Generalization of the Calogero–Cohn bound on the number of bound states

Abstract

It is shown that for the Calogero-Cohn type upper bounds on the number of bound states of a negative spherically symmetric potential V (r), in each angular momentum state, that is, bounds containing only the integral ∞ 0 |V (r)| 1/2 dr, the condition V ′ (r) ≥ 0 is not necessary, and can be replaced by the less stringent condition (d/dr)[r 1−2p (−V ) 1−p ] ≤ 0, 1/2 ≤ p < 1, which allows oscillations in the potential. The constants in the bounds are accordingly modified, depend on p and ℓ, and tend to the standard value for p = 1/2.

Critical strength of attractive central potentials

“…where K denotes the symmetric linear operator, operating on the Hilbert space L 2 (R), which is in this paper the integral operator generated by the so-called Birman-Schwinger [3,25] kernel K , equation (14).…”

“…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]). …”

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

“…To this end, an upper limit on L is needed. The best limit, which behaves linearly with the strength of potential, is obtained, not from (35a), but instead from the simple relation (33). We have…”

“…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.…”

“…The expression (37) for the upper limit on L is obtained using (33), (34) This last inequality means that the upper limit (24) would always be better than the limit (36) if its coefficient was equal to 1. Since the coefficient is greater than unity, 4πK = 1.294 > 1, there is some room for the limit (36) to be more stringent.…”