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 zeroes of determinants of the continued fractions. They are also useful in the study 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 also provided. We apply some of our results to the Schrödinger operator acting on the graphene with line and point defects.

Source waves in the uniform discrete plane with linear and point defects

“…Let us make one remark. The integration (7) over [−π, π] can be reduced to the integral over [0, π], since C j and, hence, all the matrices that expressed through C j , are even in k 1 , see (11) and (10). Thus, we can write (7) as…”

“…1. In order to do this we adapt the revised analytic technique developed for self-oscillations in discrete media with defects of various dimensions, see [8,9,10,11]. In particular, we extend some of the results from [12], where 2D discrete uniform lattice with local defects but without waveguides is considered.…”