Calculation of the Discrete Spectrum of some Two-Dimensional Schrödinger Equations with a Magnetic Field

Abstract

One of us previously obtained and integrated the first examples of two-dimensional Schrödinger equations with a magnetic field belonging to the class of quasi–exactly solvable problems. It was shown that the wave functions are expressed in terms of degenerations of the Heun function: biconfluent and confluent Heun functions. Algebraic conditions were also found that determine the discrete spectrum and wave functions. Our goal here is to solve these algebraic equations numerically. In some cases, we can find an analytic approximation of the discrete spectrum.