Self-adjoint extension procedure for a singular oscillator

TL;DR

This paper investigates the energy spectrum of a singular oscillator, demonstrating how self-adjoint extensions alter the energy level structure and introducing the concept of quantum defect.

Contribution

It presents a novel analysis of how self-adjoint extension parameters influence the energy levels of a singular oscillator, challenging traditional equidistance properties.

Findings

Energy levels depend on the self-adjoint extension parameter.

Self-adjoint extension breaks the equidistance of energy levels.

Wave function can be expressed as a single unified function.

Abstract

For a singular oscillator, the Schrodinger equation is obtained an equation of eigenvalues, and the dependence of energy on the self-adjoint extension parameter is established. It is shown that the self-adjoint extension violates the well-known property of equidistance of energy levels for the oscillatory potential, well-known in quantum mechanics. The concept of quantum defect is generally introduced, and the wave function of the problem is written as a single function.