Heat-kernel and Resolvent Asymptotics for Schrödinger Operators on Metric Graphs

Jens Bolte; Sebastian Egger; Ralf Rueckriemen

doi:10.1093/amrx/abu009

Heat-kernel and Resolvent Asymptotics for Schrödinger Operators on Metric Graphs

Jens Bolte, Sebastian Egger, Ralf Rueckriemen

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Abstract

We consider Schroedinger operators on compact and non-compact (finite) metric graphs. For such operators we analyse their spectra, prove that their resolvents can be represented as integral operators and introduce trace-class regularisations of the resolvents. Our main result is a complete asymptotic expansion of the trace of the (regularised) heat-semigroup generated by the Schroedinger operator. We also determine the leading coefficients in the expansion explicitly.

Original language	English
Pages (from-to)	129-165
Number of pages	37
Journal	Applied Mathematics Research Express
Volume	2015
Issue number	1
Early online date	3 Dec 2014
DOIs	https://doi.org/10.1093/amrx/abu009
Publication status	E-pub ahead of print - 3 Dec 2014

Access to Document

10.1093/amrx/abu009

Accepted ManuscriptAccepted author manuscript, 446 KB

http://arxiv.org/abs/1406.1045

Cite this

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title = "Heat-kernel and Resolvent Asymptotics for Schr{\"o}dinger Operators on Metric Graphs",

abstract = "We consider Schroedinger operators on compact and non-compact (finite) metric graphs. For such operators we analyse their spectra, prove that their resolvents can be represented as integral operators and introduce trace-class regularisations of the resolvents. Our main result is a complete asymptotic expansion of the trace of the (regularised) heat-semigroup generated by the Schroedinger operator. We also determine the leading coefficients in the expansion explicitly.",

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N2 - We consider Schroedinger operators on compact and non-compact (finite) metric graphs. For such operators we analyse their spectra, prove that their resolvents can be represented as integral operators and introduce trace-class regularisations of the resolvents. Our main result is a complete asymptotic expansion of the trace of the (regularised) heat-semigroup generated by the Schroedinger operator. We also determine the leading coefficients in the expansion explicitly.

AB - We consider Schroedinger operators on compact and non-compact (finite) metric graphs. For such operators we analyse their spectra, prove that their resolvents can be represented as integral operators and introduce trace-class regularisations of the resolvents. Our main result is a complete asymptotic expansion of the trace of the (regularised) heat-semigroup generated by the Schroedinger operator. We also determine the leading coefficients in the expansion explicitly.

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