DOI

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.

Original languageEnglish
Pages (from-to)1691-1714
Number of pages24
JournalCommunications on Pure and Applied Analysis
Volume21
Issue number5
DOIs
StatePublished - May 2022

    Research areas

  • Discrete Schrödinger operators, estimates of the total bandwidth, periodic graphs

    Scopus subject areas

  • Analysis
  • Applied Mathematics

ID: 100016961