2,346 research outputs found
An inverse Sturm-Liouville problem with a fractional derivative
In this paper, we numerically investigate an inverse problem of recovering
the potential term in a fractional Sturm-Liouville problem from one spectrum.
The qualitative behaviors of the eigenvalues and eigenfunctions are discussed,
and numerical reconstructions of the potential with a Newton method from finite
spectral data are presented. Surprisingly, it allows very satisfactory
reconstructions for both smooth and discontinuous potentials, provided that the
order of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of
Computational Physic
Inverse Spectral Theory for Sturm-Liouville Operators with Distributional Potentials
We discuss inverse spectral theory for singular differential operators on
arbitrary intervals associated with rather general
differential expressions of the type where the coefficients , ,
, are Lebesgue measurable on with , , , and real-valued with and a.e.\ on
. In particular, we explicitly permit certain distributional potential
coefficients.
The inverse spectral theory results derived in this paper include those
implied by the spectral measure, by two-spectra and three-spectra, as well as
local Borg-Marchenko-type inverse spectral results. The special cases of
Schr\"odinger operators with distributional potentials and Sturm--Liouville
operators in impedance form are isolated, in particular.Comment: 29 page
Inverse eigenvalue problem for discrete three-diagonal Sturm-Liouville operator and the continuum limit
In present article the self-contained derivation of eigenvalue inverse
problem results is given by using a discrete approximation of the Schroedinger
operator on a bounded interval as a finite three-diagonal symmetric Jacobi
matrix. This derivation is more correct in comparison with previous works which
used only single-diagonal matrix. It is demonstrated that inverse problem
procedure is nothing else than well known Gram-Schmidt orthonormalization in
Euclidean space for special vectors numbered by the space coordinate index. All
the results of usual inverse problem with continuous coordinate are reobtained
by employing a limiting procedure, including the Goursat problem -- equation in
partial derivatives for the solutions of the inversion integral equation.Comment: 19 pages There were made some additions (and reformulations) to the
text making the derivation of the results more precise and understandabl
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