Isospectral potentials with Dirac delta interaction: Constrained Spectra

Abstract

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.

The paper asks how a Dirac delta interaction affects supersymmetric partner potentials when it is generated by a discontinuous superpotential, rather than added to an otherwise solvable potential. It considers $V_\pm=W^2\pm W'$ with a $B\operatorname{sgn}(x)$ contribution to $W$, producing delta terms $\pm2B\delta(x)$. The calculation joins solutions on two half-lines through a narrow square well or barrier, then takes the delta-function limit and imposes continuity and the derivative jump. For $W=Ax+B\operatorname{sgn}(x)$, the paper restricts the half-line solutions to Hermite-polynomial oscillator states and derives state-dependent algebraic matching conditions. It concludes that compatibility with the ground-state condition $\Omega=2B$ leaves only the ground state for a nonzero harmonic trap. For $W=A\tanh(\alpha x)+B\operatorname{sgn}(x)$, successive shape-invariance constructions yield additional constraints. The reported surviving parameter choices are $(A,\alpha)=(3B,2B)$ and $(-4B,-3B)$, with two states assigned $E=0$; the constructed second excited state vanishes or is nonnormalizable. The paper reports only parity-even surviving states and no surviving partner-potential states. Adding an inverse-square contribution motivates a spatial cutoff: the coincidence limit with the delta interaction is found incompatible with the construction. A harmonic-confined, cutoff version retains an explicit ground state but loses shape invariance. Its excited states are not solved, and their proposed truncation is inferred rather than established. More generally, the analysis uses selected algebraically constructed states rather than a complete spectral characterization.