Perturbative and nonperturbative studies with the delta function potential

Bera, Nabakumar; Bhattacharyya, Kamal; Bhattacharjee, Jayanta K.

doi:doi:10.1119/1.2830531

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American Journal of Physics -- March 2008 -- Volume 76, Issue 3, pp. 250

Perturbative and nonperturbative studies with the delta function potential

Nabakumar Bera¹, Kamal Bhattacharyya², and Jayanta K. Bhattacharjee¹

¹Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
²Department of Chemistry, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India

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We show that the δ-function potential can be exploited along with perturbation theory to yield the result of certain infinite series. The idea is that any exactly soluble potential, if coupled with a δ function potential, remains exactly soluble. We use the strength of the δ function as an expansion parameter and express the second-order energy shift as an infinite sum in perturbation theory. The analytical solution is used to determine the second-order energy shift and hence the sum of an infinite series. By an appropriate choice of the unperturbed system, we can show the importance of the continuum in the energy shift of bound states.