About dual two-dimensional oscillator and Coulomb-like theories on a plane

TL;DR

This paper rigorously analyzes and compares the spectra and eigenfunctions of dual two-dimensional oscillator and Coulomb-like quantum theories on a plane, establishing a one-to-one correspondence between their spectral points.

Contribution

It provides a mathematically rigorous construction of self-adjoint operators and spectral solutions for dual 2D oscillator and Coulomb-like theories, revealing a spectral correspondence.

Findings

Spectral points of dual theories correspond one-to-one.

Constructed all self-adjoint Schrödinger operators for these theories.

Established a rigorous spectral analysis using Krein's method.

Abstract

We present a mathematically rigorous quantum-mechanical treatment of a two-dimensional nonrelativistic quantum dual theories (with oscillator and Coulomb like potentials) on a plane and compare their spectra and the sets of eigenfunctions. All self-adjoint Schrodinger operators for these theories are constructed and rigorous solutions of the corresponding spectral problems are presented. The first part of the problem is solved by using a method of specifying s.a. extensions by (asymptotic) s.a. boundary conditions. Solving spectral problems, we follow the Krein's method of guiding functionals. We show, that there is one to one correspondence between the spectral points of dual theories in the planes energy-coupling constants not only for discrete, but also for continuous spectra.