Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators

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