A Class of Potentials for Which Exact Semiclassical Quantization Can Be Achieved

We consider a class of potentials for which the exact semiclassical quantization is achieved by a certain modification of the quantization condition. A list of potentials for which the new quantization condition is exact coincides with the list of potentials for which the spectrum is determined by the factorization method. We construct a one-parameter family of quantization conditions including the supersymmetric WKB condition as a special case. The new condition allows considering the interrelations between different modifications of the leading approximation and their validity ranges and also allows developing new approximate methods for calculating spectra.     

The reduced semiclassical description method

We develop a new version of the semiclassical analysis of a system of bound states in centrally symmetrical potentials. The set of potentials is in a 1∶1 correspondence with a certain set of pairs of functions of the orbital momentum. The first of these functions determines the usual WKB quantization condition and groups the potentials into equivalence classes. Its Mellin transform demonstrates similar behavior for the typical potentials, which allows describing the equivalence class with a small number of parameters. We can chose these parameters as the asymptotically exact estimates of the number of states. We obtain an equation that allows classifying states in a self-consistent atomic potential without knowing the explicit form of the potential. The second of these functions distinguishes the potentials within an equivalence class and also gives the first correction to the quantization condition. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 1, pp. 99–115, July, 1999.     

Instanton Effects and Quantum Spectral Curves

We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov–Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in \({\hbar}\), and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or unrefined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string.     

The WKB method for the quantum mechanical two-Coulomb-center problem

Using a modified perturbation theory, we obtain asymptotic expressions for the two-center quasiradial and quasiangular wave functions for large internuclear distances R. We show that in each order of 1/R, corrections to the wave functions are expressed in terms of a finite number of Coulomb functions with a modified charge. We derive simple analytic expressions for the first, second, and third corrections. We develop a consistent scheme for obtaining WKB expansions for solutions of the quasiangular equation in the quantum mechanical two-Coulomb-center problem. In the framework of this scheme, we construct semiclassical two-center wave functions for large distances between fixed positively charged particles (nuclei) for the entire space of motion of a negatively charged particle (electron). The method ensures simple uniform estimates for eigenfunctions at arbitrary large internuclear distances R, including R ≥ 1. In contrast to perturbation theory, the semiclassical approximation is not related to the smallness of the interaction and hence has a wider applicability domain, which permits investigating qualitative laws for the behavior and properties of quantum mechanical systems.     

Chiral solitons in a quantum potential

We study chiral solitons in a quantum potential using a dimensional reduction of the problem for (2+1)-dimensional anyons. We show that the integrable version of the model is described by a family of the resonant derivative nonlinear Schrödinger equations. For a quantum potential strength s > 1, we show that the chiral soliton interaction has a resonance. We investigate the semiclassical quantization procedure for solitons.     

‐symmetry and Schrödinger operators. The double well case

We study a class of ‐symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double‐well potential. In the simple well case, two of the authors have proved in 6 that, when the potential is analytic, the eigenvalues stay real for a perturbation of size . We show here, in the double‐well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB‐analysis, leading to a fairly explicit quantization condition.     

Spectral properties of solutions of hypergeometric-type differential equations

The second-order differential equation σ(x)y″ + τ(x)y′ + λy = 0 is usually called equation of hypergeometric type, provided that σ, τ are polynomials of degree not higher than two and one, respectively, and λ is a constant. Their solutions are commonly known as hypergeometric-type functions (HTFs). In this work, a study of the spectrum of zeros of those HTFs for which , v , and σ, τ are independent of ν, is done within the so-called semiclassical (or WKB) approximation. Specifically, the semiclassical or WKB density of zeros of the HTFs is obtained analytically in a closed way in terms of the coefficients of the differential equation that they satisfy. Applications to the Gaussian and confluent hypergeometric functions as well as to Hermite functions are shown.     

Semiclassical Asymptotics and Gaps in the Spectra of Magnetic Schrödinger Operators

In this paper, we study an L ² version of the semiclassical approximation of magnetic Schrödinger operators with invariant Morse type potentials on covering spaces of compact manifolds. In particular, we are able to establish the existence of an arbitrary large number of gaps in the spectrum of these operators, in the semiclassical limit as the coupling constant goes to zero.     

A sharp estimate for the rate of convergence in mean of Birkhoff sums for some classes of periodic differentiable functions

For a badly approximable vector α, we obtain a sharp estimate for the rate of convergence in the space L _p (0 < p < ∞) of the Birkhoff means $\frac{1}{n}\sum\nolimits_{s = 0}^{n = 1} {f(x + s\alpha )} $ for an absolutely continuous periodic function f and for functions in spaces of Bessel potentials.     

Semiclassical Spectral Series of a Helium-like Atom in a Magnetic Field

We apply the complex WKB method (the Maslov complex germ theory) to the model of two electrons in a field with a fixed center. We construct semiclassical spectral series of the Pauli operator eigenvalues in the external magnetic field with the spin–orbital and spin–spin interactions and the quadrupole moment of the nucleus taken into account. These series correspond to a new type of closed phase trajectories, the relative equilibrium positions of the corresponding classical nonintegrable system. Explicit effective formulas are derived for the fine (Zeeman effect) and the hyperfine splitting of semiclassical energy levels of a helium-like ion with an arbitrary nucleus charge Z in the entire range of the magnetic field magnitude, including the extreme cases of weak and ultrastrong fields.     

Maslov Complex Germ Method for Systems with First-Class Constraints

We develop the semiclassical mechanics of systems with first-class constraints. A convenient quantization method is the method based on modifying the inner product used in the theory. We consider semiclassical states of the wave-packet type (with small indeterminacies in both coordinates and momenta) that appear in the theory of the Maslov complex germ at a point. We show that these states have a nonzero norm only if the classical coordinates and momenta lie on the constraint surface. The set of semiclassical states of the wave-packet type forms a (semiclassical) bundle whose base is the set of admissible classical states and whose fibers are function spaces determining the form of the wave packet. In some cases, the difference between two semiclassical states has a zero norm; it is therefore possible to introduce the gauge equivalence relation. The semiclassical gauge transformations that are automorphisms of the semiclassical bundle form a Batalin quasigroup. We also study the action of semiclassical observables and of semiclassical evolution transformations. We show that they preserve the norm and the gauge equivalence relation and that the observables coinciding on the constraint surface act on semiclassical states similarly up to the gauge invariance.     

Relativistically Covariant Quantum Field Theory of the Maslov Complex Germ

We consider an explicitly covariant formulation of the quantum field theory of the Maslov complex germ (semiclassical field theory) in the example of a scalar field. The main object in the theory is the “semiclassical bundle” whose base is the set of classical states and whose fibers are the spaces of states of the quantum theory in an external field. The respective semiclassical states occurring in the Maslov complex germ theory at a point and in the theory of Lagrangian manifolds with a complex germ are represented by points and surfaces in the semiclassical bundle space. We formulate semiclassical analogues of quantum field theory axioms and establish a relation between the covariant semiclassical theory and both the Hamiltonian formulation previously constructed and the axiomatic field theory constructions Schwinger sources, the Bogoliubov S-matrix, and the Lehmann-Symanzik-Zimmermann R-functions. We propose a new covariant formulation of classical field theory and a scheme of semiclassical quantization of fields that does not involve a postulated replacement of Poisson brackets with commutators.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 492–512, September, 2005.     

Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results

The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectra of free and periodic Schrödinger operators are preserved under all perturbations satisfying This result is optimal in the power scale. Slightly more general classes of perturbing potentials are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on maximal function and norm estimates, and on almost everywhere convergence results for certain multilinear integral operators.

     

Uniformization of WKB functions by Wigner transform

We develop a technique for uniformizing WKB functions which fail to correctly represent wave fields on caustics due to geometric singularities of ray fields. The uniformization technique is based on appropriate asymptotic surgery of the Wigner transform of the WKB functions, in different regions of the phase space. We present the details of the computation for the model example of the semiclassical Airy equation and we explain how the method can be applied to higher dimensional WKB functions for fold caustics     

Spectral distributions for long range perturbations

We study distributions which generalize the concept of spectral shift function, for pseudo-differential operators on . We call such distributions spectral distributions. Relations between relative scattering determinants and spectral distributions are established; they lead to the definition of regularized scattering phase. These relations are analogous to the usual one for the standard spectral shift function. We give several asymptotic properties in the high energy and semiclassical limits where both nontrapping and trapping cases are considered. In particular, we prove Breit-Wigner formulae for the regularized scattering phases, for semiclassical Schrödinger operators with long-range potentials.     

The Asymptotic Form of the Lower Landau Bands in a Strong Magnetic Field

The asymptotic form of the bottom part of the spectrum of the two-dimensional magnetic Schrödinger operator with a periodic potential in a strong magnetic field is studied in the semiclassical approximation. Averaging methods permit reducing the corresponding classical problem to a one-dimensional problem on the torus; we thus show the almost integrability of the original problem. Using elementary corollaries from the topological theory of Hamiltonian systems, we classify the almost invariant manifolds of the classical Hamiltonian. The manifolds corresponding to the bottom part of the spectrum are closed or nonclosed curves and points. Their geometric and topological characteristics determine the asymptotic form of parts of the spectrum (spectral series). We construct this asymptotic form using the methods of the semiclassical approximation with complex phases. We discuss the relation of the asymptotic form obtained to the magneto-Bloch conditions and asymptotics of the band spectrum.     

Schrödinger Operator Levels for a Crystal Film with a Nonlocal Potential

For a crystal film, we consider the Schrödinger operator defined on Bloch functions (with respect to two variables) in a cell. The potential is the sum of two small terms: a function decreasing with respect to the third variable and an operator of rank one. We prove the existence of two levels (eigenvalues or resonances) near the parameter value E=0 and obtain their asymptotic behavior.     

“Classical” equations of motion in quantum mechanics with gauge fields

For the Schrödinger and Dirac equations in an external gauge field with symmetry groupSU(2), we construct to any degree of accuracy in powers ofh ^1/2,h0, approximate dynamical states in the form of wave packets—semiclassical trajectory-coherent states. For the quantum expectation values calculated with respect to these semiclassical states we obtain for the operators of the coordinates, momenta, and isospin of the particle Hamiltonian equations of motion that are equivalent (in the relativistic case after transition to the proper time) to Wong's well-known equations for a non-Abelian charge with isospin 1/2.Moscow Engineering Physics Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 41–61, July, 1992.     

Small time and semiclassical asymptotics for stochastic heat equations driven by Lévy noise

The paper is devoted to the study of stochastic heat equations driven by Lévy noise. Applying the WKB method, we obtain multiplicative small time and semiclassical asymptotics for the Green functions and for solutions of the Cauchy problem for the heat equation under some natural additional assumptions on their coefficients. The first step in this construction consists in solving the corresponding stochastic Hamilton-Jacobi equations which constitute the "classical part" of the semiclassical approximation. In its turn, the corresponding Hamilton-Jacobi equations can be solved via solutions of the corresponding Hamiltonian systems, which gives rise to the method of stochastic characteristics. The relevant theory of stochastic Hamiltonian systems and stochastic Hamilton-Jacobi equations was developed in our previous papers. Here we put the final rung on the ladder: stochastic Hamiltonian systems, stochastic Hamilton-Jacobi equations, stochastic heat equations.     

Semiclassical maslov asymptotics with complex phases. I. General approach

A method of constructing semiclassical asymptotics with complex phases is presented for multidimensional spectral problems (scalar, vector, and with operator-valued symbol) corresponding to both classically integrable and classically nonintegrable Hamiltonian systems. In the first case, the systems admit families of invariant Lagrangian tori (of complete dimension equal to the dimensionn of the configuration space) whose quantization in accordance with the Bohr—Sommerfeld rule with allowance for the Maslov index gives the semiclassical series in the region of large quantum numbers. In the nonintegrable case, families of Lagrangian tori with complete dimension do not exist. However, in the region of regular (nonchaotic) motion, such systems do have invariant Lagrangian tori of dimensionk (incomplete dimension). The construction method associates the families of such tori with spectral series covering the region of intermediate quantum numbers. The construction includes, in particular, new quantization conditions of Bohr—Sommerfeld type in which other characteristics of the tori appear instead of the Maslov index. Applications and also generalizations of the theory to Lie groups will be presented in subsequent publications of the series.Moscow Institute of Electronic Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 2, pp. 215–254, August, 1992.