The asymptotic behaviour of the number of bound states of a given potential in the limit of large coupling

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

“…These results, as well as those discussed in the rest of this Letter, can be extended to higher partial waves characterized by the angular momentum quantum number , via the standard replacement of the potential V (r) with the "effective" -wave potential [2] has shown that-consistently with the "correspondence principle" relating quantum mechanics at large quantum numbers with classical mechanics-the number N of S-wave bound states (as well as the number of bound states for any fixed angular momentum ) possessed by the potential…”

“…grows asymptotically, when the strength g 2 of the potential diverges, just as the semiclassical estimate (2):…”

Critical strength of attractive central potentials

“…A fairly large number of results of this kind can be found in the literature for the Schrödinger equation (see, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]) and for results applicable to one and two dimension spaces (see, for example, [19][20][21][22][23]). …”

“…An important theorem for classifying these results was found by Chadan [8] and gives the asymptotic behaviour of the number of -wave bound states as the strength, g, of the central potential V (r) = gv(r) goes to infinity:…”

Upper limit on the number of bound states of the spinless Salpeter equation

“…Conversely to the Schrödinger equation, for which a fairly large number of results giving both upper and lower limits on the number of bound states can be found in the literature (see, for example, [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]), only one result, concerning the total number of bound states, is known for the spinless Salpeter equation [36]. After recalling in Section 2 a general method to obtain upper limits on the number of bound states due to Birman [20] and Schwinger [21], we derive such limits for the spinless Salpeter equation in Section 3.…”