Particle in a box with a δ-function potential: Strong and weak coupling limits

Abstract

A particle in a box with the δ-function potential Vλ(x)=λδ(x−x0) has been recently explored. We show that this example is solvable in the weak (λ→0±) and the strong (1∕λ→0±) coupling limits. In either limit the attractive and repulsive potentials lead to identical spectra, with the possible exception of a single negative-energy state that is present when 1∕λ→0−. We numerically obtain the spectra near the strong-coupling limit and discuss the consequences of the degeneracy that arises when 1∕λ→0±.