Application of matrix-valued integral continued fractions to spectral problems on periodic graphs with defects

Abstract

We show that spectral problems for periodic operators on lattices with embedded defects of lower dimensions can be solved with the help of matrix-valued integral continued fractions. While these continued fractions are usual in the approximation theory, they are less known in the context of spectral problems. We show that the spectral points can be expressed as zeros of determinants of the continued fractions. They are also useful in the analysis of inverse problems (one-to-one correspondence between spectral data and defects). Finally, the explicit formula for the resolvent in terms of the continued fractions is provided. We apply some of the results to the Schrödinger operator acting on graphene with line and point defects.