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Sch’nol’s theorem for strongly local forms. (English) Zbl 1188.47003

It is well-known that the spectral values of Schrödinger operators \(H\) can be characterized in terms of the existence of appropriate generalized eigenfunctions. According to the classical theorem of È.È.Šnol’ [Mat.Sb.,N. Ser.42(84), 273–286; erratum ibid.46(88), 259 (1957; Zbl 0078.27904)], the existence of an eigensolution of \(Hu = \lambda u\) with enough decay guarantees \(\lambda\in\sigma(H)\).
In the paper under review, the authors prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures (with form small negative and arbitrary positive part). It is shown that existence of a generalized eigenfunction \(Hu = \lambda u\) which is subexponentially bounded (in the intrinsic metric) implies that \(\lambda\in\sigma(H)\). Applications to quantum graphs with \(\delta \)- or Kirchhoff boundary conditions are included.

MSC:

47A07 Forms (bilinear, sesquilinear, multilinear)
35P05 General topics in linear spectral theory for PDEs
47N50 Applications of operator theory in the physical sciences

Citations:

Zbl 0078.27904
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References:

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