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Essential self-adjointness and the \(L^2\)-Liouville property. (English) Zbl 1461.35079

Summary: We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Green’s function and when it gives a non-constant harmonic function which is square integrable.

MSC:

35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R01 PDEs on manifolds
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
47B25 Linear symmetric and selfadjoint operators (unbounded)
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