77 research outputs found
Relative Oscillation Theory for Dirac Operators
We develop relative oscillation theory for one-dimensional Dirac operators
which, rather than measuring the spectrum of one single operator, measures the
difference between the spectra of two different operators. This is done by
replacing zeros of solutions of one operator by weighted zeros of Wronskians of
solutions of two different operators. In particular, we show that a Sturm-type
comparison theorem still holds in this situation and demonstrate how this can
be used to investigate the number of eigenvalues in essential spectral gaps.
Furthermore, the connection with Krein's spectral shift function is
established. As an application we extend a result by K.M. Schmidt on the
finiteness/infiniteness of the number of eigenvalues in essential spectral gaps
of perturbed periodic Dirac operators.Comment: 13 page
Inverse Scattering Problem on the Axis for the Schrödinger Operator with Triangular 2 x 2 Matrix Potential. I. Main Theorem
The necessary and sufficient conditions for solvability of ISP under consideration are obtained
Inverse Scattering Problem on the Axis for the Schrödinger Operator with Triangular 2 x 2 Matrix Potential. II. Addition of the Discrete Spectrum
The theorem of the necessary and sufficient conditions for the solvability of ISP under consideration is proved. The method of addition of the discrete spectrum to the considered matrix not self-adjoint case is developed
Necessary and Sufficient Conditions in Inverse Scattering Problem on the Axis for the Triangular 2 x 2 Matrix Potential
The characteristic properties of the scattering data for the Schr¨odinger operator on the axis with a triangular 2 × 2 matrix potential are obtained. A difference between the necessary and sufficient conditions for solvability of ISPunder consideration, contained in the previous works of the authors, is eliminated
On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
For the Sturm-Louville equation with block-triangular matrix potential that increases at infinity, both increasing and decreasing at infinity matrix solutions are found. The structure of spectrum for the differential operator with these coefficients is defined.Для уравнения Штурма-Лиувилля с блочно-треугольным растущим на бесконечности матричным потенциалом построены убывающие и растущие на бесконечности матричные решения. Установлена структура спектра дифференциального оператора с такими коэффициентами
Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function
We develop an analog of classical oscillation theory for Sturm-Liouville
operators which, rather than measuring the spectrum of one single operator,
measures the difference between the spectra of two different operators.
This is done by replacing zeros of solutions of one operator by weighted
zeros of Wronskians of solutions of two different operators. In particular, we
show that a Sturm-type comparison theorem still holds in this situation and
demonstrate how this can be used to investigate the finiteness of eigenvalues
in essential spectral gaps. Furthermore, the connection with Krein's spectral
shift function is established.Comment: 26 page
On the shape of spectra for non-self-adjoint periodic Schr\"odinger operators
The spectra of the Schr\"odinger operators with periodic potentials are
studied. When the potential is real and periodic, the spectrum consists of at
most countably many line segments (energy bands) on the real line, while when
the potential is complex and periodic, the spectrum consists of at most
countably many analytic arcs in the complex plane.
In some recent papers, such operators with complex -symmetric
periodic potentials are studied. In particular, the authors argued that some
energy bands would appear and disappear under perturbations. Here, we show that
appearance and disappearance of such energy bands imply existence of nonreal
spectra. This is a consequence of a more general result, describing the local
shape of the spectrum.Comment: 5 pages, 2 figure
The velocity of slow nuclear burning in the two-group approximation
The velocity of slow nuclear burning was obtained in the two-group approximation. Two groups of neutrons were considered: the group of thermal (slow) neutrons and the group of fast neutrons; each group being described with its diffusion equation. It was shown that in the case of heavy moderators the obtained expression for the two-group velocity had the same structure as the one-group velocity studied by authors before if new effective diffusion and multiplication coefficients were introduced. The expressions for corresponding effective coefficients are presented
Biorthogonal Quantum Systems
Models of PT symmetric quantum mechanics provide examples of biorthogonal
quantum systems. The latter incorporporate all the structure of PT symmetric
models, and allow for generalizations, especially in situations where the PT
construction of the dual space fails. The formalism is illustrated by a few
exact results for models of the form H=(p+\nu)^2+\sum_{k>0}\mu_{k}exp(ikx). In
some non-trivial cases, equivalent hermitian theories are obtained and shown to
be very simple: They are just free (chiral) particles. Field theory extensions
are briefly considered.Comment: 34 pages, 5 eps figures; references added and other changes made to
conform to journal versio
On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators
The article is devoted to the following question. Consider a periodic
self-adjoint difference (differential) operator on a graph (quantum graph) G
with a co-compact free action of the integer lattice Z^n. It is known that a
local perturbation of the operator might embed an eigenvalue into the
continuous spectrum (a feature uncommon for periodic elliptic operators of
second order). In all known constructions of such examples, the corresponding
eigenfunction is compactly supported. One wonders whether this must always be
the case. The paper answers this question affirmatively. What is more
surprising, one can estimate that the eigenmode must be localized not far away
from the perturbation (in a neighborhood of the perturbation's support, the
width of the neighborhood determined by the unperturbed operator only).
The validity of this result requires the condition of irreducibility of the
Fermi (Floquet) surface of the periodic operator, which is expected to be
satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio
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