Open AccessTwo-sided estimates of total bandwidth for Schrödinger operators on periodic graphs
Author(s) -
Evgeny Korotyaev,
Natalia Saburova
Publication year - 2022
Publication title -
communications on pure and applied analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.077
H-Index - 42
eISSN - 1553-5258
pISSN - 1534-0392
DOI - 10.3934/cpaa.2022042
Subject(s) - fourier series , mathematics , floquet theory , fourier transform , schrödinger's cat , discrete fourier series , graph , pure mathematics , fourier analysis , mathematical analysis , discrete mathematics , physics , quantum mechanics , short time fourier transform , nonlinear system
We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We obtain two-sided estimates of the total bandwidth for the Schrödinger operators in terms of geometric parameters of the graph and the potentials. In particular, we show that these estimates are sharp. It means that these estimates become identities for specific graphs and potentials. The proof is based on the Floquet theory and trace formulas for fiber operators. The traces are expressed as finite Fourier series of the quasimomentum with coefficients depending on the potentials and cycles of the quotient graph from some specific cycle sets. In order to obtain our results we estimate these Fourier coefficients in terms of geometric parameters of the graph and the potentials.