Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory
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Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory, 2024, Volume 231, Pages 74–82
DOI: https://doi.org/10.36535/2782-4438-2024-231-74-82 (Mi into1255) 		 
Treatment of symmetry in the Ritz method for the Schrödinger equation in crystals with a basis

N. B. Melnikova, B. I. Reserb

a Lomonosov Moscow State University
b Institute of Metal Physics, Ural Division of the Russian Academy of Sciences, Ekaterinburg
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DOI: https://doi.org/10.36535/2782-4438-2024-231-74-82
Abstract: This paper is devoted to treatment of symmetry in the Schrödinger equation with a periodic potential for crystals with a basis. We present a general group-theoretical approach, which yields the matrix elements of the Hamiltonian in the tight-binding approximation, using the spatial symmetry of the problem, time reversal symmetry, and the Hermitian property of the Hamiltonian. The developed mathematical theory generalizes the well-known result for crystals with two atoms in the unit cell to the case of crystals with several atoms in the unit cell.
Keywords: Schrödinger equation, periodic potential, Ritz method, crystal lattice, space group, representation theory
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 122021000039-4
This work was performed within the framework of the State assignment of the Ministry of Science and Higher Education of the Russian Federation (theme “Electron”, state registration number 122021000039-4).
Document Type: Article
UDC: 512.54, 517.958
MSC: 20C35, 81Q05
Language: Russian
Citation: N. B. Melnikov, B. I. Reser, “Treatment of symmetry in the Ritz method for the Schrödinger equation in crystals with a basis”, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 2, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 231, VINITI, Moscow, 2024, 74–82
Citation in format AMSBIB
\Bibitem{MelRes24}
\by N.~B.~Melnikov, B.~I.~Reser
\paper Treatment of symmetry in the Ritz method for the Schr\"odinger equation in crystals with a basis
\inbook Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 2
\serial Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz.
\yr 2024
\vol 231
\pages 74--82
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/into1255}
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