Isospectral potentials with Dirac delta interaction: Constrained Spectra
We study quantum potentials containing Dirac delta function (DDF) within its supersymmetric (isospectral) construction via a discontinuous superpotential, instead of being simply an added part. This construction comprises of two independently solvable sectors joined at the singular point by matching conditions imposed by the corresponding self-adjoint extension. Needing careful regularization, these boundary conditions impose independent algebraic constraints on each of the eigenstates, that restrict and truncate the spectrum. The number of surviving bound states, which are required to be parity-even, is thus fixed by the available system parameters rather than by the usual quantization rule. The specific cases of the Harmonic oscillator and the Rosen-Morse potential isospectrally infused with the DDF demonstrate such highly restrictive spectra, while the similar combination of the DDF with a Calogero-type singularity turns out to be largely incompatible as the regularization breaks down. Therefore, localized singularities are capable of controlling and engineering the discrete spectrum of quantum systems, with possible use in myriads of physical systems with very localized interactions.