Matrix representation of the nonlocal kinetic energy operator, the spinless Salpeter equation and the Cornell potential

Abstract

A new procedure for solving the spinless Salpeter equation is developed. This procedure is implemented with the Cornell potential, where all of the required matrix elements can be calculated from analytic expressions in a convenient basis. Beginning with analytic results for the square of the momentum operator, the matrix elements of the nonlocal kinetic energy operator are obtained from an algorithm that computes the square root of the square of the relativistic kinetic energy operator. Results calculated with the spinless Salpeter equation are compared with those obtained from Schr\"odinger's equation for heavy-quark systems, heavy-light systems, and light-quark systems. In each case the Salpeter energies agree with experiment substantially better than the Schr\"odinger energies.