It is well known that the standard flat torus T^2=R^2/Z^2 has arbitrarily large Laplacian-eigenvalue multiplicities. Consider the discrete torus C_N * C_N with the discrete Laplacian operator; we prove, however, that the eigenvalue multiplicities are uniformly bounded for any N, except for the eigenvalue one when N is even. In fact, similar phenomena also hold for higher-dimensional discrete tori and abelian Cayley graphs. In this talk, we will outline a proof of the uniformly bounded multiplicity result.
This is a joint work with Bing Xie and Yigeng Zhao.
