21,220 research outputs found
Asymptotic properties of eigenvalues and eigenfunctions of a Sturm-Liouville problem with discontinuous weight function
In this paper, by using the similar methods of [O. Sh. Mukhtarov and M.
Kadakal, Some spectral properties of one Sturm-Liouville type problem with
discontinuous weight, Siberian Mathematical Journal, 46 (2005) 681-694] we
extend some spectral properties of regular Sturm-Liouville problems to those
which consist of a Sturm-Liouville equation with discontinuous weight at two
interior points together with spectral parameter-dependent boundary conditions.
We give an operator-theoretic formulation for the considered problem and obtain
asymptotic formulas for the eigenvalues and eigenfunctions.Comment: 11 page
Black-hole concept of a point-like nucleus with supercritical charge
The Dirac equation for an electron in the central Coulomb field of a
point-like nucleus with the charge greater than 137 is considered. This
singular problem, to which the fall-down onto the centre is inherent, is
addressed using a new approach, based on a black-hole concept of the singular
centre and capable of producing cut-off-free results. To this end the Dirac
equation is presented as a generalized eigenvalue boundary problem of a
self-adjoint operator. The eigenfunctions make complete sets, orthogonal with a
singular measure, and describe particles, asymptotically free and
delta-function-normalizable both at infinity and near the singular centre
. The barrier transmission coefficient for these particles responsible for
the effects of electron absorption and spontaneous electron-positron pair
production is found analytically as a function of electron energy and charge of
the nucleus. The singular threshold behaviour of the corresponding amplitudes
substitutes for the resonance behaviour, typical of the conventional theory,
which appeals to a finite-size nucleus.Comment: 22 pages, 5 figures, LATEX requires IOPAR
Black Hole Scattering from Monodromy
We study scattering coefficients in black hole spacetimes using analytic
properties of complexified wave equations. For a concrete example, we analyze
the singularities of the Teukolsky equation and relate the corresponding
monodromies to scattering data. These techniques, valid in full generality,
provide insights into complex-analytic properties of greybody factors and
quasinormal modes. This leads to new perturbative and numerical methods which
are in good agreement with previous results.Comment: 28 pages + appendices, 2 figures. For Mathematica calculation of
Stokes multipliers, download "StokesNotebook" from
https://sites.google.com/site/justblackholes/techy-zon
Sturm Liouville Problem with Moving Discontinuity Points
In this paper, we present a new discontinuous Sturm Liouville problem with
symmetrically located discontinuities which are defined depending on a
neighborhood of a midpoint of the interval. Also the problem contains an
eigenparameter in one of the boundary conditions and has coupled transmission
conditions at the discontinuity points. We investigate the properties of the
eigenvalues, obtain asymptotic formulas for the eigenvalues and the
corresponding eigenfunctions and construct Green's function of this problem.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1210.4350 by
other author
Transmission eigenvalues and thermoacoustic tomography
The spectrum of the interior transmission problem is related to the unique
determination of the acoustic properties of a body in thermoacoustic imaging.
Under a non-trapping hypothesis, we show that sparsity of the interior
transmission spectrum implies a range separation condition for the
thermoacoustic operator. In odd dimension greater than or equal to three, we
prove that the transmission spectrum for a pair of radially symmetric
non-trapping sound speeds is countable, and conclude that the ranges of the
associated thermoacoustic maps have only trivial intersection
- …