Two new integrable cases of two-dimensional quantum mechanics with a magnetic field

TL;DR

This paper introduces two new exactly solvable models of two-dimensional quantum systems with magnetic fields, utilizing special functions and deriving quantization rules for each case.

Contribution

It presents novel integrable cases of the 2D Schrödinger equation with magnetic fields, solved via Heun functions, expanding the set of exactly solvable quantum models.

Findings

Solutions expressed through Biconfluent and Confluent Heun functions

Derived quantization rules for both systems

Identified new integrable cases in 2D quantum mechanics

Abstract

Two integrable cases of two-dimensional Schr\"odinger equation with a magnetic field are proposed. Using the polar coordinates and the symmetrical gauge, we will obtain solutions of these equation through Biconfluent and Confluent Heun functions. The quantization rules will be derived for both systems under consideration.