Exact spectrum for quantum oscillator in spaces of constant curvature from WKB-quantization

TL;DR

This paper develops a WKB-quantization approach to find exact energy spectra of quantum oscillators in curved 3-spaces, extending the method to Euclidean, Riemann, and Lobachevsky geometries.

Contribution

It introduces a generalized covariant Schrödinger equation and constructs special WKB series to obtain exact energy levels in curved spaces.

Findings

Exact energy spectra for quantum oscillators in curved spaces

WKB-method extended to complex variable function theory

Unified approach for Euclidean, Riemann, and Lobachevsky geometries

Abstract

Quantum-mechanical WKB-method is elaborated for the known quantum oscillator problem in curved 3-spaces models Euclid, Riemann, and Lobachevsky E_{3}, H_{3}, S_{3} in the framework of the complex variable function theory. Generalized generally covariant Schr\"{o}dinger equation is considered. In all three space models, exact energy levels are found with the help of constructing special formal WKB-sieries.