Schr\"{o}dinger particle in magnetic and electric fields in Lobachevsky and Riemann spaces

TL;DR

This paper solves the Schrödinger equation in curved 4-spaces with magnetic and electric fields, deriving generalized Landau levels affected by curvature, and analyzes the spectrum's structure in Lobachevsky and Riemann models.

Contribution

It provides analytical solutions for the Schrödinger equation in Lobachevsky and Riemann 4-spaces with external fields, revealing how curvature modifies Landau levels and energy spectra.

Findings

In Lobachevsky space, the spectrum has discrete and continuous parts with finite bound states.

In Riemann space, the spectrum is entirely discrete.

External electric fields allow analytical determination of the energy spectrum in Riemann space.

Abstract

Schr\"{o}dinger equation in Lobachevsky and Riemann 4-spaces has been solved in the presence of external magnetic field that is an analog of a uniform magnetic field in the flat space. Generalized Landau levels have been found, modified by the presence of the space curvature. In Lobachevsky4-model the energy spectrum contains discrete and continuous parts, the number of bound states is finite; in Riemann 4-model all energy spectrum is discrete. Generalized Landau levels are determined by three parameters, the magnitude of the magnetic field $B$, the curvature radius $\rho$ and the magnetic quantum number $m$. It has been shown that in presence of an additional external electric field the energy spectrum in the Riemann model can be also obtained analytically.