Wave equation with energy-dependent potentials for confined systems

Abstract

We study the properties of the wave equation for potentials depending on the energy with emphasis on confining potentials. In this case, for a linear energy dependence, the spectrum shows a saturation effect: the eigenvalues reach a finite limit as the quantum numbers increase. The harmonic oscillator and the linear potentials are studied as examples admitting analytical solutions. We apply such a model to the description of heavy quark systems. We first present a toy model based on the harmonic oscillator and show its ability to reproduce the experimental spectra of charmonium and bottomium. In more realistic calculations, use is made of the Cornell potential for the radial shape and an energy dependence more general than the linear assumption. Comparing the results with those of conventional potentials, we discuss to what extent energy-dependent potentials can bring new features in the description of heavy quark systems. Finally, we show that the energy dependence of the potential has a clear influence on the saturation of the spectrum.