JournalsprimsVol. 9 , No. 2DOI 10.2977/prims/1195192566

Bargmann's Inequalities In Spaces of Arbitrary Dimension

  • Noriaki Seto

    Kyoto University, Japan
Bargmann's Inequalities In Spaces of Arbitrary Dimension cover

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.