Rational deformations of conformal mechanics

We study deformations of the quantum conformal mechanics of de Alfaro-Fubini-Furlan with a 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 nonisospectral deformations of the conformal mechanics model with special values of the coupling constant g=m(m+1), mN, 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 that 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