Discrete supersymmetries of the Schrodinger equation and non-local exactly solvable potentials

Abstract

Using an isomorphism between Hilbert spaces $L^2$ and ${\ell }^2$ we consider Hamiltonians which have tridiagonal matrix representations (Jacobi matrices) in a discrete basis and an eigenvalue problem is reduced to solving a three term difference equation. Technique of intertwining operators is applied to creating new families of exactly solvable Jacobi matrices. It is shown that any thus obtained Jacobi matrix gives rise to a new exactly solvable non-local potential of the Schroedinger equation. We also show that the algebraic structure underlying our approach corresponds to supersymmetry. Supercharge operators acting in the space ${\ell }^2\times {\ell }^2$ are introduced which together with a matrix form of the superhamiltonian close the simplest superalgebra.

12 pages