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Expansion in generalized eigenfunctions for Laplacians on graphs and metric measure spaces


Authors: Daniel Lenz and Alexander Teplyaev
Journal: Trans. Amer. Math. Soc. 368 (2016), 4933-4956
MSC (2010): Primary 81Q35, 05C63, 28A80; Secondary 31C25, 60J45, 05C22, 31C20, 35P05, 39A12, 47B25, 58J35, 81Q10
DOI: https://doi.org/10.1090/tran/6639
Published electronically: June 24, 2015
MathSciNet review: 3456166
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Abstract: We consider an arbitrary selfadjoint operator in a separable Hilbert space. To this operator we construct an expansion in generalized eigenfunctions, in which the original Hilbert space is decomposed as a direct integral of Hilbert spaces consisting of general eigenfunctions. This automatically gives a Plancherel type formula. For suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions. For Laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the corresponding difference equation.


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Additional Information

Daniel Lenz
Affiliation: Mathematisches Institut, Friedrich Schiller Universität Jena, D-07743 Jena, Germany
Email: daniel.lenz@uni-jena.de

Alexander Teplyaev
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: alexander.teplyaev@uconn.edu

DOI: https://doi.org/10.1090/tran/6639
Received by editor(s): June 1, 2014
Published electronically: June 24, 2015
Additional Notes: The first author gratefully acknowledges partial support by the German Research Foundation (DFG) as well as enlightening discussions with Gunter Stolz and Peter Stollmann. He also takes this opportunity to thank the mathematics departments of the University of Lyon and of the University of Geneva for their hospitality
The second author is deeply thankful to Peter Kuchment and Robert Strichartz for interesting and helpful discussions related to this work. His research was supported in part by NSF grant DMS-0505622
Article copyright: © Copyright 2015 American Mathematical Society