Symmetries of the Self-Similar Potentials

TL;DR

This paper explores the symmetries of self-similar potentials using nonlinear operator algebras, revealing connections to q-deformations of known quantum systems and integrable equations.

Contribution

It introduces a framework linking self-similar potentials with nonlinear algebras and q-deformations, extending the understanding of spectral problems in quantum mechanics.

Findings

Identifies algebraic structures associated with self-similar potentials.

Connects q-deformations to finite-gap and related potentials.

Describes deformations of Painleve IV transcendent.

Abstract

An application of the particular type of nonlinear operator algebras to spectral problems is outlined. These algebras are associated with a set of one-dimensional self-similar potentials, arising due to the q-periodic closure f_{j+N}(x)=qf_j(qx), k_{j+N}=q^2 k_j of a chain of coupled Riccati equations (dressing chain). Such closure describes q-deformation of the finite-gap and related potentials. The N=1 case corresponds to the q-oscillator spectrum generating algebra. At N=2 one gets a q-conformal quantum mechanics, and N=3 set of equations describes a deformation of the Painleve IV transcendent.