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Spectral decomposition of square-tiled surfaces. (English) Zbl 1156.58012
A square-tiled surface is a collection of \(N\) unit squares \(S_0\) with identifications of the opposite sides, which can be endowed with a flat metric with some conical points. In this paper, the author deals with the Laplacian associated to such a metric and shows that it belongs to a particular class of operators: the selfadjoint operators defined (uniquely) from a certain quadratic form, which is considered on the set of points of \(\bigl[H^1(S_0)\bigr]^N\) that satisfy a boundary condition given by two unitary matrices \(H,V\) acting on \({\mathbb C}^N\). Here, \(H^1(S_0)\) denotes the usual Sobolev space.
The author proves a general spectral decomposition theorem and derives conditions on the matrices \(H,V\) in order to get a precise decomposition of the spectrum of the Laplacian operator; in particular, if \(H,V\) are permutation matrices, it gives the decomposition of the spectrum of the square-tiled surface defined by the corresponding permutations.

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
47B25 Linear symmetric and selfadjoint operators (unbounded)
58J53 Isospectrality
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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