  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.
