# Spectra of Cayley graphs of the lamplighter group and random Schrodinger operators

## TL;DR

This paper demonstrates that the lamplighter group has a Cayley graph with a spectrum consisting of an interval plus isolated points, revealing the first example of a group with infinitely many spectral gaps, using spectral analysis of convolution operators.

## Contribution

It introduces the first example of a group with infinitely many spectral gaps in its Cayley graph spectrum, through detailed spectral analysis of convolution operators on the lamplighter group.

## Key findings

- Spectrum is a union of an interval and isolated points
- Eigenvalues are solutions to algebraic equations with Chebyshev polynomials
- Spectral bifurcation occurs at parameter values 1 and -1

## Abstract

We show that the lamplighter group L has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the spectrum of its Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family of convolution operators on L. Our results show that the spectrum is a pure point spectrum for each value of the parameter, the eigenvalues are solutions of algebraic equations involving Chebyshev polynomials of the second kind, and the topological structure of the spectrum makes a bifurcation when the parameter passes the points 1 and -1.