The Wave Equation with Energy Dependent Potential - the Linear Case

Abstract

We summarize recent works devoted to energy dependent potentials. It concerns the class of potentials having a coupling constant depending linearly on the energy. Introduced in the Schrödinger equation, it produces non-linear effects. Few cases admit analytical solutions. They are of great help to get acquainted with this non-linearity. The harmonic oscillator and the Coulomb potential are presented as typical examples. Applications concern heavy. quark-anti-quark systems, as well as the many-body problem with harmonic interactions. Finally, we show that the energy dependence does not modify the number of bound states of attractive potentials. It can regularize some potentials singular at the origin. For instance, the ground state energy of the -1/r2 potential in D=3 dimensional space becomes finite for finite energy dependence.