Expansion in generalized eigenfunctions for Laplacians on graphs and metric measure spaces
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- by Daniel Lenz and Alexander Teplyaev PDF
- Trans. Amer. Math. Soc. 368 (2016), 4933-4956 Request permission
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.References
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Additional Information
- Daniel Lenz
- Affiliation: Mathematisches Institut, Friedrich Schiller Universität Jena, D-07743 Jena, Germany
- MR Author ID: 656508
- Email: daniel.lenz@uni-jena.de
- Alexander Teplyaev
- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
- MR Author ID: 361814
- Email: alexander.teplyaev@uconn.edu
- 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 - © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3456166