Elliptic operators on infinite graphs

Details Elliptic operators on infinite graphs Elliptic operators on infinite graphs

2005-09-08

arxiv.org/abs/math/0509193v1

Mathematics

Spectral Theory

14 pages, submitted to Proceedings of the conference "Krzysztof Wojciechowski 50 years - Analysis and Geometry of Boundary Val

Scientific paper

We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of "elliptic operators" as defined by Y. Colin de Verdiere. For such operators, we discuss analogs of inequalities of Cheeger and Harnack and of the maximum principle (in both elliptic and parabolic versions), and apply them to study spectral theory, the ground state and the heat semigroup associated to these operators.

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