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Threshold conditions and bound states for locally periodic delta potentials

100%

Dharani M., Sahu B., Shastry C.

Open Physics

2014

vol. 12

issue 10

755-766

We present a systematic study of the conditions for the generation of threshold energy eigen states and also the energy spectrum generated by two types of locally periodic delta potentials each having the same strength λV and separation distance parameter a: (a) sum of N attractive potentials and (b) sum of pairs of attractive and repulsive potentials. Using the dimensionless parameter g = λV a in case (a) the values of g = g n, n = 1, 2, …, N at which threshold energy bound state gets generated are shown to be the roots of Nth order polynomial D 1(N, g) in g. We present an algebraic recursive procedure to evaluate the polynomial D 1(N, g) for any given N. This method obviates the need for the tedious mathematical analysis described in our earlier work to generate D 1(N, g). A similar study is presented for case (b). Using the properties of D 1(N, g) we establish that in case (a) the critical minimum value of g which guarantees the generation of the maximum possible number of bound states is g = 4. The corresponding result for case (b) is g = 2. A typical set of numerical results showing the pattern of variation of g n as a function of n and several interesting features of the energy spectrum for different values of g and N are also described.

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

100%

Dharani M., Sahu B., Shastry C.

Open Physics

2013

vol. 11

issue 8

995-1005

This 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.

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2 Open Physics

1 2014

1 2013

2 Dharani M.

2 Sahu B.

2 Shastry C.

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Threshold conditions and bound states for locally periodic delta potentials

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