On the orthogonal polynomials in several variables

We study the connection between orthogonal polynomials in several variables and families of commuting symmetric operators of a special form.     

Spectral analysis of the Direct Sum Hamiltonian Operators

In this paper we investigate the deficiency indices theory and the selfad-joint and nonselfadjoint (dissipative, accumulative) extensions of the minimal symmetric direct sum Hamiltonian operators. In particular using the equivalence of the Lax-Phillips scattering matrix and the Sz.-Nagy-Foia¸s characteristic function, we prove that all root (eigen and associated) vectors of the maximal dissipative extensions of the minimal symmetric direct sum Hamiltonian operators are complete in the Hilbert spaces.     

Spectral functions of singular differential operators with matricial coefficients

We study the asymptotic behavior of the Harish-Chandra function associated to a singular second order differential operator with matricial coefficients. The study is based on a detailed analysis of the asymptotic behavior of some eigenvectors of the operator from which results on the asymptotic behavior of the spectral function and the scattering matrix are derived.     

A relationship between the Nehari and the Carathéodory-Toeplitz extension problems

The solutions of the two extension problems mentioned in the title are described via linear fractional transformations. In this paper, we relate the coefficient matrix functions of these transformations.     

To the spectral theory of Krein systems

We consider the Krein systems. For the set of Stummel class coefficients, we establish the criterion in terms of these coefficients for the system to satisfy the Szegö-type estimate on the spectral measure.     

ON THE TITCHMARSH-WEYL THEORY OF ORDINARY SYMMETRIC ODD-ORDER DIFFERENTIAL EXPRESSIONS AND A DIRECT CONVERGENCE THEOREM

Abstract

In this paper the odd-order differential equation M[y] λ wy on the interval (O,∞), associated with the symmetric differential expression M of (2k-1)st order (k ≥ 2) with w a positive weight function and λ a complex number, is shown to possess k-Titchmarsh-Weyl solutions for every non-real λ in the underlying Hilbert space L² _w(O, ∞) having identical representation for every non-real λ. In terms of these solutions the Green's function associated with the singular boundary value problem is shown to possess identical representation for all non-real λ which has been further made use of in the third-order case to establish a direct convergence eigenfunction expansion theorem. The symmetric spectral matrix appearing in the expansion theorem has been characterized in terms of the Titchmarsh-Weyl m-coefficients.     

Dissipative q-Dirac operator with general boundary conditions

In this paper, we consider the symmetric q-Dirac operator. We describe dissipative, accumulative, self-adjoint and the other extensions of such operators with general boundary conditions. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.     

Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials

In this paper we prove two consequences of the subnormal character of the Hessenberg matrix D when the hermitian matrix M of an inner product is a moment matrix. If this inner product is defined by a measure supported on an algebraic curve in the complex plane, then D satisfies the equation of the curve in a noncommutative sense. We also prove an extension of the Krein theorem for discrete measures on the complex plane based on properties of subnormal operators.     

Some remarks on the asymptotic behavor of toeplitz determinants

In order to optimize the stability regions we construct families of k multi-step methods with free parameters and of order k+1. We exhibit parameters for which the real interval of stability is arbitrarly large and using the Flexible Tolerance Method we show how these parameters optimize - under a predefined criterion - the stability region. Computational experience, while limited, has indicated that the procedure is efficient     

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szeg? polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szeg? polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.     

Riccati inequality and functional properties of differential operators on the half line

Given a piecewise continuous function and a projection P₁ onto a subspace X₁ of CN, we investigate the injectivity, surjectivity and, more generally, the Fredholm properties of the ordinary differential operator with boundary condition . This operator acts from the “natural” space into L²×X₁. A main novelty is that it is not assumed that A is bounded or that has any dichotomy, except to discuss the impact of the results on this special case. We show that all the functional properties of interest, including the characterization of the Fredholm index, can be related to the existence of a selfadjoint solution H of the Riccati differential inequality . Special attention is given to the simple case when H=A+A^∗ satisfies this inequality. When H is known, all the other hypotheses and criteria are easily verifiable in most concrete problems.     

On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm-Liouville operators with square summable potential

We prove that there exist some Sturm-Liouville operators with square summable potentials such that the singular continuous component of the spectral measure lies on the positive half-line. The Hausdorff dimension of the support of this singular measure can be arbitrary number from 0 to 1.     

A note on the existence of singular integrals

Abstract. In this note the existence of a singular integral operator T acting on Lipo(R“) spacesis studied. Suppose     

Slowing down and reflection of waves in truncated periodic media

We consider wave propagation and scattering governed by 1-D Schrödinger operators with truncated periodic potentials. The propagation of wave packets with narrow frequency supports is studied. The goal is to describe potentials for which the group velocity (for a periodic problem) is small and the transmission coefficient for the truncated potential is not too small, i.e. to find media where a slowing down of the wave packets coexists with a transparency.     

Principal functions of non-selfadjoint matrix Sturm-Liouville equations

In this paper we investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem

     

1-D Schrödinger operators with local point interactions on a discrete set

Spectral properties of 1-D Schrödinger operators with local point interactions on a discrete set are well studied when d_∗:=infn,k∈N|xn−xk|>0. Our paper is devoted to the case d_∗=0. We consider HX,α in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions.We show that the spectral properties of HX,α like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators HX,α to be self-adjoint, lower semibounded, and discrete in the case d_∗=0.The operators with δ^′-type interactions are investigated too. The obtained results demonstrate that in the case d_∗=0, as distinguished from the case d_∗>0, the spectral properties of the operators with δ- and δ^′-type interactions are substantially different.