Abstract
A class of heat operators over non-archimedean local fields acting on L_2-function spaces on holed discs in the local field are developed and seen as being operators previously introduced by Zúñiga-Galindo, and if the underlying trees are regular, they are associated here with certain finite Kronecker product graphs. L_2-spaces and integral operators invariant under the action of a finite group acting on a holed disc are studied, and then applied to Mumford curves. It is found that the spectral gap in families of Mumford curves can become arbitrarily small.
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