On negative eigenvalues of generalized laplace operators

The negative eigenvalues problem for the generalized Laplace operator − Δ = −Δ+ + α T, α < 0, where T is a positive operator singular in L 2 and acting from the Sobolev space W 1 2 to its dual W −1 2, is investigated. The question, whether the number of negative eigenvalues N_(− Δ ) is finite or...

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Bibliographic Details
Published in:Reports on mathematical physics 2001-12, Vol.48 (3), p.359-387
Main Authors: Albeverio, S, Karwowski, W, Koshmanenko, V
Format: Article
Language:English
Subjects:
generalized Laplace operator
negative eigenvalues problem
singular perturbations
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Summary:The negative eigenvalues problem for the generalized Laplace operator − Δ = −Δ+ + α T, α < 0, where T is a positive operator singular in L 2 and acting from the Sobolev space W 1 2 to its dual W −1 2, is investigated. The question, whether the number of negative eigenvalues N_(− Δ ) is finite or infinite is answered. Under the assumption that the not necessarily compact operator T = ( I − Δ) −1 T in W 1 2 has a discrete spectrum, different conditions leading to N_(− Δ ) = ∞, as well as to N_(− Δ ) < ∞ are found and the corresponding examples are given.
ISSN:0034-4877
1879-0674
DOI:10.1016/S0034-4877(01)80095-1