Spectral reciprocity and matrix representations of unbounded operators

2009/11/02 by Jorgensen, Palle E. T., Pearse, Erin P. J. · 1 citation
#46E22 #47B25 #47B32 #47B39 #60J10. Secondary: 42C25 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Primary: 05C50 #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.0911.0185

Abstract

Motivated by potential theory on discrete spaces, we study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. These operators are discrete analogues of the classical conformal Laplacians and Hamiltonians from statistical mechanics. For an infinite discrete set X, we consider operators acting on Hilbert spaces of functions on X, and their representations as infinite matrices; the focus is on ℓ²(X), and the energy space H_{\mathcal E}. In particular, we prove that these operators are always essentially self-adjoint on ℓ²(X), but may fail to be essentially self-adjoint on H_{\mathcal E}. In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the H_{\mathcal E} operators with the use of a new approximation scheme.

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