Lower limit in semiclassical form for the number of bound states in a central potential

Abstract

We identify a class of potentials for which the semiclassical estimateof the number N of (S-wave) bound states provides a (rigorous) lower limit: N {{N (semi) }}, where the double braces denote the integer part. Higher partial waves can be included via the standard replacement of the potential V (r) with the effective -wave potential V (eff) (r) = V (r) + ( + 1)/r 2 . An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted.

Transition states and the critical parameters of central potentials

“…For l > 0, the asymptotic boundary condition (1) enables us to neglect the potential V (r) = −βv(r) in equations ( 9)- (11) at large enough r. The general solution of the resulting equation has a form…”

“…Since then many authors have contributed to the study of this problem. Among them we would like to refer to the works [3][4][5][6][7] and to emphasize particularly the contribution of F. Calogero and F. Brau [8][9][10][11][12].…”

Upper and lower bounds on the mean square radius and criteria for the occurrence of quantum halo states

“…This effect will be attenuated (suppressed in practice) if we use upper limits on g c N . Accurate upper and lower limits on the value of g c 1 and on g c NϾ1 , involving only the potential, can be found in the literature [3,[11][12][13][14][15][16][17][38][39][40].…”

Local approximation to the critical parameters of quantum wells

“…The accurate calculation of the number of bound states supported by a finite quantum-mechanical potential well is of great physical and mathematical importance and for this reason there has been considerable interest in 1 e-mail: fernande@quimica.unlp.edu.ar the derivation of upper and lower bounds [1][2][3][4][5][6][7][8][9][10][11][12][13]. Most of those bounds are given in terms of the potential-energy function.…”