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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 016 852 173
Author(s):
Makin, A. S.:
Shorttitle:
On many-point spectral boundary value problems.
Source:
Differ. Equ. 36, No. 10, 1461 - 1468 (2000); translation from Differ. Uravn. 36, No. 10, 1324 - 1330 (2000).
Classification:
34L15Estimation of eigenvalues, upper and lower bounds
Primary Classification:
34B10Multipoint boundary value problems
Secondary Classification:
34L20Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
Keywords:
Review:

The growing role of the non-Hermitian Hamiltonians in physics (the
most fresh Los Alamos preprint arXiv: math-ph/0205002 by B. Bagchi
and C. Quesne may be recalled for review and typical illustration)
is paralleled by a perceivable intensification of their rigorous
studies. This represents a strong motivation for the study of the
Laplace operator on a finite interval with the Dirichlet boundary
condition at the mere left end. In the letter in question this
operator is made non-self-adjoint via the generalized right-end
boundary condition, complementing its usual mixed form by a
strongly non-local term (viz, by a superposition of the first
derivatives at an m-plet of internal points).

For the resulting Bitsadze-Samarskii (or generalized
Samarskii-Ionkin) solutions (forming a bi-orthogonal basis in the
corresponding Hilbert space) the author proves a bound for the
norms and (sizes of) eigenvalues. An appeal of this result stems
from the fact that the m-plet of internal points must be assumed
rational. Otherwise, the estimate is shown to cease to be valid.
The author also outlines a
few further improvements of his/her estimate in the rational cases.
Remarks to the editors:

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