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Spectral sequences for quadratic pencils and the inverse spectral problem for the damped wave equation. (English) Zbl 0956.47006

Let \(L(\lambda)=\lambda^2A+\lambda B+C\) be a quadratic operator pencil with selfadjoint operators \(A\), \(B\) and \(C\) defined on a dense subset of a Hilbert space. It is supposed that the spectrum of \(L(\lambda)\) is pure discrete (i.e. the spectrum consists of normal eigenvalues only). A sequence \(\Lambda\) of complex numbers is said to be admissible spectral sequence (a.s.s.) for \(L(\lambda)\) if there exists a triple of operators \(A\), \(B\), \(C\) (within a certain class) such that the spectrum of \(L(\lambda)\) coincides with \(\Lambda\). The author establishes conditions (necessary in the case of infinite dimensional Hilbert space) for a sequence to be a.s.s. These conditions are both necessary and sufficient in the case of finite dimensional space and weakly damped pencil (i.e. pencil such that \((By,y)^2<4(Ay,y)(Cy,y)\) for all \(y\) from the domain of \(L(\lambda)\)). In particular, the author gives necessary conditions for a sequence to be a.s.s. for the case of a weakly damped wave operator. For a complete collection see [G. M. Gubreev and V. N. Pivovarchik, Funct. Anal. Appl. 31, No. 1, 54-57 (1997); translation from Funkts. Anal. Prilozh. 31, No. 1, 70-74 (1997; Zbl 0907.34069), V. N. Pivovarchik, J. Oper. Theory 38, No. 2, 243-263 (1997; Zbl 0895.34013), V. N. Pivovarchik, ibid. 42, 189-220 (1999)].

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
34A55 Inverse problems involving ordinary differential equations
47F05 General theory of partial differential operators
47N20 Applications of operator theory to differential and integral equations
15A22 Matrix pencils
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