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Elliptic operators on planar graphs: Unique continuation for eigenfunctions and nonpositive curvature

S. Klassert·D. Lenz·N. Peyerimhoff·P. Stollmann
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Abstract

This paper is concerned with elliptic operators on plane tessellations. We show that such an operator does not admit a compactly supported eigenfunction, if the combinatorial curvature of the tessellation is nonpositive. Furthermore, we show that the only geometrically finite, repetitive plane tessellations with nonpositive curvature are the regular $(3,6), (4,4)$ and $(6,3)$ tilings.

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S. Klassert, D. Lenz, N. Peyerimhoff, P. Stollmann. 2004-10-07. Elliptic operators on planar graphs: Unique continuation for eigenfunctions and nonpositive curvature. https://arxiv.org/abs/math-ph/0410022

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