Spectral Theory Of Operator Pencils, Hermite-biehler Functions, And Their Applications (operator Theory: Advances And Applications)
by Manfred Möller /
2015 / English / PDF
4.1 MB Download
The theoretical part of this monograph examines the distribution
of the spectrum of operator polynomials, focusing on quadratic
operator polynomials with discrete spectra. The second part is
devoted to applications. Standard spectral problems in Hilbert
spaces are of the form A-λI for an operator A, and self-adjoint
operators are of particular interest and importance, both
theoretically and in terms of applications. A characteristic
feature of self-adjoint operators is that their spectra are real,
and many spectral problems in theoretical physics and engineering
can be described by using them. However, a large class of
problems, in particular vibration problems with boundary
conditions depending on the spectral parameter, are represented
by operator polynomials that 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 focus of the present volume. For some classes
of quadratic operator polynomials, inverse problems are also
considered. The connection between the spectra of such quadratic
operator polynomials and generalized Hermite-Biehler functions is
discussed in detail.
The theoretical part of this monograph examines the distribution
of the spectrum of operator polynomials, focusing on quadratic
operator polynomials with discrete spectra. The second part is
devoted to applications. Standard spectral problems in Hilbert
spaces are of the form A-λI for an operator A, and self-adjoint
operators are of particular interest and importance, both
theoretically and in terms of applications. A characteristic
feature of self-adjoint operators is that their spectra are real,
and many spectral problems in theoretical physics and engineering
can be described by using them. However, a large class of
problems, in particular vibration problems with boundary
conditions depending on the spectral parameter, are represented
by operator polynomials that 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 focus of the present volume. For some classes
of quadratic operator polynomials, inverse problems are also
considered. The connection between the spectra of such quadratic
operator polynomials 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, which is
supplemented with detailed proofs. With regard to the reader’s
background knowledge, only the basic properties of operators in
Hilbert spaces and well-known results from complex analysis are
assumed.
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, which is
supplemented with detailed proofs. With regard to the reader’s
background knowledge, only the basic properties of operators in
Hilbert spaces and well-known results from complex analysis are
assumed.