A sufficient condition for the existence of bound states for scalar spherically symmetric potentials

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

“…A simple upper limit, that we denote by BSN, can be obtained from the BS upper limit, see (17); it reads…”

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]). …”

“…This relation yields a lower limit on the critical value g ( ) c , by making a minimization over p 1, which was shown to be very accurate (see, for example, [9,16,28]). …”

“…The most stringent conditions are obtained by minimizing the left-hand sides of (3) and (4) over all positive values of R. Some other sufficient conditions for the existence of bound states can be found in the literature (see, for example, [16,28] and for reviews see [10,17,18]). A sufficient condition which does not require the spherical symmetry for the potential V was proposed in 1980 by Chadan [29] (see also [30]).…”

“…The methods proposed by Chadan enter this category of iterative convergent procedures. Also, Lassaut and Lombard [16] worked in this sense some years ago, but the procedure was constrained by the condition g [32,33] in what concerns the upper limits, is that there is no need to start from a conveniently chosen wavefunction. Indeed, the improvements are simple for these sequences: one just needs to calculate the next order.…”