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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 018 728 86X
Author(s):
Behncke, Horst:
Shorttitle:
The spectrum of differential operators with almost constant coefficients
Source:
J. Comput. Appl. Math. 148, No. 1, 287-305 (2002)
Classification:
34L20Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
Primary Classification:
34B20Weyl theory and its generalizations
Secondary Classification:
47E05Ordinary differential operators
Keywords:
Review:

The practical importance of Schroedinger equation on half-axis is one
of the key roots of the author's interest in the determination of the
absolutely continuous spectrum of the more general operators H
related to the linear and formaly symmetric differential expressions
L of the order 2n. Under standard assumptions on the smoothness and
asymptotic decay of coefficients it is proved and illustrated that
for any self-adjoint extension H of L, its absolutely continuous part
is unitarily equivalent to the operator of multiplication derived
from the asymptotics of coefficients. The method of asymptotic
integration (in a way continuing the previous work [5] and inspired
by a twenty years old work [1] by Ahlbrandt et al) is used. Imagining
that the associated first-order system is Hamiltonian, it is shown
most natural to employ the generalized (= matrix) form of the
Titchmarsh-Weyl concept of the m-function (cf. [12]). Its
interpretation as a Borel transform of the spectral measure leads to
the desired results via a repeated Kummer-Liouville transformations
(= ``diagonalizations") modulo ``Levinson" terms (having no influence
on asymptotics).
Remarks to the editors:

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