# Bargmann's Inequalities In Spaces of Arbitrary Dimension

### Noriaki Seto

Kyoto University, Japan

## Abstract

An inequality is derived which gives an upper bound of the number of bound states in the $l$th partial wave ($l=0,1,…$) of the two-body Schrödinger equation with a spherically symmetric potential function in the $n$-dimensional space ($n=1,2,…$). This is a generalization of Bargmann’s inequality for the case $n=3$. The generalization is straightforward for the case $l≥1$ with $n=2$ and $l≥0$ with $n≥3$. 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=–21 $, $n=3$, which is equivalent to the case $l=0$, $n=2$, is reobtained.

## Cite this article

Noriaki Seto, Bargmann's Inequalities In Spaces of Arbitrary Dimension. Publ. Res. Inst. Math. Sci. 9 (1973), no. 2, pp. 429–461

DOI 10.2977/PRIMS/1195192566