Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications

The theoretical part of this monograph considers the distribution of the spectrum of polynomial operator pencils, focussing on quadratic operator pencils with discrete spectrum.§§Standard spectral problems in Hilbert spaces are of the form A- I for an operator A, and self-adjoint operators are of particular interest and importance, both theoretically and in applications. A characteristic feature of self-adjoint operators is that their spectra are real. Many spectral problems in theoretical physics and engineering can be described by such operators. However, a large class of problems, in particular, vibration problems with boundary conditions depending on the spectral parameter, are represented by operator polynomials which are quadratic in the eigenvalue parameter and whose coefficients are self-adjoint operators. The spectra of such operator polynomials are in general no more real, but still exhibit certain patterns. The distribution of these spectra is the main theme of the present volume. For some classes of quadratic operator pencils, also inverse problems are considered. The connection between the spectra of such quadratic operator pencils and generalized Hermite-Biehler functions is discussed in detail.§§Many applications are thoroughly investigated, such as the Regge problem and damped vibrations of smooth strings, Stieltjes strings, beams, star graphs of strings and quantum graphs. Some chapters summarize advanced background material with detailed proofs. Only basic properties of operators in Hilbert spaces and well-known results from complex analysis will be assumed.§