Bilender P. Allahverdiev

Spectral analysis of dissipative Dirac operators with general boundary conditions

Abstract A space of boundary values is constructed for minimal symmetric Dirac operator in L A 2 ((−∞,∞); C 2 ) with defect index (2,2) (in Weyls limit-circle cases at ±∞). A description of all maximal dissipative (accretive), selfadjoint, and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate maximal dissipative operators with, generally speaking, nonseparated (nondecomposed) boundary conditions. In particular, if we consider separated boundary conditions, at ±∞ the nonselfadjoint (dissipative) boundary conditions are prescribed simultaneously. We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function. We prove the theorem on completeness of the system of eigenvectors and associated vectors of the dissipative Dirac operators.

Dissipative eigenvalue problems for a singular Dirac system

Abstract Dissipative singular Dirac operators are studied in the space L A 2 ([a,b); C 2 ) (−∞ , that the extensions of a minimal symmetric operator in Weyls limit-point case. We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We construct a functional model of the dissipative operator and specify its characteristic function in terms of the Titchmarsh–Weyl function of selfadjoint operator. We prove theorems on completeness of the system of eigenvectors and associated vectors of the dissipative Dirac operators.

Dissipative Schrödinger Operators with Matrix Potentials

Maximal dissipative Schrödinger operators are studied in L2((−∞,∞);E) (dimE=n<∞) that the extensions of a minimal symmetric operator with defect index (n,n) (in limit-circle case at −∞ and limit point-case at ∞). We construct a selfadjoint dilation of a dissipative operator, carry out spectral analysis of a dilation, use the Lax–Phillips scattering theory, and find the scattering matrix of a dilation. We construct a functional model of the dissipative operator, determine its characteristic function in terms of the Titchmarsh–Weyl function of selfadjoint operator and investigate its analytic properties. Finally, we prove a theorem on completeness of the eigenvectors and associated vectors of a dissipative Schrödinger operators.

Dissipative q-Dirac operator with general boundary conditions

Abstract 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.

Tauberian conditions with controlled oscillatory behavior

Abstract Let ( u n ) be a sequence of real numbers. In this paper, we give new Tauberian conditions imposed on the general control modulo of the oscillatory behavior of integer order m ≥ 1 of a sequence ( u n ) , under which, convergence follows from the Abel summability. These are generalizations of some of the well-known classical Tauberian conditions.

Limit-point criteria for q-Sturm-Liouville equations

Abstract In this article, the deficiency index problem of a singular q-Sturm-Liouville problem is studied. We establish some criteria under which the q-Sturm-Liouville equation is of limit-point case at infinity.

Spectral analysis of the direct sum Hamiltonian operators

Abstract 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.

Extensions of symmetric infinite Jacobi operator

A space of positive boundary values is constructed for a positive definite minimal symmetric operator generated by an infinite Jacobi matrix in limit-circle case. A description of all solvable, maximal dissipative (accumulative) solvable, self-adjoint solvable, positive definite self-adjoint and non-positive definite self-adjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity.

Extensions of symmetric second-order difference operators with matrix coefficients

A space of boundary values is constructed for a minimal symmetric second-order difference operator with matrix coefficients on the whole-line. A description of all maximal dissipative, maximal accretive, self-adjoint and other extensions of such operators is given in terms of boundary conditions at ± ∞.

Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases

We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space l2𝑤(ℤ) (ℤ:={0,±1,±2,…}), that is, the extensions of a minimal symmetric operator with defect index (2,2) (in the Weyl-Hamburger limit-circle cases at ±∞). We investigate two classes of maximal dissipative operators with separated boundary conditions, called “dissipative at −∞” and “dissipative at ∞.” In each case, we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also establish a functional model of the maximal dissipative operator and determine its characteristic function through the Titchmarsh-Weyl function of the self-adjoint operator. We prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.