Abstract
Inequalities on eigenvalues of the Schrödinger operator are re-examined in the case of spherically symmetric potentials. In particular, we obtain:
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i)
A connection between the moments of order (n − 1)/2 of the eigenvalues of a one-dimensional problem and the total number of bound statesN n, inn space dimensions;
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ii)
optimal bounds on the total number of bound states below a given energy in one dimension;
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iii)
alower bound onN 2;
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iv)
a self-contained proof of the inequality for α ≧ 0,n ≧ 3, leading to the optimalC 04,C 3;
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v)
solutions of non-linear variation equations which lead, forn ≧ 7, to counter examples to the conjecture thatC 0n is given either by the one-bound state case or by the classic limit; at the same time a conjecture on the nodal structure of the wave functions is disproved.
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Communicated by R. Stora
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Glaser, V., Grosse, H. & Martin, A. Bounds on the number of eigenvalues of the Schrödinger operator. Commun.Math. Phys. 59, 197–212 (1978). https://doi.org/10.1007/BF01614249
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DOI: https://doi.org/10.1007/BF01614249