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Some properties of eigenfunctions of linear pencils. (English. Russian original) Zbl 0759.47013
Sib. Math. J. 30, No. 4, 587-597 (1989); translation from Sib. Mat. Zh. 30, No. 4(176), 111-124 (1989).
Let \(E\) be a Hilbert space, \(B\) a selfadjoint operator in \(E\) and \(L\) a symmetric operator which is positive on a complement of a finite- dimensional subspace. The spectral problem \(Lu=\lambda Bu\) is considered in an interpolation scale of Hilbert spaces \(H_ s\) (\(s\in[-1,1]\)), associated with this problem, where \(H_ 1\) is the form domain of the Friedrichs extension of \(L\) and \(H_ 1\) is embedded in \(H_{-1}\). Under an appropriate positivity assumption on \(L\), and compactness assumption on \(B\) and an additional assumption on the scale \(H_ s\), it is shown that the normalized eigenfunctions of the eigenvalue problem in \(H_ 1\) form a Riesz basis in the space \(H_ 0\), which is the completion of \(E\) with respect to the norm \(\| | B|^{1/2} u\|\). The abstract theory is applied to the case where \(L\) is a (possibly degenerate) elliptic operator on a bounded domain \(\Omega\) and \(B\) is the operator of multiplication by a function \(g\); the space \(E\) is \(L^ 2(\Omega)\). Both \(g\) and \(1/g\) can be unbounded; the zero set of \(g\) is assumed to be a smooth manifold of dimension \(n-1\). The verification of the assumptions from the abstract theory is only partly performed; certain assumptions reduce to embedding and compactness theorems for function spaces which are not known in the desired generality; therefore some of the assumptions on the function \(g\) and the coefficients of \(L\) are implicit. The conclusion is that the eigenfunctions form a Riesz basis in the space \(L_{| g|}^ 2\) and moreover (under certain conditions) in some weighted Sobolev spaces.

MSC:
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
35P20 Asymptotic distributions of eigenvalues in context of PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46M35 Abstract interpolation of topological vector spaces
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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