SPECTRA OF CAYLEY GRAPHS OF THE LAMPLIGHTER GROUP AND RANDOM SCHRODINGER OPERATORS Academic Article uri icon

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.

published proceedings

  • TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

author list (cited authors)

  • Grigorchuk, R., & Simanek, B.

citation count

  • 3

complete list of authors

  • Grigorchuk, Rostislav||Simanek, Brian

publication date

  • January 2021