Related papers: The $J$-matrix method
We present a generic operator $J$ simply defined as a linear map not increasing the degree from the vectorial space of polynomial functions into itself and we address the problem of finding the polynomial sequences that coincide with the…
We consider Schr\"odinger operators with complex-valued decreasing potentials on the half-line. Such operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the…
For each irreducible module of the symmetric group on $N$ objects there is a set of parametrized nonsymmetric Jack polynomials in $N$ variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative…
We construct efficient approximations for the eigenfunctions of non-self-adjoint Schroedinger operators in one dimension. The same ideas also apply to the study of resonances of self-adjoint Schroedinger operators which have dilation…
We introduce a perturbative formulation for a nonlinear extension of the J-matrix method of scattering in two dimensions. That is, we obtain the scattering matrix for the time-independent nonlinear Schr\"odinger equation in two dimensions…
We propose a new method in the spectral analysis of noisy time-series data for damped oscillators. From the Jacobi three terms recursive relation for the denominators of the Pad\'e Approximations built on the well-known Z-transform of an…
The Jaynes-Cummings model is solved with the raising and lowering (shift) operators by using the matrix-diagonalizing technique. Bell nonlocality is also found present ubiquitously in the excitations states of the model.
The single-channel Jost function is calculated with the computational R-matrix on a Lagrange-Jacobi mesh, in order to study its behaviour at complex wavenumbers. Three potentials derived from supersymmetric transformations are used to test…
The paper is largely concerned with the possibility of obtaining a series representation for a compact linear map $T$ acting between Banach spaces. It is known that, using the notions of $j-$eigenfunctions and $j-$% eigenvalues, such a…
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…
The motion of holes on the triangular lattice is studied using the t-J model. Within the Born self-consistent approximation and the exact Lanczos diagonalization, the single hole physics is first analyzed. Then the spiral theory of Shraiman…
We utilize group-theoretical methods to develop a matrix representation of differential operators that act on tensors of any rank. In particular, we concentrate on the matrix formulation of the curl operator. A self-adjoint matrix of the…
In this article we obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the self-adjoint operator generated by a system of Sturm-Liouville equations with summable coefficients and the quasiperiodic boundary conditions.…
Schr\"{o}dinger operators with nonlocal $\delta$-interaction are studied with the use of the Lax-Phillips scattering theory methods. The condition of applicability of the Lax-Phillips approach in terms of non-cyclic functions is…
The relativistic version of the J-matrix method for a scattering problem on the potential vanishing faster than the Coulomb one is formulated. As in the non-relativistic case it leads to a finite algebraic eigenvalue problem. The derived…
The aim of this paper is to show some examples of matrix-valued orthogonal functions on the real line which are simultaneously eigenfunctions of a second-order differential operator of Schr\"{o}dinger type and an integral operator of…
Given a set of $p$ symmetric (real) matrices, the Orthogonal Joint Diagonalization (OJD) problem consists of finding an orthonormal basis in which the representation of each of these $p$ matrices is as close as possible to a diagonal…
We introduce a class of Jacobi operators with discrete spectra which is characterized by a simple convergence condition. With any operator J from this class we associate a characteristic function as an analytic function on a suitable…
Unlike in complex linear operator theory, polynomials or, more generally, Laurent series in antilinear operators cannot be modelled with complex analysis. There exists a corresponding function space, though, surfacing in spectral mapping…
We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials.…