Multiple minimality of the root vectors of a polynomial operator pencil perturbed by an analytic outside a disc operator-function S(?) with S(?)=0

1989 ◽  
Vol 40 (5) ◽  
pp. 511-519
Author(s):  
G. V. Radzievskii
Keyword(s):  
Operator Function ◽  
Operator Pencil ◽  
Polynomial Operator ◽  
Polynomial Operator Pencil

The Laplace transform and spectral representations of polynomial operator pencils

1993 ◽  
Vol 443 (1918) ◽  
pp. 333-342
Keyword(s):  
Laplace Transform ◽  
Spectral Representation ◽  
Eigenvalue Problems ◽  
Initial Conditions ◽  
Classical Problem ◽  
Operator Pencil ◽  
Linear Operators ◽  
Inverse Laplace Transform ◽  
Polynomial Operator ◽  
The Laplace Transform

The spectral representation associated with the polynomial operator pencil L 0 + λL 1 + λ 2 L 2 +. . . λ N L N , where L n ( n = 0,1,2,..., N ) are linear operators and λ is a complex parameter, is derived formally using the Laplace transform. The derivation involves a conversion of the eigenvalue problem for the operator pencil into an initial-value problem by replacing λ with ∂/∂t and introducing N -1 initial conditions. This procedure yields the spectral representation in the form of an inverse Laplace transform of the Green’s operator associated with the operator pencil. The results of this paper are illustrated with examples and provide a simple but powerful and systematic approach to non-standard eigenvalue problems for linear operators. These examples are a 2 x 2 matrix problem which has three eigenvalues, a Sturm-Liouville-Rossby type wave equation discussed recently by A. Masuda ( Q. appl. Math . 47, 435-445 (1989)), and a classical problem in which the eigenvalue parameter appears not only in the differential equation but also in the boundary conditions.

Trace formula for polynomial operator pencils and its applications

1992 ◽  
Vol 25 (4) ◽  
pp. 306-308 ◽  
Author(s):  
K. Kh. Boimatov ◽  
A. G. Kostyuchenko
Keyword(s):  
Trace Formula ◽  
Polynomial Operator ◽  
Operator Pencils

scholarly journals Normal modes of a waveguide as eigenvectors of a self-adjoint operator pencil

2021 ◽  
Vol 29 (1) ◽  
pp. 14-21
Author(s):  
Mikhail D. Malykh
Keyword(s):  
Electromagnetic Field ◽  
Differential Equations ◽  
Wave Equations ◽  
Adjoint Operator ◽  
Normal Modes ◽  
Operator Pencil ◽  
Homogeneous Case ◽  
Permittivity And Permeability ◽  
Simply Connected ◽  
Magnetic Potentials

A waveguide with a constant, simply connected section S is considered under the condition that the substance filling the waveguide is characterized by permittivity and permeability that vary smoothly over the section S, but are constant along the waveguide axis. Ideal conductivity conditions are assumed on the walls of the waveguide. On the basis of the previously found representation of the electromagnetic field in such a waveguide using 4 scalar functions, namely, two electric and two magnetic potentials, Maxwells equations are rewritten with respect to the potentials and longitudinal components of the field. It appears possible to exclude potentials from this system and arrive at a pair of integro-differential equations for longitudinal components alone that split into two uncoupled wave equations in the optically homogeneous case. In an optically inhomogeneous case, this approach reduces the problem of finding the normal modes of a waveguide to studying the spectrum of a quadratic self-adjoint operator pencil.

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