Isospectral deformation of quantum potentials and the Liouville equation

Abstract

A quantum problem on an isospectral deformation of one-dimensional potentials (and of corresponding wave functions) is considered. The isospectral deformation defined in the form of a phase flow is shown to obey a system of coupled Liouville equations. In a simple case of an individual flow the well-known integrable Liouville equation arises; its solution provides known families of isospectral potentials. Operators performing this deformation are studied; their unitary property is proved. An evolution of spectral shift operators is determined using those unitary operators. An asymptotical behavior of both a potential and wave functions under this isospectral deformation is studied. It is shown, in particular, that the deformation of the Rosen-Morse potential and that of the harmonic oscillator's potential have common analytical properties. The approach used in the paper can be extended to the case of a deformation leading to a shift of one selected energy level. In the case of the simplest individual flow we get a generalization of the integrable Liouville equation.