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Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications (Hardcover, 2015 ed.)
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Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications (Hardcover, 2015 ed.)
Series: Operator Theory: Advances and Applications, 246
Expected to ship within 12 - 17 working days
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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- I 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.
General
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