179 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
Toda Hierarchy with Indefinite Metric
We consider a generalization of the full symmetric Toda hierarchy where the
matrix of the Lax pair is given by , with a full
symmetric matrix and a nondegenerate diagonal matrix . The key feature
of the hierarchy is that the inverse scattering data includes a class of
noncompact groups of matrices, such as . We give an explicit formula
for the solution to the initial value problem of this hierarchy. The formula is
obtained by generalizing the orthogonalization procedure of Szeg\"{o}, or the
QR factorization method of Symes. The behaviors of the solutions are also
studied. Generically, there are two types of solutions, having either sorting
property or blowing up to infinity in finite time. The -function
structure for the tridiagonal hierarchy is also studied.Comment: 26 pages, LaTe
On Weyl-Titchmarsh Theory for Singular Finite Difference Hamiltonian Systems
We develop the basic theory of matrix-valued Weyl-Titchmarsh M-functions and
the associated Green's matrices for whole-line and half-line self-adjoint
Hamiltonian finite difference systems with separated boundary conditions.Comment: 30 pages, to appear in J. Comput. Appl. Mat
Manipulation of Semiclassical Photon States
Gabriel F. Calvo and Antonio Picon defined a class of operators, for use in
quantum communication, that allows arbitrary manipulations of the three lowest
two-dimensional Hermite-Gaussian modes {|0,0>,|1,0>,|0,1>}. Our paper continues
the study of those operators, and our results fall into two categories. For
one, we show that the generators of the operators have infinite deficiency
indices, and we explicitly describe all self-adjoint realizations. And secondly
we investigate semiclassical approximations of the propagators. The basic
method is to start from a semiclassical Fourier integral operator ansatz and
then construct approximate solutions of the corresponding evolution equations.
In doing so, we give a complete description of the Hamilton flow, which in most
cases is given by elliptic functions. We find that the semiclassical
approximation behaves well when acting on sufficiently localized initial
conditions, for example, finite sums of semiclassical Hermite-Gaussian modes,
since near the origin the Hamilton trajectories trace out the bounded
components of elliptic curves.Comment: 30 pages, 3 figures. Small corrections, mostly in Section V. To
appear in the Journal of Mathematical Physic
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
A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line
Szego's procedure to connect orthogonal polynomials on the unit circle and
orthogonal polynomials on [-1,1] is generalized to nonsymmetric measures. It
generates the so-called semi-orthogonal functions on the linear space of
Laurent polynomials L, and leads to a new orthogonality structure in the module
LxL. This structure can be interpreted in terms of a 2x2 matrix measure on
[-1,1], and semi-orthogonal functions provide the corresponding sequence of
orthogonal matrix polynomials. This gives a connection between orthogonal
polynomials on the unit circle and certain classes of matrix orthogonal
polynomials on [-1,1]. As an application, the strong asymptotics of these
matrix orthogonal polynomials is derived, obtaining an explicit expression for
the corresponding Szego's matrix function.Comment: 28 page
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
Discrete supersymmetries of the Schrodinger equation and non-local exactly solvable potentials
Using an isomorphism between Hilbert spaces and we consider
Hamiltonians which have tridiagonal matrix representations (Jacobi matrices) in
a discrete basis and an eigenvalue problem is reduced to solving a three term
difference equation. Technique of intertwining operators is applied to creating
new families of exactly solvable Jacobi matrices. It is shown that any thus
obtained Jacobi matrix gives rise to a new exactly solvable non-local potential
of the Schroedinger equation. We also show that the algebraic structure
underlying our approach corresponds to supersymmetry. Supercharge operators
acting in the space are introduced which together
with a matrix form of the superhamiltonian close the simplest superalgebra.Comment: 12 page
Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials
In this work we characterize a high-order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to study the solution of this dynamical system, we give explicit expressions for the Weyl function, generalized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high-order Toda lattice. We also present a Lax-type theorem for the point spectrum of the Jacobi operator associated with a Toda-type lattic
On Inverse Scattering at a Fixed Energy for Potentials with a Regular Behaviour at Infinity
We study the inverse scattering problem for electric potentials and magnetic
fields in \ere^d, d\geq 3, that are asymptotic sums of homogeneous terms at
infinity. The main result is that all these terms can be uniquely reconstructed
from the singularities in the forward direction of the scattering amplitude at
some positive energy.Comment: This is a slightly edited version of the previous pape
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