Threshold conditions and bound states for locally periodic delta potentials

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 = gn, n = 1, 2, …, N at which threshold energy bound state gets generated are shown to be the roots of Nth order polynomial D1(N, g) in g. We present an algebraic recursive procedure to evaluate the polynomial D1(N, g) for any given N. This method obviates the need for the tedious mathematical analysis described in our earlier work to generate D1(N, g). A similar study is presented for case (b). Using the properties of D1(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 gn as a function of n and several interesting features of the energy spectrum for different values of g and N are also described.