Rational deformations of conformal mechanics

TL;DR

This paper explores rational deformations of quantum conformal mechanics using Darboux-Crum-Krein-Adler transformations, revealing infinite families of spectral modifications and nonlinear symmetry deformations.

Contribution

It introduces a systematic method to generate and analyze isospectral and non-isospectral deformations of conformal mechanics with detailed spectral and symmetry structures.

Findings

Infinite families of deformations with special coupling constants.

Identification of spectrum-generating ladder operators.

Discovery of nonlinear deformations of conformal symmetry.

Abstract

We study deformations of the quantum conformal mechanics of De Alfaro-Fubini-Furlan with rational additional potential term generated by applying the generalized Darboux-Crum-Krein-Adler transformations to the quantum harmonic oscillator and by using the method of dual schemes and mirror diagrams. In this way we obtain infinite families of isospectral and non-isospectral deformations of the conformal mechanics model with special values of the coupling constant $g=m(m+1)$, $m\in {\mathbb N}$, in the inverse square potential term, and for each completely isospectral or gapped deformation given by a mirror diagram, we identify the sets of the spectrum-generating ladder operators which encode and coherently reflect its fine spectral structure. Each pair of these operators generates a nonlinear deformation of the conformal symmetry, and their complete sets pave the way for investigation of the associated superconformal symmetry deformations.