Spectral decomposition of square-tiled surfaces

Abstract

We consider N copies of a square S0 and define selfadjoint extensions of the Euclidean Laplacian acting on \(\left (\mathcal {C}_0^{\infty}(S_0) \right )^N\) by choosing some boundary conditions that are parametrized by two unitary matrices H and V acting on \({\mathbb{C}}^N.\) Denoting by Sp(ΔN,H,V) the spectrum of such an operator we derive conditions on H and V so that the following spectral decomposition holds: $${\rm Sp}(\Delta_N, H, V) = \bigcup_{1\leq i\leq k} {\rm Sp}(\Delta_{N_i}, {H_i}, {V_i}) \quad {\rm with}\, \sum_{1}^{k} N_i = N.$$ If H and V are permutation matrices this gives a spectral decomposition of the spectrum of the square-tiled surface defined by the corresponding permutations. We apply this to derive examples related to isospectrality and to high multiplicity.