ISOSPECTRAL DEFORMATION OF QUANTUM POTENTIALS AND THE LIOUVILLE EQUATION

A quantum problem on an isospectral deformation of one-dimensional pot entials (and of corresponding wave functions) is considered. The isosp ectral deformation defined in the form of a phase flow is shown to obe y a system of coupled Liouville equations. In a simple case of an indi vidual flow the well-known integrable Liouville equation arises; its s olution provides known families of isospectral potentials. Operators p erforming this deformation are studied; their unitary property is prov ed. An evolution of spectral shift operators is determined using those unitary operators. An asymptotical behavior of both a potential and w ave functions under this isospectral deformation is studied. It is sho wn, in particular, that the deformation of the Rosen-Morse potential a nd 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 th e integrable Liouville equation.