The Criteria For The Number Of Bound States with l = 0 for A Non-relativistic Single-Particle Potential

Abstract

We have studied some criteria for bound state energies in the non-relativistic regime by using the 3D Schrodinger equation. In relations with these criteria, we examined: The number of bound states, the critical conditions, eigenvalues, and infinite versus finite number of eigenvalues, and the fixed number expression, which determines the number of bound states. We have studied these criteria by solving the Schrodinger equation in 3D for l=0 for many central potentials: the finite spherical potential, the spherical potential shell, the Yukawa potential, the cutoff and regular triangular potential, the Woods-Saxon potential, the regular and cutoff Coulomb potential, the cutoff square and cubic inverse potentials. We mean by a cutoff potential just a potential cutoff near the origin by connecting a potential with the finite spherical potential. Then, we have used some estimating methods to compare their results with the exact results. The estimating methods are expressions that give the lower and upper limits of the number of bound state energies for a given potential. We have considered the most accurate and recent expressions and we have compared them with the exact results.