56 research outputs found
An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated weights
We consider self-adjoint unbounded Jacobi matrices with diagonal q_n=n and
weights \lambda_n=c_n n, where c_n is a 2-periodical sequence of real numbers.
The parameter space is decomposed into several separate regions, where the
spectrum is either purely absolutely continuous or discrete. This constitutes
an example of the spectral phase transition of the first order. We study the
lines where the spectral phase transition occurs, obtaining the following main
result: either the interval (-\infty;1/2) or the interval (1/2;+\infty) is
covered by the absolutely continuous spectrum, the remainder of the spectrum
being pure point. The proof is based on finding asymptotics of generalized
eigenvectors via the Birkhoff-Adams Theorem. We also consider the degenerate
case, which constitutes yet another example of the spectral phase transition
Lowering and raising operators for the free Meixner class of orthogonal polynomials
We compare some properties of the lowering and raising operators for the
classical and free classes of Meixner polynomials on the real line
On a two variable class of Bernstein-Szego measures
The one variable Bernstein-Szego theory for orthogonal polynomials on the
real line is extended to a class of two variable measures. The polynomials
orthonormal in the total degree ordering and the lexicographical ordering are
constructed and their recurrence coefficients discussed.Comment: minor change
Spectral and Localization Properties for the One-Dimensional Bernoulli Discrete Dirac Operator
A 1D Dirac tight-binding model is considered and it is shown that its
nonrelativistic limit is the 1D discrete Schr?odinger model. For random
Bernoulli potentials taking two values (without correlations), for typical
realizations and for all values of the mass, it is shown that its spectrum is
pure point, whereas the zero mass case presents dynamical delocalization for
specific values of the energy. The massive case presents dynamical localization
(excluding some particular values of the energy). Finally, for general
potentials the dynamical moments for distinct masses are compared, especially
the massless and massive Bernoulli cases.Comment: no figure; 24 pages; to appear in Journal of Mathematical Physic
Exact Boundary Conditions at an Artificial Boundary for Partial Differential Equations in Cylinders
Intertwining technique for a system of difference Schroedinger equations and new exactly solvable multichannel potentials
The intertwining operator technique is applied to difference Schroedinger
equations with operator-valued coefficients. It is shown that these equations
appear naturally when a discrete basis is used for solving a multichannel
Schroedinger equation. New families of exactly solvable multichannel
Hamiltonians are found
The class of n-entire operators
We introduce a classification of simple, regular, closed symmetric operators
with deficiency indices (1,1) according to a geometric criterion that extends
the classical notions of entire operators and entire operators in the
generalized sense due to M. G. Krein. We show that these classes of operators
have several distinctive properties, some of them related to the spectra of
their canonical selfadjoint extensions. In particular, we provide necessary and
sufficient conditions on the spectra of two canonical selfadjoint extensions of
an operator for it to belong to one of our classes. Our discussion is based on
some recent results in the theory of de Branges spaces.Comment: 33 pages. Typos corrected. Changes in the wording of Section 2.
References added. Examples added. arXiv admin note: text overlap with
arXiv:1104.476
Spectral averaging techniques for Jacobi matrices with matrix entries
A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal
matrix with invertible blocks on the off-diagonals. Averaging over boundary
conditions leads to explicit formulas for the averaged spectral measure which
can potentially be useful for spectral analysis. Furthermore another variant of
spectral averaging over coupling constants for these operators is presented
Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))
Diagonalization of a certain operator in irreducible representations of the
positive discrete series of the quantum algebra U_q(su(1,1)) is studied.
Spectrum and eigenfunctions of this operator are found in an explicit form.
These eigenfunctions, when normalized, constitute an orthonormal basis in the
representation space. The initial U_q(su(1,1))-basis and the basis of
eigenfunctions are interrelated by a matrix with entries, expressed in terms of
big q-Laguerre polynomials. The unitarity of this connection matrix leads to an
orthogonal system of functions, which are dual with respect to big q-Laguerre
polynomials. This system of functions consists of two separate sets of
functions, which can be expressed in terms of q-Meixner polynomials
M_n(x;b,c;q) either with positive or negative values of the parameter b. The
orthogonality property of these two sets of functions follows directly from the
unitarity of the connection matrix. As a consequence, one obtains an
orthogonality relation for q-Meixner polynomials M_n(x;b,c;q) with b<0. A
biorthogonal system of functions (with respect to the scalar product in the
representation space) is also derived.Comment: 15 pages, LaTe
Boundary relations and generalized resolvents of symmetric operators
The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint
exit space extensions of a, not necessarily densely defined, symmetric
operator, in terms of maximal dissipative (in \dC_+) holomorphic linear
relations on the parameter space (the so-called Nevanlinna families). The new
notion of a boundary relation makes it possible to interpret these parameter
families as Weyl families of boundary relations and to establish a simple
coupling method to construct the generalized resolvents from the given
parameter family. The general version of the coupling method is introduced and
the role of boundary relations and their Weyl families for the
Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page
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