terms of the scattering amplitude is determined. The theory is extended to states with nonvanishing angular momenta.

Author: Eletsky VL. Popov VS. Weinberg VM.
Institution: Inst. of Theoretical & Experimental Phys., Moscow, USSR.
Title: Logarithmic perturbation theory for the screened Coulomb potential.
Source: Physics Letters A, vol.84A, no.5, 3 Aug. 1981, pp.235-8. Netherlands.
Abstract: Logarithmic perturbation theory (PT) is formulated for the screened Coulomb potential in powers of the screening parameter mu . Recurrence relations are obtained which allow one to compute successively the PT series coefficients E/sub k/ for the ground state energy E=E( mu ). On a number of examples the authors demonstrate that Pade summation of the PT series makes it possible to obtain the energy in the strong coupling region.

Author: Alliluev SP. Vainberg VM. Eletskii VL. Popov VS.
Title: Logarithmic perturbation theory for excited levels.
Source: Doklady Akademii Nauk SSSR, vol.254, no.4-6, Oct. 1980, pp.1392-6. USSR.
Abstract: A significant simplification of higher-order perturbation-theory (PT) calculations in quantum mechanics has recently been achieved. For perturbations of the form V(x)= Sigma /sub -2[left angle bracket]or=k[left angle bracket]or=N/c/sub k/x/sup k/, which are frequently encountered in practice, the calculation of the coefficients in the PT series for the energy levels is reduced to recurrence relations that are convenient for numerical calculation. In this way about 150 orders of PT were found for the anharmonic gr/sup 2N/-oscillator and the Stark effect in hydrogen. The authors continue the study of this method and generalize it to the case of excited states whose wave functions have nodes.

Author: Kudryavtsev AE. Lisin VI. Popov VS.
Title: Coulomb systems with short-action distance: Sigma /sup -/p atom and the resonance states of light nuclei.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.34, no.5, 5 Sept. 1981, pp.292-6. USSR.
Abstract: The equation connecting the shift and breadth of atomic levels with strong scattering length, is generalised for the case of complex pulses k and arbitrary values of momentum l and zeta =-Z/sub 1/Z/sub 2/. Estimates are presented for the shift and breadth of the s-levels for Sigma /sup -//sub p/ atom. The connection between the position and width of resonances in the case of Coulombic scattering is investigated.

Author: Veinberg VM. Eletskii VL. Popov VS.
Title: Logarithmic perturbation theory for screened Coulomb potential and charmonium potential.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.81, no.5, 1981, pp.1567-82. USSR.
Abstract: Logarithmic perturbation theory (PT) for screened Coulomb potential is considered. Calculation of higher PT orders of the level energy E/sub k/ is reduced to recurrent relations which are valid for any type of screening and are very convenient for computer calculations. They are employed to calculate the PT coefficient of E/sub k/ up to k=100 for a number of physically interesting cases such as the Yukawa potential, quarkonium potential etc. The approach of E/sub k/ at k to infinity to the asymptotics. E/sub k/, defined by the quasiclassical approximation, is discussed. The problem of summation of the divergent PT series is considered. By employing the Pade approximants and the asymptotics E/sub k/ it is possible to determine with a high accuracy the energy level E(g) for the potentials mentioned above in a range which considerably exceeds the range of applicability of PT, including the strong coupling region.

Author: Popov VS.
Title: Quantum electrodynamics of superforce fields.
Source: Priroda, no.10, Oct. 1981, pp.14-22. USSR.
Abstract: The author firstly describes the progress and results of investigations of the phenomena of quantum electrodynamics, i.e. the electromagnetic interactions of elementary particles, since Dirac's formulation in 1928 of his relativistic wave equation. Physical effects in superheavy nuclei are then described, and it is shown how these effects may be analysed in greater detail by applying the so-called 'method of effective potential', in which the Dirac equation is replaced by the simpler one of Schroedinger. Finally an account is given of the mechanism of formation of positrons (positively charged electrons) by the high-speed interaction of heavy nuclei, as in the Darmstadt experiment, the results of which confirm the applicability of the Dirac equation for describing the movement of particles in very powerful electromagnetic fields.

Author: Alliluev SP. Veinberg VM. Eletskii VL. Popov VS.
Title: Logarithmic perturbation theory and its application to the Stark effect in the hydrogen atom.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.82, no.1, 1982, pp.77-90. USSR.
Abstract: The technique considered by S.P. Alliluev et al. (see Phys. Lett. A, vol.78, p.43, 1980) for calculating of higher orders in perturbation theory (PT) in quantum mechanics is extended to excited states whose wave functions have nodes. Calculation of the E/sub k/ coefficients of the PT series for the level energies reduces to recurrent relations which are convenient for numerical computation. As an example the Stark effect in the hydrogen atom is considered for states with n=1 or 2 and also for all states with n/sub 1/=n/sub 2/=0. The asymptotic value for E/sub k/ for k to infinity which depends on the level width in weak fields is obtained. Summation of the divergent PT series makes it possible to determine the Stark shift and the level width in a strong electric field. The asymptotic value of the energy E(E) in extremely strong fields is found for a model which can be solved exactly (Stark effect for a one-dimensional delta -potential).

Author: Popov VS. Weinberg VM.
Institution: Inst. of Theoretical & Experimental Phys., Moscow, USSR.
Title: On summation of perturbation series in quantum mechanics.
Source: Physics Letters A, vol.90A, no.3, 28 June 1982, pp.107-9. Netherlands.
Abstract: Summation of a divergent perturbation series for a funnel-like potential is considered. It is done by the Pade-Borel method with the transformation of the variable which moves the nearest singularity of the Borel transform to infinity. The method makes it possible to compute the energy E(g) with a good accuracy for all physical values of the coupling constant g, including the behaviour of E(g) as g to infinity .

Author: Vainberg VM. Popov VS.
Title: The summation of perturbation series.
Source: Doklady Akademii Nauk SSSR, vol.264, no.1-3, May 1982, pp.330-3. USSR.
Abstract: The authors consider the summation of the perturbation series for the potential V(r)=-r/sup -1/+gr, h(cross)=m=e=1, They show that, if one knows exactly the first few coefficients of the perturbation series and their asymptotic form E/sub k/ for k to infinity , one can determine the energy E(g) not only for g to 0 but also in the strong coupling regime.

Author: Popov VS. Kudryavtsev AE. Lisin VI. Mur VD.
Institution: Inst. of Theoretical & Experimental Phys., Moscow, USSR.
Title: Relation between scattering lengths and effective radii in the charged and neutral channels.
Source: Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.36, no.6, 20 Sept. 1982, pp.207-10. USSR.
Abstract: The Coulomb correction to the strong scattering length a/sub l//sup (s)/ and the effective radius r/sub l//sup (s)/ are calculated for states with arbitrary angular momentum l. This allows relating the position and width near threshold resonances in charged and neutral channels for the lightest nuclei (/sup 5/He and /sup 5/Li, etc.). The Coulomb correction to a/sub l//sup (s)/ is especially large for the s wave, while the correction to r/sub l//sup (s)/ is large for the p wave, when they contain the large logarithm ln(a/sub B//r/sub 0/), where a/sub B/ is the Bohr radius and r/sub 0/ is the range of nuclear forces.

Author: Alliluev SP. Vainberg VM. Popov VS.
Title: Higher orders of perturbation theory for the screened Coulomb potential.
Source: Doklady Akademii Nauk SSSR, vol.265, no.1-3, July 1982, pp.597-601. USSR.
Abstract: The calculation of the higher orders of perturbation theory in quantum mechanics has been significantly advanced recently by the development of logarithmic perturbation theory. The authors consider its application to the screen Coulomb potential.

Author: Mur VD. Kudryavtsev AE. Popov VS.
Institution: Inst. of Theoretical & Experimental Phys., State Comm. on Use of Atomic Energy, USSR.
Title: Coulomb corrections to scattering length and effective range.
Source: Yadernaya Fizika, vol.37, no.6, June 1983, pp.1417-28. USSR.
Abstract: The authors consider the problem of determination of the purely nuclear scattering length a/sub s/ (pertaining to a strong-interaction potential V/sub s/ with the Coulomb interaction switched off) from the nuclear-Coulomb scattering length a/sub cs/ that is directly measured in experiment. The difference between a/sub s/ and a/sub cs/ is especially large if the potential V/sub s/ has a level (real or virtual) with energy close to zero. For such a case they have obtained the formulae relating the scattering lengths a/sub s/ and a/sub cs/ and the effective ranges r/sub s/ and r/sub ss/. The results are generalized to states with arbitrary angular momentum l. They show that the Coulomb correction to the coefficient of k/sup 2l/ in the effective-range expansion is especially large because of the large logarithm ln(a/sub B//r/sub 0/). The Coulomb renormalization of the other terms in the effective-range expansion is of the order r/sub 0//a/sub B/, where r/sub 0/ is the range of the nuclear force and a/sub B/ is the Bohr radius. The formulae obtained are illustrated for several model potentials V/sub s/ used in nuclear physics.

Author: Vainberg VM. Popov VS.
Title: Summing the perturbation series for the Yukawa potential.
Source: Doklady Akademii Nauk SSSR, vol.272, no.1-3, Sept. 1983, pp.335-40. USSR.
Abstract: Over the last few years, significant progress has been made in computing higher orders of perturbation theory (PT) in quantum mechanics and field theory. Very high orders of PT for a screened Coulomb potential have been computed-in particular, for the Yukawa potential V(r)=-e/sup - mu r/r/sup -1/ (up to k=100). Since the PT series for the energy levels E/sup (nl)/( mu )= Sigma /sub k=0//sup infinity /E/sub k//sup (nl)/ mu /sup k/ has zero radius of convergence, the computation of higher orders of the PT does not by itself solve the problem of finding the energy eigenvalues E/sup (nl)/ in the strong-coupling region. The authors consider the summation of the divergent series, taking the Yukawa potential as an example.

Author: Kudrjavtsev AE. Mur VD. Popov VS.
Institution: Inst. of Theoretical & Experimental Phys., Moscow, USSR.
Title: Possible existence of K/sup -/-light nucleus bound states.
Source: Physics Letters B, vol.143B, no.1-3, 9 Aug. 1984, pp.41-4. Netherlands.
Abstract: Experimental data on the shift of the K/sup -/ /sup 4/He atom 2p-level evidence a possible existence of a loosely bound nuclear p-state in this system, its binding energy and width being epsilon approximately J approximately 0.5 MeV. The probabilities of radiative transitions to this state and its production cross section in nuclear reaction on /sup 6/Li are calculated.

Author: Popov VS. Weinberg VM.
Institution: Inst. of Theor. & Exp. Phys., Moscow, USSR.
Title: On summation of perturbation series for screened Coulomb potentials.
Source: Physics Letters A, vol.107A, no.8, 25 Feb. 1985, pp.371-5. Netherlands.
Abstract: Higher orders of perturbation theory (PT) in quantum mechanics and field theory rise factorially at k to infinity (k is the PT order). The authors discuss the computation results of higher orders of PT and summation methods for divergent perturbation series (PS) for quantum mechanical problems: the Yukawa and Hulthen potentials, the funnel-shaped potential, etc. Criteria of PS summability by Pade approximants are examined.

Author: Popov VS. Vainberg VM. Mur VD.
Institution: Inst. of Theor. & Exp. Phys., USSR.
Title: Diverging perturbation-theory series and classical mechanics.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.41, no.10, 25 May 1985, pp.439-42. USSR.
Abstract: A summation of the diverging perturbation-theory series yields expressions for the energies of levels (real, virtual, and quasistationary) in terms of the coefficients of the perturbation theory, E/sub k/. Taking the limit of classical mechanics reveals the asymptotic behavior of the highly excited levels with l=n-1[right angle bracket][right angle bracket]1, confirming the results of the summation of the perturbation-theory series.

Author: Vainberg VM. Popov VS.
Title: Summation of divergent series in perturbation theory for screened Coulomb potentials.
Source: Doklady Akademii Nauk SSSR, vol.279, no.1-3, Nov. 1984, pp.59-63. USSR.
Abstract: As a rule, the higher orders of perturbation theory (PT) in quantum mechanics and field theory increase factorially as k to infinity , and so the PT series diverge however small the coupling constant g (the Dyson phenomenon). The authors consider the summation of the PT series for the screened Coulomb potential: V(r)=-r/sup -1/f(ur). In the limit mu to 0 the energies of the levels can be expanded in formal power series: E/sup (nl)/( mu )= Sigma /sub k=0//sup infinity /E/sub k//sup (nl)/ mu /sup k/. The method of 'logarithmic perturbation theory', which has been developed in recent years, can be used effectively to calculate the PT coefficients E/sub k/. Using this method the authors have calculated E/sub k/ for various screened potentials. In a number of cases (the Yukawa, Hulthen, and funnel potentials) the terms of PT series alternate in sign, while in other cases its structure is more complicated. Because of the factorial growth of the higher orders of PT, the point mu =0 is an essential singularity for the energy E( mu ). This restricts the use of the usual polynomials of PT to the region of very small mu .

Author: Karnakov BM. Mur VD. Popov VS.
Institution: Eng. Phys. Inst., Moscow, USSR.
Title: Atomic-spectrum restructuring when I not=0.
Source: Doklady Akademii Nauk SSSR, vol.279, no.1-3, Nov. 1984, pp.345-9. USSR.
Abstract: The authors discuss the restructuring of the atomic spectrum for states with l[right angle bracket]or=1. They give simple formulas that describe the shifts of l-level everywhere, including the region of intersection of terms. This analysis is, in the authors' opinion, of interest in itself, since 'atomic-spectrum restructuring' has a general quantum mechanical nature and can manifest itself not only in hadronic atoms. The phenomenon of restructuring of an atomic spectrum (for states with an arbitrary angular momentum l) is called the Zel'dovich effect.

Author: Popov VS.
Title: Fast-converging perturbation theory.
Source: Doklady Akademii Nauk SSSR, vol.279, no.4-6, Dec. 1984, pp.1362-7. USSR.
Abstract: The authors analyse nonstandard perturbation theory (NPT). A detailed study of the convergence of the higher approximations of the NPT is presented. As an example, they consider the NPT for a funnel potential.

Author: Popov VS. Kudryavtsev AE. Lisin VI. Mur VD.
Institution: Inst. of Theor. & Exp. Phys., USSR.
Title: Shallow nuclear levels and radiative transitions in hadronic atoms.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.88, no.3, March 1985, pp.715-27. USSR.
Abstract: The position of a nuclear level perturbing the Coulomb spectrum can be calculated in the analytic theory of nuclear level shifts from the level shift of a hadronic atom. The is illustrated by the example of the K/sup -4/He atom. Experimental 2p-level shifts suggest that this system may have a weakly-bound p-state with binding energy and width epsilon approximately gamma approximately 0.5 MeV. The probabilities of radiative transitions to this level and the cross section for its creation in nuclear reactions on /sup 6/Li are calculated. The possible existence of states in which K/sup -/ and p are weakly bound to other light nuclei is examined. An exact solution is obtained for a model Coulomb problem with short-range interaction, and is used to determine the range of validity of the initial approximations.

Author: Karnakov BM. Mur VD. Popov VS.
Institution: Moscow Eng.-Phys. Inst., USSR.
Title: Zel'dovich effect for states with l=0.
Source: Yadernaya Fizika, vol.41, no.5, May 1985, pp.1258-71. USSR.
Abstract: In systems bound by Coulomb potential distorted at short distances (r approximately r/sub 0/[left angle bracket][left angle bracket]a/sub B/) by the strong interaction a rearrangement of the atomic spectrum (Zel'dovich effect) can arise. Such systems are encountered in nuclear physics (the lightest hadronic atoms: pp, Sigma /sup -/p, K/sup -4/ He, etc.), in solid state physics, etc. Qualitative features of the Zel'dovich effect are discussed for states with nonzero angular momentum l related to the existence of two regions of attraction separated by a centrifugal barrier. Simple expressions are obtained which describe atomic level shifts including the region of 'crossing terms'. The influence of the Coulomb interaction on the location of nuclear levels is considered. Expressions of the improved perturbation theory are proposed for the calculation of scattering lengths and it is shown that in the case l[right angle bracket]or=1 they are very accurate.

Author: Mur VD. Popov VS.
Institution: Inst. of Theor. & Exp. Phys., State Comm. for Use of Atomic Energy, Moscow, USSR.
Title: Coulomb problem with short-range forces (exact solution).
Source: Yadernaya Fizika, vol.42, no.6, Dec. 1985, pp.1470-9. USSR.
Abstract: The nonrelativistic Coulomb problem with short-range interaction is considered. The model chosen for the strong potential V/sub s/(r) is the delta -function interaction on a sphere r-r/sub 0/, and this makes it possible to solve the Schrodinger equation exactly and to find explicit analytical expressions for the scattering lengths, effective ranges, phase shifts, etc. Detailed comparison of the exact solutions with approximate formulas obtained previously for an arbitrary potential V/sub s/ makes it possible to establish the limits of the range of applicability of these approximations.

Author: Mur VD. Popov VS.
Institution: Moscow Eng. Phys. Inst., USSR.
Title: Coulomb problem with short-range interaction: exactly solvable model.
Source: Teoreticheskaya i Matematicheskaya Fizika, vol.65, no.2, Nov. 1985, pp.238-49. USSR.
Abstract: The nonrelativistic Coulomb problem with short-range interaction is considered. The strong potential V/sub s/(r) is modeled by a delta-function interaction on a sphere r=r/sub 0/. Exact solutions to the Schrodinger equation are obtained for states with arbitrary angular momentum l together with explicit analytic expressions for the scattering lengths, effective ranges, etc. Comparison of the exact solutions with the approximate formulas established earlier for arbitrary short-range potential V/sub s/(r) makes it possible to determine the limits of applicability of these approximations.

Author: Mur VD. Popov VS.
Institution: Moscow Eng. Phys. Inst., USSR.
Title: Coulomb correction to low-energy scattering.
Source: Doklady Akademii Nauk SSSR, vol.284, no.1-3, Sept. 1985, pp.586-91. USSR.
Abstract: The calculation of the Coulomb corrections to low-energy scattering parameters (scattering length a/sub l/, effective radius r/sub l/, etc.) is important in applications and has been widely discussed in the literature. The authors demonstrate that the renormalization of the coefficients in the expansion of the effective radius generally contains a 'Coulomb logarithm' lna/sub B//r/sub 0/, with the singular term being independent of the strong potential V/sub s/(r) and calculable in explicit form. This circumstance is of interest in quantum scattering theory, and is also applicable to cases where r/sub 0/[left angle bracket][left angle bracket]a/sub B/ (for example, for the lightest hadron atoms).

Author: Vainberg VM. Mur VD. Popov VS. Sergeev AV.
Institution: Inst. of Theor. & Exp. Phys., USSR.
Title: Strong-field Stark effect.
Source: Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.44, no.1, 10 July 1986, pp.9-12. USSR.
Abstract: The Stark shifts and the widths of the ground and excited states of the hydrogen atom are calculated. Two independent calculation methods are used: a summation of a divergent perturbation-theory series and a 1/n expansion.

Author: Karnakov BM. Kudryavtsev AE. Mur VD. Popov VS.
Institution: Moscow Phys. Eng. Inst., USSR.
Title: Zel'dovich effect in quantum mechanics.
Source: Nuovo Cimento della Societa Italiana di Fisica B-General Physics Relativity Astronomy & Mathematical Physics & Methods, vol.98B, ser.2, no.1, 11 March 1987, pp.63-86. Italy.
Abstract: In systems bound by the Coulomb potential with a short-range distortion, the reconstruction of the atomic spectrum (or Zel'dovich effect) can appear. Some peculiarities of this phenomenon for the state of nonzero angular momentum are discussed. The analytical properties of the basic equation appearing in the presence of absorption are investigated. The values of absorption at which the reconstruction of atomic levels is replaced by an oscillation regime are given. Some applications of the Zel'dovich effect for hadronic atoms and mesomolecules are considered.

Author: Popov VS. Vainberg VM. Mur VD.
Institution: Inst. of Theor. & Exp. Phys., State Comm. on Use of Atomic Energy, USSR.
Title: Higher orders of perturbation theory, classical mechanics, and the 1/n expansion.
Source: Yadernaya Fizika, vol.44, no.4, Oct. 1986, pp.1103-13. USSR.
Abstract: The higher orders of perturbation theory in quantum mechanics and field theory increase as k] for k to infinity , where k is the order. The authors discuss the results of calculations of higher orders of perturbation theory and techniques of summing divergent perturbation series for a large class of quantum mechanical problems (including the Yukawa and Hulthen potentials, the V-funnel potential, and so on). It is shown that in this way it is possible to obtain very accurate values of the energy levels (real, virtual, and quasistationary) using the perturbation series coefficients E/sub k/. By going to classical mechanics the authors find the asymptotic form of the energies of highly excited states and calculate their quantum corrections. These calculations confirm the results of summation of the perturbation series.

Author: Vainberg VM. Popov VS.
Title: Summation of perturbation theory series and classical mechanics.
Source: Doklady Akademii Nauk SSSR, vol.289, no.4-6, Aug. 1986, pp.1095-9. USSR.
Abstract: The problem of calculating virtual and resonance levels is constantly encountered in atomic and nuclear physics, scattering, theory, etc. The authors present an approach to this problem based on the summation of perturbation theory (PT) series. They use the results of higher-order calculations for screened Coulomb potentials.

Author: Mur VD. Popov VS.
Institution: Inst. of Theor. & Exp. Phys., USSR.
Title: 1/n expansion and wave functions.
Source: Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.45, no.7, 10 April 1987, pp.323-6. USSR.
Abstract: The 1/n expansion is used to calculate wave functions. Analytic expressions are derived. They are valid for an arbitrary smooth potential V(r), in particular, for the asymptotic coefficients in the limits r to 0 and r to infinity . Comparison with numerical calculations shows that the wave function is calculated very accurately at the origin.

Author: Popov VS. Mur VD. Shcheblykin AV. Weinberg VM.
Institution: Inst. of Theor. & Exp. Phys., Moscow, USSR.
Title: 1/n-expansion for the Stark effect in strong field.
Source: Physics Letters A, vol.124, no.1-2, 14 Sept. 1987, pp.77-80. Netherlands.
Abstract: Stark shifts and widths of the ground and excited states of the hydrogen atom in a strong field E have been computed using the 1/n-expansion (up to E values comparable to the field on an electron orbit, i.e. E[right angle bracket]or approximately=n/sup -4/, and for n=1-30). Computation results for the states with parabolic quantum numbers n/sub 1/=n/sub 2/=0, m=n-1 are given and the Rydberg limit, n to infinity , is also considered.

Author: Vainberg VM. Orlov YuV. Popov VS. Tupovtsev VV.
Title: Virtual and resonance levels in the Yukawa potential.
Source: Ukrainskii Fizicheskii Zhurnal, vol.32, no.8, Aug. 1987, pp.1125-9. Ukrainian SSR, USSR.
Abstract: Two methods for computation of real, virtual and resonance energy eigenvalues are considered: summation of divergent perturbation series and solution of integral equations for t-matrix on physical and unphysical sheets of energy. It is shown on the example of the Yukawa potential that the energy eigenvalues given by both methods coincide with high precision. As an illustration the positions of real (deuteron) and virtual (singlet deuteron) poles of the neutron-proton scattering amplitude were calculated.

Author: Popov VS. Vainberg VM. Mur VD. Shcheblykin AV.
Institution: Inst. of Theor. & Exper. Phys., Moscow, USSR.
Title: 1/n expansion in quantum mechanics (H atom Stark effect).
Source: Doklady Akademii Nauk SSSR, vol.293, no.4-6, April 1987, pp.851-5. USSR.
Abstract: The authors consider the 1/n expansion in quantum mechanics by including the Stark effect in the H atom. They limit the discussion to the Bohr-atom model.

Author: Vainberg VM. Mur VD. Popov VS. Sergeev AV.
Institution: Inst. of Theor. & Exp. Phys., Moscow, USSR.
Title: Stark effect for the Rydberg states of the hydrogen atom.
Source: Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.46, no.5, 10 Sept. 1987, pp.178-82. USSR.
Abstract: The Stark shifts and the widths of highly excited (n=15-30) states of a hydrogen atom in a strong electric field, which is higher than the classical ionization threshold of an atom, are calculated. The results of the calculations are in agreement with the experiment.

Author: Vainberg VM. Mur VD. Popov VS. Sergeev AV.
Institution: Inst. of Theor. & Exp. Phys., USSR.
Title: The hydrogen atom in a strong electric field.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.93, no.2, Aug. 1987, pp.450-62. USSR.
Abstract: A calculation is reported of Stark shifts and widths of the ground and first excited states of the hydrogen atom in a strong electric field E up to values of E comparable with the field strength on an atomic orbit. Two independent methods are used, namely, summation of divergent perturbation-theory series and the 1/n expansion. The two methods have a wide region of overlap (in the parameters n/sub 1/, n/sub 2/, m, and E). The calculations are confirmed by considering the Rydberg limit as n to infinity .

Author: Mur VD. Popov VS.
Institution: Inst. of Theor. & Exp. Phys., State Comm. for Utilization of Atomic Energy, Moscow, USSR.
Title: The 1/n expansion and wave functions.
Source: Yadernaya Fizika, vol.47, no.3, March 1988, pp.697-704. USSR.
Abstract: The 1/n expansion is applied to the calculation of wave functions. Analytical formulas valid for an arbitrary smooth potential V(r) are obtained, in particular, for the asymptotic coefficients for r to 0 and r to infinity . Comparison with exact solutions (for power potentials) and with a numerical calculation (for a funnel potential) shows that the wave function at the origin is calculated with good accuracy ( approximately 1%). An approximate formula is also obtained for the effective range r/sub 1/ at which a nodeless bound state with orbital angular momentum l arises.

Author: Vainberg VM. Mur VD. Popov VS. Sergeev AV. Shcheblykin AV.
Institution: Inst. of Theor. & Exp. Phys., USSR.
Title: The 1/n expansion in quantum mechanics.
Source: Teoreticheskaya i Matematicheskaya Fizika, vol.74, no.3, March 1988, pp.399-411. USSR.
Abstract: The classical approximation (l=n-1 to infinity ) for the energy in /sup (0)/ and the semiclassical expansion in quantum mechanics are discussed. A recursive method is proposed for calculating the quantum corrections of arbitrary order to in /sup (0)/, this being valid for both bound and quasistationary states. The generalization of the method to states with an arbitrary number of nodes and the possibility of a more general choice of the parameter of the semiclassical expansion are considered. The method is illustrated by the example of the Yukawa and 'funnel' potentials and for the Stark effect in the hydrogen atom. These examples demonstrate the rapid convergence of the 1/n expansion even for small quantum numbers.

Author: Mur VD. Popov VS.
Institution: Inst. of Exper. & Theor. Phys., USSR.
Title: Scaling for the Stark effect in the Rydberg atoms.
Source: Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.48, no.2, 25 July 1988, pp.67-70. USSR.
Abstract: Equations determining the shift and widths of the Rydberg states in a strong electric field (for an arbitrary atom) and the scaling ratios for near-threshold Stark resonances with n/sub 1/ approximately n[right angle bracket][right angle bracket]1, n/sub 2/ and m approximately 1 have been obtained. These scaling ratios have been confirmed experimentally.

Author: Vainberg VM. Mur VD. Popov VS. Shcheblykin AV.
Institution: Int. of Theor. & Exp. Phys., Moscow, USSR.
Title: 1/n expansion for the Stark effect in a strong field.
Source: Doklady Akademii Nauk SSSR, vol.299, no.4-5, April 1988, pp.1116-20. USSR.
Abstract: The 1/n expansion is a promising method with applications in quantum mechanics, field theory, and other areas of theoretical physics. The authors present results of applying this method to the theory of the Stark effect in a strong field using the hydrogen atom as an example.

Author: Karnakov VM. Kudryavtsev AE. Mur VD. Popov VS.
Institution: Inst. of Theor. & Exp. Phys., USSR.
Title: Zel'dovich effect in atomic and nuclear physics.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.94, no.7, July 1988, pp.65-79. USSR.
Abstract: Restructuring of the atomic spectrum (the Zel'dovich effect) can occur in systems coupled by a Coulomb interaction that is distorted at short distances. The features of this phenomenon are discussed for states with arbitrary angular momentum l. The analytic properties of the solution of the basic equation for this effect in the case hadronic-atom spectra are investigated in the presence of absorption. The conditions under which the spectrum restructuring is replaced by an oscillatory regime are obtained. Some manifestations of the Zel'dovich effect in the physics of hadronic atoms and mesic molecules are considered.

Accession Number: 003416864
Author: Popov VS.
Institution: Inst. of Theor. & Exp. Phys., Moscow, USSR.
Editor: Fodor Z.
Title: Zel'dovich effect in atomic and nuclear physics.
Source: Eighth Balaton Conference on Nuclear Physics. Intermediate Energy Nuclear Physics. Proceedings of the International Conference on High Energy Nuclear Physics. Central Res. Inst. Phys. 1987, pp.257-65. Budapest, Hungary.
Conference Information: Balatonfured, Hungary. 6-11 June 1987.
Abstract: In systems bound by the coulomb interaction distorted at short distances there may arise a rearrangement of the atomic spectrum (or the Zel'dovich effect). Specific features of this effect are considered both in the absence and presence of absorption. The Coulomb renormalization of scattering lengths and effective ranges for any l is briefly discussed as well as some examples of the Zel'dovich effect in the physics of hadronic atoms and mesomolecules.

Author: Mur VD. Popov VS.
Institution: Inst. of Theor. & Exp. Phys., Moscow, USSR.
Title: Stark effect exhibited by Rydberg atoms and the near-threshold resonances of the photoionization cross section.
Source: Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.94, no.10, Oct. 1988, pp.125-34. USSR.
Abstract: The Stark shifts and the widths of the Rydberg states of atoms in a strong electric field are calculated in the vicinity of E approximately=0. The scaling relationships are obtained for above-barrier resonances. The theoretical results are compared with the experimental data on the photoionization of hydrogen and rubidium in the presence of a static electric field.

Author: Mur VD. Pozdnyakov SG. Popov VS.
Title: 1/n-expansion and calculation of wave functions.
Source: Doklady Akademii Nauk SSSR, vol.303, no.4-6, Dec. 1988, pp.1102-7. USSR.<