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.
Similar content being viewed by others
References
M. Planck, “Über eine Verbesserung der Wienschen Spectralgleichung,” Verhandl. Dtsc. Phys. Ges., 2, 202–204 (1900).
E. Schrödinger, “Quantisierung als Eigenwertproblem,” Ann. Phys., 384, 361–376 (1926).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics [in Russian], Vol. 3, Quantum Mechanics: Nonrelativistic Theory, Nauka, Moscow (1989); English transl. prev. ed., Pergamon, Oxford (1977).
V. G. Marikhin, “Two new integrable cases of two-dimensional quantum mechanics with a magnetic ield,” JETP Lett., 103, 489–493 (2016).
A. Turbiner, “Quasi-exactly-solvable problems and sl(2) algebra,” Commun. Math. Phys., 118, 467 (1988).
A. Ushveridze, “Quasi–exactly solvable models in quantum mechanics,” Sov. J. Part. Nucl., 20, 504–528 (1990).
A. M. Ishkhanyan, “Exact solution of the Schrödinger equation for the inverse square root potential V0/√ x,” Europhys. Lett., 112, 10006 (2015).
A. Ronveaux, ed., Heun’s Differential Equations, Oxford Univ. Press, New York (1995).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 197, No. 2, pp. 464–474, December, 2018.
The research of V. G. Marikhin was supported by the Russian Foundation for Basic Research (Grant No. 16-01-00289).
Rights and permissions
About this article
Cite this article
Marikhina, A.V., Marikhin, V.G. Calculation of the Discrete Spectrum of some Two-Dimensional Schrödinger Equations with a Magnetic Field. Theor Math Phys 197, 1797–1805 (2018). https://doi.org/10.1134/S0040577918120097
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577918120097