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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 017 587 319
Author(s):
Andrianov, A. Yu.
Shorttitle:
A spectral theorem for Sturm Liouville operators
Source:
Differ. Equ. 37, No. 8, 1074 - 1087 (2001); translation from Russian: Differ. Uravn. 37, No. 8, 1028 - 1040 (2001).
Classification:
30B05 Sorry no match found within MSC2000
Primary Classification:
34L10Eigenfunction expansions, completeness of eigenfunctions
Secondary Classification:
42A38Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Keywords:
Schroedinger equation; potentials that are finite sums of exponentials;
Review:

A finite sum of exponentials exp(igx) (over some K positive g's)
defines a potential V(x) and Hamiltonian H which is not
self-adjoint. The author proves that the set of the related
(right) Hilbert-space eigenfunctions may still be understood as
playing the same eigenfunction expansion role as their standard
Sturm-Liouville analogs do in the self-adjoint cases. The precise
meaning of this observation is formulated and proved as a theorem
giving the expansion of a function f in terms of the eigenstates
of H in the form of the Fourier-type pricipal-value integral over
the spectrum. The author adds a remark which recommends an
alternative arrangement of the eigenfunction expansion in the form
with separates the contribution of the continuous spectrum from
that of the isolated spectral singularities. Marginally, it is
also worth noting that on the certain modified domains specified
by the so called PT symmetry requirement in quantum mechanics, the
Hamiltonians H in question may still possess the pure point
spectrum [cf. F. Cannata et al, Phys. Lett. A 246 (1998) 219 where
K = 1 and M. Znojil, Phys. Lett. A 264 (1999) 108 where K = 2] and
could attract a further study motivated by the physics of bound
states.
Remarks to the editors:

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