Upper limit on the number of bound states of the spinless Salpeter equation

TL;DR

This paper derives upper bounds on the number of bound states in the spinless Salpeter equation using the Birman-Schwinger method, providing conditions for the existence of bound states in the ultrarelativistic limit.

Contribution

It introduces new upper limits on bound states and a simple existence condition for ultrarelativistic cases in the spinless Salpeter equation.

Findings

Upper bounds on total and $ ext{l}$-wave bound states are established.

A simple integral condition for bound state existence at zero mass.

The results apply to semirelativistic quantum systems.

Abstract

We obtain, using the Birman-Schwinger method, upper limits on the total number of bound states and on the number of $\ell$-wave bound states of the semirelativistic spinless Salpeter equation. We also obtain a simple condition, in the ultrarelativistic case ($m=0$), for the existence of at least one $\ell$-wave bound states: $C(\ell,p/(p-1))$ $\int_0^{\infty}dr r^{p-1} |V^-(r)|^p\ge 1$, where $C(\ell,p/(p-1))$ is a known function of $\ell$ and $p>1$.