Deformation of Supersymmetric and Conformal Quantum Mechanics Through Affine Transformations

Affine transformations (dilatations and translations) are used to define a deformation of one-dimensional $N=2$ supersymmetric quantum mechanics. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are $q$-isospectral, i.e. the spectrum of one can be obtained from another (with possible exception of the lowest level) by $q^2$-factor scaling. This construction allows easily to rederive a special self-similar potential found by Shabat and to show that for the latter a $q$-deformed harmonic oscillator algebra of Biedenharn and Macfarlane serves as the spectrum generating algebra. A general class of potentials related to the quantum conformal algebra $su_q(1,1)$ is described. Further possibilities for $q$-deformation of known solvable potentials are outlined. Talk presented at the workshop on Harmonic Oscillators, College Park, 25-28 March 1992.