Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms. (English) Zbl 1248.35133
Janas, Jan (ed.) et al., Spectral theory and analysis. Conference on operator theory, analysis and mathematical physics (OTAMP) 2008, Bedlewo, Poland. Berlin: Springer (ISBN 978-3-7643-9993-1/hbk; 978-94-007-0431-2/ebook). Operator Theory: Advances and Applications 214, 83-106 (2011).
It is well known that the spectral values of second order self-adjoint partial differential operators \(H\) can be characterized in terms of the existence of appropriate generalized eigenfunctions (“Shnol theorem”). Another known result (“Allegretto-Piepenbrink theorem”) relates solutions and spectra of the operator \(H\). Recently A. Boutet De Monvel and the authors of the paper under review [Doc. Math., J. DMV 14, 167–189 (2009; Zbl 1172.35045); Isr. J. Math. 173, 189–211 (2009; Zbl 1188.47003)] proved the above-mentioned results for the generator of a strongly local Dirichlet form.
In the present paper, the authors present an introduction to the framework of strongly local Dirichlet forms and discuss the Shnol theorem, the Allegretto-Piepenbrink theorem, the Harnack principle, and some other spectral properties within this framework in a way that is accessible to the non-specialist. Applications to the singular interaction Hamiltonians and quantum graphs are included.
For the entire collection see [Zbl 1213.35010].
In the present paper, the authors present an introduction to the framework of strongly local Dirichlet forms and discuss the Shnol theorem, the Allegretto-Piepenbrink theorem, the Harnack principle, and some other spectral properties within this framework in a way that is accessible to the non-specialist. Applications to the singular interaction Hamiltonians and quantum graphs are included.
For the entire collection see [Zbl 1213.35010].
Reviewer: Michael Perelmuter (Kyïv)
MSC:
| 35P05 | General topics in linear spectral theory for PDEs |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
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