On the spectrum of the Schr\"odinger Hamiltonian of the one-dimensional conic oscillator perturbed by a point interaction

TL;DR

This paper exactly solves and analyzes how point interactions of various types affect the energy spectrum of a one-dimensional conic oscillator, revealing invariances, level shifts, and crossings in the energy levels.

Contribution

It provides explicit solutions and spectral analysis for the conic oscillator with delta, delta prime, and nonlocal delta prime interactions, advancing understanding of these models.

Findings

Even energy levels remain invariant under delta interactions.

All energy levels decrease with increasing delta interaction strength.

Level crossings occur for the nonlocal delta prime interaction.

Abstract

We decorate the one-dimensional conic oscillator $\frac{1}{2} \left[-\frac{d^{2} }{dx^{2} } + \left|x \right| \right]$ with a point impurity of either $\delta$-type, or local $\delta'$-type or even nonlocal $\delta'$-type. All the three cases are exactly solvable models, which are explicitly solved and analysed, as a first step towards higher dimensional models of physical relevance. We analyse the behaviour of the change in the energy levels when an interaction of the type $-\lambda\,\delta(x)$ or $-\lambda\,\delta(x-x_0)$ is switched on. In the first case, even energy levels (pertaining to antisymmetric bound states) remain invariant with $\lambda$ although odd energy levels (pertaining to symmetric bound states) decrease as $\lambda$ increases. In the second, all energy levels decrease when the form factor $\lambda$ increases. A similar study has been performed for the so called nonlocal $\delta'$ interaction, requiring a coupling constant renormalization, which implies the replacement of the form factor $\lambda$ by a renormalized form factor $\beta$. In terms of $\beta$, even energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of $\beta$ the energy of each odd level, with the natural exception of the first one, becomes lower than the constant energy of the previous even level. Finally, we consider an interaction of the type $-a\delta(x)+b\delta'(x)$, and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian.