Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs

By Peter Kuchment. 2004

279 citations recorded by the source corpus.

The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics and other areas. A Schnol-type theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an sub-exponential growth integral estimate is available. A theorem on spectral gap opening for 'decorated' quantum graphs is established (its analogue is known for the combinatorial case). It is also shown that if a periodic combinatorial or quantum graph has a point spectrum, it is generated by compactly supported eigenfunctions ('scars').

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ba5ea21afd5d7d3f35ff2dac3274eb8ce27a5d66

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In Geometry and Topology

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

The Essential Norm of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions (2012, methodology): “For a discussion of composition operators on classical spaces of analytic functions we refer the reader to the excellent monographs [5] and [6].”
Dirac operators and spectral triples for some fractal sets built on curves (2006, background): “ and on each component, the topology induced by the metric is the same as the one the component inherits from the ambient compact space. A main emphasis of Kuchement’s research on quantum graphs (see [23, 24]) is the study of the spectral properties of a second order differential operator on a quant”