Shape Invariant Potentials in ‘Discrete’ Quantum Mechanics

This article is part of the special issue published in honour of Francesco Calogero on the occasion of his 70th birthday Shape invariance is an important ingredient of many exactly solvable quantum me-chanics. Several examples of shape invariant “discrete quantum mechanical systems” are introduced and discussed in some detail. They arise in the problem of describ-ing the equilibrium positions of Ruijsenaars-Schneider type systems, which are “dis-crete ” counterparts of Calogero and Sutherland systems, the celebrated exactly solv-able multi-particle dynamics. Deformed Hermite and Laguerre polynomials are the typical examples of the eigenfunctions of the above shape invariant discrete quantum mechanical systems.