Conditions governing the generalisation of threshold bound states by N attractive delta potentials in one and three dimensions

Abstract

AbstractThis paper proves that for N attractive delta function potentials the number of bound states (Nb) satisfies 1 ≤ N b ≤ N in one dimension (1D), and is 0 ≤ N b ≤ N in three dimensions (3D). Algebraic equations are obtained to evaluate the bound states generated by N attractive delta potentials. In particular, in the case of N attractive delta function potentials having same separation a between adjacent wells and having the same strength λV, the parameter g=λVa governs the number of bound states. For a given N in the range 1–7, both in 1D and 3D cases the numerical values of gn, where n=1,2,..N are obtained. When g=gn, Nb ≤ n where Nb includes one threshold energy bound state. Furthermore, gn are the roots of the Nth order polynomial equations with integer coefficients. Based on our numerical calculations up to N=40, even when N becomes large, 0 ≤ g n ≤ 4 and $\frac{{\Sigma g_n }} {N} \simeq 2 $ and this result is expected to be generally valid. Thus, for g > 4 there will be no threshold or zero energy bound state, and if g≈ 2 for a given large N, the number of bound states will be approximately N/2. The empirical formula gn = 4/[1+exp((N 0 − n)/β)] gives a good description of the variation of gn as a function of n. This formula is useful in estimating the number of bound states for any N and g both in 1D and 3D cases.