Exactly solvable models for two-dimensional quantum systems

Abstract

A wide class of two-dimensional exactly solvable models is constructed on the basis of the inverse scattering problem in the adiabatic representation. Exactly solvable models with prescribed spectral properties are constructed by using the generalized method of Bargmann potentials. Two-dimensional potentials are presented within the consistent formulation of both mutually connected inverse problems to which the initial task is reduced: the parametric problem and the multichannel problem for the system of gauge equations. The algebraic technique is elaborated for the reconstruction of time-dependent and time-independent two-dimensional Bargmann potentials and corresponding solutions in a closed analytic form on the basis of the nonstandard parametric inverse problem with scattering data depending on a coordinate variable. Specific examples of exactly solvable models are given within the parametric problem on the entire line and on the half-line. In particular, transparent symmetric and nonsymmetric potentials, parametric family of phase-equivalent potentials, two-dimensional potentials without and with bound states are presented with the corresponding solutions of the parametric problem.