TY - JOUR

T1 - TWO-SIDED ESTIMATES OF TOTAL BANDWIDTH FOR SCHRÖDINGER OPERATORS ON PERIODIC GRAPHS

AU - Korotyaev, Evgeny

AU - Saburova, Natalia

N1 - Publisher Copyright: © 2022 American Institute of Mathematical Sciences. All rights reserved.

PY - 2022/5

Y1 - 2022/5

N2 - 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.

AB - 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.

KW - Discrete Schrödinger operators

KW - estimates of the total bandwidth

KW - periodic graphs

UR - http://www.scopus.com/inward/record.url?scp=85128313665&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/8fd8142b-fde6-3378-b2ca-52997cd506df/

U2 - 10.3934/cpaa.2022042

DO - 10.3934/cpaa.2022042

M3 - Article

AN - SCOPUS:85128313665

VL - 21

SP - 1691

EP - 1714

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

SN - 1534-0392

IS - 5

ER -