About dual two-dimensional oscillator and Coulomb-like theories on pseudosphere

TL;DR

This paper rigorously analyzes dual 2D oscillator and Coulomb-like quantum theories on a pseudosphere, comparing their spectra and eigenfunctions using advanced spectral theory methods.

Contribution

It provides a rigorous mathematical framework for dual quantum theories on pseudosphere, including self-adjoint operator construction and spectral analysis.

Findings

Spectral points of dual theories correspond in energy-coupling space.

Constructed all self-adjoint Schrödinger operators for these theories.

Established one-to-one spectral correspondence for discrete and continuous spectra.

Abstract

We present a mathematically rigorous quantum-mechanical treatment of a two-dimensional nonrelativistic quantum dual theories (with oscillator and Coulomb like potentials) on pseudosphere and compare their spectra and the sets of eigenfunctions. We construct all self-adjoint Schrodinger operators for these theories and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use 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.