Approximating Spectral Invariants of Harper Operators on Graphs
Copyright policy )Approximating Spectral Invariants of Harper Operators on Graphs
We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. A main result in this paper is that the spectral density function of DMLs associated to rational weight functions on graphs with a free action of an amenable discrete group, can be approximated by the average spectral density function of the DMLs on a regular exhaustion, with either Dirichlet or Neumann boundary conditions. This then gives a criterion for the existence of gaps in the spectrum of the DML, as well as other interesting spectral properties of such DMLs. The technique used incorporates some results of algebraic number theory.
20 pages, Latex2e, final version
- University of Adelaide Australia
- Massachusetts Institute of Technology United States
High Energy Physics - Theory, algebraic number theory, discrete magnetic Laplacians, von Neumann algebras, maximal ideals, FOS: Physical sciences, 514, approximation theorems, Fuglede–Kadison determinant, FOS: Mathematics, Difference operators, Quantum equilibrium statistical mechanics (general), Spectrum, resolvent, graphs, Linear difference operators, Applications of operator theory in statistical physics, Functional Analysis (math.FA), Harper operator, Mathematics - Functional Analysis, Banach spaces, High Energy Physics - Theory (hep-th), amenable groups, Ideals and subalgebras, rational weight functions, Discrete version of topics in analysis, Analysis, 58J22, 46L85, 39A12 (Primary) 46L60 (Secondary)
High Energy Physics - Theory, algebraic number theory, discrete magnetic Laplacians, von Neumann algebras, maximal ideals, FOS: Physical sciences, 514, approximation theorems, Fuglede–Kadison determinant, FOS: Mathematics, Difference operators, Quantum equilibrium statistical mechanics (general), Spectrum, resolvent, graphs, Linear difference operators, Applications of operator theory in statistical physics, Functional Analysis (math.FA), Harper operator, Mathematics - Functional Analysis, Banach spaces, High Energy Physics - Theory (hep-th), amenable groups, Ideals and subalgebras, rational weight functions, Discrete version of topics in analysis, Analysis, 58J22, 46L85, 39A12 (Primary) 46L60 (Secondary)
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