Spectral sequences for quadratic pencils and the inverse spectral problem for the damped wave equation
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 [\textit{G. M. Gubreev} and \textit{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), \textit{V. N. Pivovarchik}, J. Oper. Theory 38, No. 2, 243-263 (1997; Zbl 0895.34013), \textit{V. N. Pivovarchik}, ibid. 42, 189-220 (1999)].
scientific article; zbMATH DE number 1463857
scientific article