Luis O. Silva

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.

Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis

For a two-parameter family of Jacobi matrices exhibiting first-order spectral phase transitions, we prove discreteness of the spectrum in the positive real axis when the parameters are in one of the transition boundaries. To this end, we develop a method for obtaining uniform asymptotics, with respect to the spectral parameter, of the generalized eigenvectors. Our technique can be applied to a wide range of Jacobi matrices.

On the Two Spectra Inverse Problem for Semi-infinite Jacobi Matrices

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg–Marchenko theorem for Schrödinger operators on the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions.

Uniform Levinson Type Theorems for Discrete Linear Systems

For the solutions of parametric discrete linear systems we obtain an asymptotic formula whose remainder is estimated uniformly with respect to the parameter. This result is an extension of the discrete Levinson theorem and allows us, in particular, to study the parametric discrete linear systems that arise when dealing with the generalized eigenvectors of Jacobi matrices.

Uniform and Smooth Benzaid-Lutz Type Theorems and Applications to Jacobi Matrices

Uniform and smooth asymptotics for the solutions of a parametric system of difference equations are obtained. These results are the uniform and smooth generalizations of the Benzaid-Lutz theorem (a Levinson type theorem for discrete linear systems) and are used to develop a technique for proving absence of accumulation points in the pure point spectrum of Jacobi matrices. The technique is illustrated by proving discreteness of the spectrum for a class of unbounded Jacobi operators.

Functional model for extensions of symmetric operators and applications to scattering theory

On the basis of the explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices, we derive a new representation for the scattering matrix for pairs of such extensions. We use this representation to explicitly recover the coupling constants in the inverse scattering problem for a finite non-compact quantum graph with

The Spectra of Selfadjoint Extensions of Entire Operators with Deficiency Indices (1,1)

\delta

The Two-Spectra Inverse Problem for Semi-infinite Jacobi Matrices in The Limit-Circle Case

-type vertex conditions.

Bounded rank-one perturbations in sampling theory

We give necessary and sufficient conditions for real sequences to be the spectra of selfadjoint extensions of an entire operator whose domain may be non-dense. For this spectral characterization we use de Branges space techniques and a generalization of Krein’s functional model for simple, regular, closed, symmetric operators with deficiency indices (1,1). This is an extension of our previous work in which similar results were obtained for densely defined operators.

On dB spaces with nondensely defined multiplication operator and the existence of zero-free functions

We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be the spectra of two different self-adjoint extensions of a Jacobi operator in the limit circle case.