Bargmann's Inequalities In Spaces of Arbitrary Dimension

Abstract

An inequality is derived which gives an upper bound of the number of bound states in the $l$th partial wave ($l=0,1,\dots$) of the two-body Schrödinger equation with a spherically symmetric potential function in the $n$-dimensional space ($n=1,2,\dots$). This is a generalization of Bargmann’s inequality for the case $n=3$. The generalization is straightforward for the case $l\ge 1$ with $n=2$ and $l\ge 0$ with $n\ge 3$. After a mathematically rigorous justification of his heuristic argument, Schwinger’s method in his simple proof of Bargmann’s inequality is employed here. Newton’s result for the case $l=–\frac{1}{2}$,  $n=3$, which is equivalent to the case $l=0$, $n=2$, is reobtained.