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) |

##### Keywords:

selfadjoint operator; symmetric operator; spectral problem; interpolation scale of Hilbert spaces; normalized eigenfunctions of the eigenvalue problem; Riesz basis; weighted Sobolev spaces
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\textit{S. G. Pyatkov}, Sib. Math. J. 30, No. 4, 587--597 (1989; Zbl 0759.47013); translation from Sib. Mat. Zh. 30, No. 4(176), 111--124 (1989)

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