Spectral theory of operator-valued functions (OFVs) envelops different solution
techniques for operator eigenvalue problems with nonlinear dependence on the spectral
parameter. This theory offers a universal approach to the solution of linear and nonlinear
partial differential equations (PDEs), integral equations (IEs), and boundary value
problems (BVP)s, and often when traditional methods fail, e.g. when nonselfadjoint
BVPs are considered in unbounded domains and the spectral parameter enters the
conditions at infinity in a nonlinear manner.
Ideas and methods of the spectral theory of OFVs are now used in quite different areas:
from qualitatively new approaches to the existence and uniqueness theorems for linear
and nonlinear PDEs to new numerical methods and mathematical models (nature of
resonance scattering, interaction phenomena, and others).
The first studies date back to the works of Hilbert [1] and Giraud [2] published in the
1920s and 1930s. They considered in particular integral operators (and infinite
determinants) with kernels depending on a complex parameter. Many important results
were obtained by Keldysh [3] and other mathematicians in the 1950s who studied
abstract operator pencils. A breakthrough was made by Gohberg [4] in the 1950s; he
introduced main definitions and constructed foundations of the modern spectral theory of
OFVs and proved on this basis the so-called analytical Fredholm theorem, according to
which a Fredholm OVF A(z) acting in a Hilbert space and analytical with respect to the
complex spectral parameter z may have not more than a finite set of characteristic
numbers (CNs) in every bounded subdomain. This result was then generalized to the case
of analytical Fredholm OVFs acting on a pair of Banach spaces. It was shown [5] that
many statements of the theory of functions of one complex variable remain valid in the
'operator' case, in particular, the Rouchet's theorem [6]. Another significant contribution
was made by Vainikko and coauthors in the 1970s [7] who performed a fine analysis of
the spectrum of operator eigenvalue problems and justified in particular the approximate
determination of CNs.
Spectral theory of nonselfadjoint operators [8, 9] has been successfully applied to the
solution of nonselfadjoint BVPs since the early 1950s. Beginning from the second half of
the 1970s the methods of spectral theory of OFVs, which was a rapidly developing
branch of the general nonselfadjoint spectral theory, found broad application as a
powerful solution technique.
Nonselfadjoint BVPs arise naturally in many branches of mathematical physics, first of
all in the mathematical theory of wave propagation and diffraction (electromagnetis), and
also in mechanics, elasticity, fluid dynamics [10—13]. BVPs for the Maxwell and
Helmholtz equations in unbounded domains occupy a specific place: first, they are
nonselfadjoint, and, second, the spectral parameter enters the boundary conditions in a
nonlinear manner.
Already correct statements of BVPs, especially eigenvalue problems meet substantial
difficulties. The proof of unique solvability and determination of the spectrum require
alternative techniques. Therefore specific methods are needed to investigate these
problems. Some of them are reviewed in [14, 15]. In this respect, the spectral theory of
OVFs clearly demonstrated advantages during the last two decades.
Perturbation theory. The analysis of the dependence of eigenvalues of differential
operators on parameters is a specific part of the perturbation theory [16] which also can
be studied using the methods of the spectral theory of OVFs. This dependence can be
described [16, 17] by functions of several complex variables that have critical points of
various nature. Degeneration of eigenvalues as well as many important physical
phenomena (e.g., resonance scattering and interaction of oscillations and waves) are
shown to be connected with such points. The knowledge of critical points enables one to
solve inverse problems because it gives a specific alternative description of the object (a
characterization of the domain where the eigenvalue problem is considered and the
equation parameters). The determination of critical points with respect to the different
problem parameters and the analysis of the interaction phenomena are reviewed in [14]
and recently, in [15].
Nonlinear equations. Development of the methods of solution to singular BVPs for the
linear and nonlinear Helmholtz and Schrödinger equations with complex-valued
coefficients is another field of application of the spectral theory of OVFs [18—20]. These
problems can be reduced to nonlinear Volterra and Fredholm integral equations and their
CNs are determined using contraction.
Application to mathematical methods in electromagnetics. In the 1980s, the specific
problem settings and mathematical models related to BVPs arising in electromagnetics
and acoustics that are solved on the basis of the spectral theory of OVFs have become
known under the common name of the spectral theory of open structures [14]; this
theory was constructed and developed during the last two decades on the basis of the
spectral theory of OVFs and specially developed methods using pseudodifferential
equations [21].
References
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Leipzig (1924).
2. G. Giraud, Equations a integrales principales, Ann. Sci. Ec. Norm. Sup., 51, 251--272
(1934).
3. M. Keldysh, On the Characteristic Values and Characteristic Functions of Certain
Classes of Nonelfadjoint Equations, Doklady Akad. Nauk SSSR Maths., 77, 11--14 1951).
4. I. Gohberg, On Linear Operators Depending Analytically on a Parameter, Doklady
Akad. Nauk SSSR Maths., 78, 629--632 (1951).
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Indexes of Linear Operators, Uspehi Mat. Nauk, 12, 43--118 (1957).
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17. V. Arnold, S. Gusein-Zade, A. Varchenko, Singularities of Differentiable Maps,
Birkhäuser, Boston (1988).
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Lossless Nonlinear Three-layer Structure, Physical Review E, 58, 1, pp. 1040--1050
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the Propagation of TE-Waves in a Nonlinear Dielectric Cylindrical Waveguide, J.
Nonlin. Math. Phys., 11, 2, pp. 256--268 (2004).
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cylindrical nonlinear dielectric waveguides, Phys. Rev. E, 71, pp. 0166141-10 (2005).
21. A. Il'inski, Y. Smirnov, Electromagnetic Wave Diffraction by Conducting Screens, Y.
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