3,070 research outputs found
The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality
We study two problems. The first one is the similarity problem for the
indefinite Sturm-Liouville operator A=-(\sgn\, x)\frac{d}{wdx}\frac{d}{rdx}
acting in . It is assumed that w,r\in L^1_{\loc}(-b,b) are
even and positive a.e. on .
The second object is the so-called HELP inequality
where the coefficients \tilde{w},\tilde{r}\in L^1_{\loc}[0,b) are
positive a.e. on .
Both problems are well understood when the corresponding Sturm-Liouville
differential expression is regular. The main objective of the present paper is
to give criteria for both the validity of the HELP inequality and the
similarity to a self-adjoint operator in the singular case. Namely, we
establish new criteria formulated in terms of the behavior of the corresponding
Weyl-Titchmarsh -functions at 0 and at . As a biproduct of this
result we show that both problems are closely connected. Namely, the operator
is similar to a self-adjoint one precisely if the HELP inequality with
and is valid.
Next we characterize the behavior of -functions in terms of coefficients
and then these results enable us to reformulate the obtained criteria in terms
of coefficients. Finally, we apply these results for the study of the two-way
diffusion equation, also known as the time-independent Fokker-Plank equation.Comment: 42 page
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
Solution for the BFKL Pomeron Calculus in zero transverse dimensions
In this paper the exact analytical solution is found for the BFKL Pomeron
calculus in zero transverse dimensions, in which all Pomeron loops have been
included. The comparison with the approximate methods of the solution is given,
and the kinematic regions are discussed where they describe the behaviour of
the scattering amplitude quite well. In particular, the semi-classical approach
is considered, which reproduces the main properties of the exact solution at
large values of rapidity (). It is shown that the mean field
approximation leads to a good description of the scattering amplitude only if
the amplitude at low energy is rather large. However, even in this case, it
does not lead to the correct asymptotic behaviour of the scattering amplitude
at high energies.Comment: 37 pages,19 figures and one table, the revised versio
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
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