Geodesic
Treatment of symmetry in the Ritz method for the Schr\"odinger equation in crystals with a basis
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, 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, Tome 231 (2024), pp. 74-82.

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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, representation theory
Mots-clés : space group 
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N. B. Melnikov; B. I. Reser. Treatment of symmetry in the Ritz method for the Schr\"odinger equation in crystals with a basis. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, 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, Tome 231 (2024), pp. 74-82. http://geodesic.mathdoc.fr/item/INTO_2024_231_a6/

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