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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 018 217 573
Author(s):
Arov, D. Z.:
Shorttitle:
Scattering matrix and impedance of a canonical differential system with a dissipative boundary condition
Source:
St. Petersbg. Math. J. 13, No. 4, 527-547 (2002); translation from Algebra Anal. 13, No. 4, 26-53 (2001).
Classification:
34A55Inverse problems
34L40Particular operators Dirac, one-dimensional Schroedinger, etc.
Primary Classification:
34B07Linear boundary value problems with nonlinear dependence on the spectral parameter
Secondary Classification:
34L25Scattering theory
Keywords:
Volterra operators; canonical differential system; dissipative boundary condition; rational matrix-valued friction; impedance; scattering matrix; dynamical pliability; inverse problem; string with friction; Darlington factorization; electrical chains; monodromy matrix;
Review:

The paper is devoted to a study of certain systems of linear
differential equations on a finite interval (0,L). I recommend
that the interested reader glimpses at eq. (5.1) first. This is
the second-order equation for the amplitude of oscillations of a
string with an arbitrary distribution of its mass and with a
certain frequency-dependent friction r at point L (=right end of
the string) and/or impedance c (i.e., the Krein's ``pliability
of the velocity") at zero. At this point, the author's older
study of this problem (reference [1] where r was assumed
independent of frequences) emerges as the direct predecessor and
guide to the present text (where r is assumed to be a rational
real function of frequences). The methods and proofs (based,
basically, on the Darligton's factorization approach) remain
fairly similar, and the results characterize again the class of
impedances by transition to the system of the two first-order
equations. One is prepared to start reading the text carefully
from the start and to get acquainted with the problem in its
natural and sufficiently general matrix formulation. Its key
ingredient is the representation of the matrix of the canonical
first-order differential-equation system in question in the form
of a product of a matrix square root J of the unit matrix I with
a positive matrix factor H called ``Hermitian". A broad area of
further applications is covered in this way in principle (e.g.,
c may be related to ``impedance" or ``scattering matrix" etc).
Thus, the reader now reveals the meaning of the title of the
paper and is prepared to endulge the wealth of the structure
theorems for c in various special cases, rewarded by the final
understanding of eq. (5.1) once more, being given the nice and
elegant necessary and sufficient conditions for c in Theorem
5.1, a guide to the construction of the related monodromy matrix
etc.
Remarks to the editors:

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