# 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 *n* ≥ 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* = –1/2, *n* = 3, which is equivalent to the case *l* = 0, *n* = 2, is reobtained.