V. G. Bagrov

Darboux transformation, factorization, and supersymmetry in one-dimensional quantum mechanics

We introduce an N-order Darboux transformation operator as a particular case of general transformation operators. It is shown that this operator can always be represented as a product of N first-order Darboux transformation operators. The relationship between this transformation and the factorization method is investigated. Supercharge operators are introduced. They are differential operators of order N. It is shown that these operators and super-Hamiltonian form a superalgebra of order N. For N=2, we have a quadratic superalgebra analogous to the Sklyanin quadratic algebras. The relationship between the transformation introduced and the inverse scattering problem in quantum mechanics is established. An elementary N-parametric potential that has exactly N predetermined discrete spectrum levels is constructed. The paper concludes with some examples of new exactly soluble potentials.

Darboux transformation of the Schrodinger equation

The recent developments in the theory of the generation of potentials for which the Schrodinger equation has an exact solution are discussed. The generalization of the Darboux transformation to the nonstationary Schrodinger equation is studied in detail. The supersymmetric generalization of the nonstationary Schrodinger equation is formulated. Versions corresponding to exact and spontaneously broken supersymmetry are discussed. New, exactly solvable nonstationary potentials are obtained as examples. The stationary Darboux transformation is viewed as a special case of the new transformation. Families of isospectral potentials with the spectra of the harmonic oscillator and the hydrogen-like atom are obtained. The effectiveness of these methods for describing the coherent states of the transformed Hamiltonians is demonstrated.

Quasiclassical trajectory‐coherent states of a particle in an arbitrary electromagnetic field

In this paper we show that for a nonrelativistic charged particle moving in an arbitrary external electromagnetic field there exist approximate solutions of the Schrodinger equation, such that the quantum‐mechanical averages of the coordinates and the momenta with respect to these states are general exact solutions of the classical Hamiltonian equations. Such states are called trajectory‐coherent states. The wave functions of the trajectory‐coherent states are obtained by the complex germ method by V. P. Maslov. The simplest properties of these states are studied.

Supersymmetry of a nonstationary Schrödinger equation

Abstract The supersymmetry of a one-dimensional time-dependent Schrodinger equation is established. It is intimately connected with the time-dependent Darboux transformation. With the help of this transformation new exactly solvable time-dependent potentials are generated.

Spin equation and its solutions

The aim of the present article is to study in detail the so-called spin equation (SE) and present both the methods of generating new solution and a new set of exact solutions. We recall that the SE with a real external field can be treated as a reduction of the Pauli equation to the (0 + 1)-dimensional case. Two-level systems can be described by an SE with a particular form of the external field. In this article, we also consider associated equations that are equivalent or (in one way or another) related to the SE. We describe the general solution of the SE and solve the inverse problem for this equation. We construct the evolution operator for the SE and consider methods of generating new sets of exact solutions. Finally, we find a new set of exact solutions of the SE.

Solutions of relativistic wave equations in superpositions of Aharonov–Bohm, magnetic, and electric fields

We present new exact solutions (in 3+1 and 2+1 dimensions) of relativistic wave equations (Klein–Gordon and Dirac) in external electromagnetic fields of special form. These fields are combinations of Aharonov–Bohm solenoid field and some additional electric and magnetic fields. In particular, as such additional fields, we consider longitudinal electric and magnetic fields, some crossed fields, and some special nonuniform fields. The solutions obtained can be useful to study the Aharonov–Bohm effect in the corresponding electromagnetic fields.

Darboux transformation and elementary exact solutions of the Schrödinger equation

Darboux transformation is applied to three classical potentials, namely the harmonic oscillator, effective Coulomb and Morse potentials to generate exactly solvable potentials of elementary form. For every potential, the isospectral families of potentials are constructed. For almost all potentials, a set of normalized discrete spectrum wave functions is given.

Coherent states for anharmonic oscillator Hamiltonians with equidistant and quasi-equidistant spectra

Two kinds of transformation for the time-dependent Schrodinger equation, i.e. the differential and integral transformations, are introduced. If one considers only stationary solutions of this equation, both transformations reduce to the well known Darboux transformation for the stationary Schrodinger equation. When applied to non-stationary solutions, they give different results. Both transformations are invertible in appropriate spaces. With the help of these transformations alternative systems of coherent states to those in the literature are obtained for isospectral Hamiltonians with equidistant spectra. These transformations are also applied to the construction of coherent states for Hamiltonians whose spectrum consists of an equidistant part and one separately disposed level with an energy gap equal to the k skipped levels.

New solutions of relativistic wave equations in magnetic fields and longitudinal fields

We demonstrate how one can describe explicitly the present arbitrariness in solutions of relativistic wave equations in external electromagnetic fields of special form. This arbitrariness is connected to the existence of a transformation, which effectively reduces the number of variables in the initial equations. Then we use the corresponding representations to construct new sets of exact solutions, which may have a physical interest. Namely, we present new sets of stationary and nonstationary solutions in magnetic field and in some superpositions of electric and magnetic fields.

The Dirac equation and its solutions

The Dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics, representing the one-particle wave equation of motion for electrons in an external electromagnetic field. In this monograph, all propagators of a particle, i.e., the various Greens functions, are constructed in a certain way by using exact solutions of the Dirac equation.