Upper and lower limits on the number of bound states in a central potential

Abstract

In a recent paper new upper and lower limits were given, in the context of the Schrödinger or Klein-Gordon equations, for the number N 0 of S-wave bound states possessed by a monotonically nondecreasing central potential vanishing at infinity. In this paper these results are extended to the number N of bound states for the th partial wave, and results are also obtained for potentials that are not monotonic and even somewhere positive. New results are also obtained for the case treated previously, including the remarkably neat lower limit N {{[σ/(2 +1)+1]/2}} with σ = (2/π ) max 0 r<∞ [r|V (r)| 1/2 ] (valid in the Schrödinger case, for a class of potentials that includes the monotonically nondecreasing ones), entailing the following lower limit for the total number N of bound states possessed by a monotonically nondecreasing central potential vanishing at infinity: N {{(σ +1)/2}}{{(σ +3)/2}}/2 (here the double braces denote the integer part).

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]). …”

Necessary and sufficient conditions for existence of bound states in a central potential

“…Note that we use the standard quantum-mechanical units such ash = 2m = 1, where m is the mass of the particle. This upper limit (1), called the Bargmann-Schwinger upper limit in the literature, was the starting point of intensive studies and a fairly large number of upper and lower limits on the number of bound states for various classes of potentials was found, see for example [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

“…Upper and lower limits featuring the correct g 1/2 dependence were first obtained in [7]. Upper and lower limits featuring the correct asymptotic behaviour (2) were first derived in [19,20]. In practice, the asymptotic regime is reached very quickly when the strength of the potential is large enough to bind two or three bound states.…”

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].…”

“…There exists a fairly large number of upper and lower limits on the energy of eigenstates in the literature [2][3][4][5][6][7][8][9][10] as well as on the number of bound states supported by central potentials [3,[11][12][13][14][15][16][17]. Similar results concerning the rms radius are scarcer.…”