Complex WKB Method Research Articles

Bohr–Sommerfeld quantization condition for self-adjoint Dirac operators

We study the eigenvalue problem for a self-adjoint 1D Dirac operator. It is known that, near an energy level where the square of the potential makes a simple well, the eigenvalues are approximated by a Bohr–Sommerfeld type quantization rule. A remarkable difference from the Schrödinger case appears in the Maslov correction term. This fact was recently found by Hirota [Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov–Shabat operator, J. Math. Phys. 58 (2017) 102–108] under the analyticity condition of the potential using a complex WKB method. In this paper, we approach this problem with a microlocal technique focusing on the asymptotic behavior of the eigenfunction along the characteristic set to generalize the result to [Formula: see text] potentials.

Complex WKB Method (One-Dimensional Linear Problems on the Complex Plane)

Complex WKB Method (One-Dimensional Linear Problems on the Complex Plane)

Complex WKB method for a system of two linear difference equations

Analytic solutions of the difference equationΨ(z+h)=M(z)Ψ(z)\Psi (z+h)=M(z)\Psi (z)are explored. Herezzis a complex variable,h>0h>0is a parameter, andMMis a givenSL(2,C)SL(2,\mathbb {C})-valued function. It is assumed thatMMeither is analytic in a bounded domain or is a trigonometric polynomial. A simple method to derive the asymptotics of solutions ash→0h\to 0is described.

Magnetic WKB constructions on surfaces

This article is devoted to the description of the eigenvalues and eigenfunctions of the magnetic Laplacian in the semiclassical limit via the complex WKB method. Under the assumption that the magnetic field has a unique and non-degenerate minimum, we construct the local complex WKB approximations for eigenfunctions on a general surface. Furthermore, in the case of the Euclidean plane, with a radially symmetric magnetic field, the eigenfunctions are approximated in an exponentially weighted space.

WKB asymptotics of meromorphic solutions to difference equations

We consider the difference Schrödinger equation where z is a complex variable, E is a spectral parameter, and h is a small positive parameter. If the potential v is an analytic function, then, for h sufficiently small, the analytic solutions to this equation have standard semi-classical behavior that can be described by means of an analog of the complex WKB method for differential equations. We assume that v has a simple pole and, in its neighborhood, we study the asymptotics of meromorphic solutions to the difference Schrödinger equation.

The Complex WKB Method for Difference Equations and Airy Functions

We consider the difference Schrodinger equation $\psi(z+h)+\psi(z-h)+ v(z)\psi(z)=0$, where $z$ is a complex variable, $h>0$ is a parameter, and $v$ is an analytic function. As $h\to0$, analytic so...

Complex WKB method for a difference Schrödinger equation with the potential being a trigonometric polynomial

Complex WKB method for a difference Schrödinger equation with the potential being a trigonometric polynomial

The Complex WKB Method for Difference Equations in Bounded Domains

We study entire solutions to the difference Schrodinger equation ψ(z + h) + ψ(z − h) + v(z)ψ(z) = Eψ(z), z ∊ C, where h > 0 and E ∊ C are parameters, and v is a trigonometric polynomial. We describe asymptotic behavior of the solutions as h tends to zero.

Spectral series of the Schrödinger operator in a thin waveguide with a periodic structure. 2. Closed three-dimensional waveguide in a magnetic field

In the paper, which is the second part of the paper by J. Bruning, S. Dobrokhotov, S. Sekerzh-Zenkovich, T. Tudorovskiy, “Spectral series of the Schrodinger operator in thin waveguides with a periodic structure. 1,” Russ. J. Math. Phys. 13 (4), 401–420 (2006), using the adiabatic approximation, diverse quantum states of the stationary Schrodinger equation for a particle in a thin waveguide in a magnetic field are constructed. The problems of “destruction” of the adiabatic approximation as the value of energy increases and of replacing this approximation by the approximation of V. P. Maslov’s theory of complex germ (the complex WKB method) are studied.

Complex WKB method for adiabatic perturbations of a periodic Schrödinger operator

This review is devoted to an analog of the complex WKB method for studying solutions of a one-dimensional periodic Schrodinger equation with an adiabatic perturbation. We illustrate with examples how to use the method. Bibliography: 30 titles.

Poles of integrále tritronquée and anharmonic oscillators. A WKB approach**Ai miei zii Ivan e Silvia.

Poles of solutions to the Painlevé-I equations are intimately related to the theory of the cubic anharmonic oscillator. In particular, poles of integrále tritronquée are in bijection with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We analyze this pair of quantization conditions by means of a suitable version of the complex WKB method.

Klein paradox and scattering theory for the semi-classical Dirac equation

We study the Klein paradox for the semi-classical Dirac operator on R with potentials having constant limits, possibly different at infinity. Using the complex WKB method, the time-independent scattering theory in terms of incoming and outgoing plane wave solutions is established. The corresponding scattering matrix is unitary. We obtain an asymptotic expansion, with respect to the semi-classical parameter h, of the scattering matrix in the cases of the Klein paradox, the total transmission and the total reflection. Finally, we treat the scattering problem in the zero mass case.

Complex WKB analysis of a -symmetric eigenvalue problem

The spectra of a particular class of -symmetric eigenvalue problems has previously been studied, and found to have an extremely rich structure. In this paper, we present an explanation for these spectral properties in terms of quantization conditions obtained from the complex WKB method. In particular, we consider the relation of the quantization conditions to the reality and positivity properties of the eigenvalues. The methods are also used to examine further the pattern of eigenvalue degeneracies observed by Dorey et al (2001 J. Phys. A: Math. Gen. 34 5679 (Preprint hep-th/0103051), 2001 J. Phys. A: Math. Gen. 34 L391 (Preprint hep-th/0104119)).

Fermion Particle Production in a Periodic Potential

In this paper we study fermion particle production in the early universe. The present work is motivated to restudy the fermion particle production from the basics and compare the results in the literature through another method developed by one of the present author. One of the authors (SB) has developed a method, known as complex trajectory WKB method, to study particle production in curved as well as flat spacetime. In the present work we have tried to compare the CWKB method with that of other works, current in the literature. In this work we have obtained the particle production amplitude starting from the basics and test our results through both analytical and numerical calculations. For fermion particle production, we first do analytical calculations with a toy example to calculate the production amplitude and verify the same doing fourth order Runge-Kutta calculation. As most problems relevant to early universe are not amenable to analytical calculations, we then take up to study the particle production in periodic potential, generally used in inflationary cosmology. We recheck two recent approaches and obtain almost identical results as that obtained by Greene and Kofman. We also verify the result through CWKB method. Boson particle production has been discussed elsewhere, we discuss it briefly in connection with CWKB. In the present work we generalize the CWKB results of boson production to fermion production. Our works will enable one to understand the various phenomena in early universe related to particle production. Using CWKB we calculate the occupation number and some other results for fermion particle production. The present work will help us clarify the variant results of fermion production current in the literature.

LOCALIZED FAMILIES OF BENDING WAVES IN A THIN MEDIUM-LENGTH CYLINDRICAL SHELL UNDER PRESSURE

The initial-boundary-value problem for the equations describing motion of a thin, medium-length, non-circular cylindrical shell is examined. The shell edges are not necessarily plane curves, with the conditions of a joint support, a rigid clamp or a free edge being considered as the boundary conditions. The shell is supposed to experience normal internal (or external) dynamic pressure which may be non-uniform in the circumferential direction. It is assumed that the initial displacements and velocities of the points at the shell middle surface are functions decreasing rapidly away from some generatrix. Using the complex WKB method the asymptotic solution of the governing equations is constructed by superimposing localized families (wave packets) of bending waves running in the circumferential direction. The dependence of frequencies, group velocities, amplitudes and other dynamic characteristics upon variable pressure and geometrical parameters of the shell are studied. As an example, the wave forms of motion of a circular cylindrical shell with sloping edges under growing dynamic pressure are considered. The effect of localization of bending vibrations near the longest generator as well as the effects of reflection, focusing and increasing amplitude in the running wave packets are revealed.

Anderson Transitions for a Family of Almost Periodic Schrödinger Equations in the Adiabatic Case

This work is devoted to the study of a family of almost periodic one-dimensional Schrodinger equations. Using results on the asymptotic behavior of a corresponding monodromy matrix in the adiabatic limit, we prove the existence of an asymptotically sharp Anderson transition in the low energy region. More explicitly, we prove the existence of energy intervals containing only singular spectrum, and of other energy intervals containing absolutely continuous spectrum; the zones containing singular spectrum and those containing absolutely continuous are separated by asymptotically sharp transitions. The analysis may be viewed as utilizing a complex WKB method for adiabatic perturbations of periodic Schrodinger equations. The transition energies are interpreted in terms of phase space tunneling.

On the Complex WKB Method for a Secondary Turning Point Problem

On the Complex WKB Method for a Secondary Turning Point Problem

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.

Wave packets in a thin cylindrical shell under a non-uniform axial load

The propagation of localized families of flexural plane waves in a thin elastic shell under an unsteady axial compression, which is non-uniform with respect to the peripheral coordinate, is considered. The shell can be non-circular and open in the peripheral direction. Conditions of hinged support are specified at the edges, which are optionally plane curves. It is assumed that the initial perturbations of the shell (non-zero initial displacements and velocities) are functions which are localized in the neighbourhood of a certain generatrix. The solution is constructed using the complex WKB-method developed in [1] in the form of a superposition of packets of flexural plane waves propagating in the peripheral direction of the shell. This paper differs from [1] in that in addition to taking account of axial stresses, the solution in the direction of the axis of the shell is assumed to be extremely variable. It is shown that it is possible to use the results obtained to investigate the free vibrations of the shell. The unsteady localized vibrations of a cylindrical shell with an elliptic cross-section under axial forces which increase linearly with time are considered as an example.

Free vibrations of a viscoelastic noncircular cylindrical shell under homogeneous axial loading

Free low-frequency vibrations of a noncircular cylindrical shell subject to an axial static load are investigated. The case where the vibrations are localized near a certain generatrix, called the weakest, is considered. The complex WKB-method is used to determine the natural modes of vibrations that damp with time and drop down far from the weakest generatrix. An example on the vibration of a viscoelastic shell is considered with the relaxation rate function given and in the presence of a static axial force.