Schur functions and their realizations in the slice hyperholomorphic setting

we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels, positive definite functions in this setting and we show how they can be obtained in our setting using the extension operator and the slice regular product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges-Rovnyak space

Wiener algebra for the quaternions

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-L\'evy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators

On the class SI of J-contractive functions intertwining solutions of linear differential equations

In the PhD thesis of the second author under the supervision of the third author was defined the class SI of J-contractive functions, depending on a parameter and arising as transfer functions of overdetermined conservative 2D systems invariant in one direction. In this paper we extend and solve in the class SI, a number of problems originally set for the class SC of functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned signature matrix J. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence between elements in a given subclass of SI and elements in SC. Another important tool in the arguments is a new result pertaining to the classical tangential Schur algorithm.Comment: 46 page

Bitangential interpolation in generalized Schur classes

Bitangential interpolation problems in the class of matrix valued functions in the generalized Schur class are considered in both the open unit disc and the open right half plane, including problems in which the solutions is not assumed to be holomorphic at the interpolation points. Linear fractional representations of the set of solutions to these problems are presented for invertible and singular Hermitian Pick matrices. These representations make use of a description of the ranges of linear fractional transformations with suitably chosen domains that was developed in a previous paper.Comment: Second version, corrected typos, changed subsection 5.6, 47 page

Strain induced variations in band offsets and built-in electric fields in InGaN/GaN multiple quantum wells

The band structure, quantum confinement of charge carriers, and their localization affect the optoelectronic properties of compound semiconductor heterostructures and multiple quantum wells (MQWs). We present here the results of a systematic first-principles based density functional theory (DFT) investigation of the dependence of the valence band offsets and band bending in polar and non-polar strain-free and in-plane strained heteroepitaxial In x Ga1- xN(InGaN)/GaN multilayers on the In composition and misfit strain. The results indicate that for non-polar m-plane configurations with [12¯10]InGaN // [12¯10]GaN and [0001]InGaN // [0001]GaN epitaxial alignments, the valence band offset changes linearly from 0 to 0.57 eV as the In composition is varied from 0 (GaN) to 1 (InN). These offsets are relatively insensitive to the misfit strain between InGaN and GaN. On the other hand, for polar c-plane strain-free heterostructures with [101¯0]InGaN // [101¯0]GaN and [12¯10]InGaN // [12¯10]GaN epitaxial alignments, the valence band offset increases nonlinearly from 0 eV (GaN) to 0.90 eV (InN). This is significantly reduced beyond x ≥ 0.5 by the effect of the equi-biaxial misfit strain. Thus, our results affirm that a combination of mechanical boundary conditions, epitaxial orientation, and variation in In concentration can be used as design parameters to rapidly tailor the band offsets in InGaN/GaN MQWs. Typically, calculations of the built-in electric field in complex semiconductor structures often must rely upon sequential optimization via repeated ab initio simulations. Here, we develop a formalism that augments such first-principles computations by including an electrostatic analysis (ESA) using Maxwell and Poisson\u27s relations, thereby converting laborious DFT calculations into finite difference equations that can be rapidly solved. We use these tools to determine the bound sheet charges and built-in electric fields in polar epitaxial InGaN/GaN MQWs on c-plane GaN substrates for In compositions x = 0.125, 0.25,…, and 0.875. The results of the continuum level ESA are in excellent agreement with those from the atomistic level DFT computations, and are, therefore, extendable to such InGaN/GaN MQWs with an arbitrary In composition

Restrictions and extensions of semibounded operators

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure

Translation Representations and Scattering By Two Intervals

Studying unitary one-parameter groups in Hilbert space (U(t),H), we show that a model for obstacle scattering can be built, up to unitary equivalence, with the use of translation representations for L2-functions in the complement of two finite and disjoint intervals. The model encompasses a family of systems (U (t), H). For each, we obtain a detailed spectral representation, and we compute the scattering operator, and scattering matrix. We illustrate our results in the Lax-Phillips model where (U (t), H) represents an acoustic wave equation in an exterior domain; and in quantum tunneling for dynamics of quantum states

de Branges-Rovnyak spaces: basics and theory

For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$ and related extension space ${\mathcal D(S)}$ consisting of pairs of analytic functions on the unit disk ${\mathbb D}$. This survey describes three equivalent formulations (the original geometric de Branges-Rovnyak definition, the Toeplitz operator characterization, and the characterization as a reproducing kernel Hilbert space) of the de Branges-Rovnyak space ${\mathcal H}(S)$, as well as its role as the underlying Hilbert space for the modeling of completely non-isometric Hilbert-space contraction operators. Also examined is the extension of these ideas to handle the modeling of the more general class of completely nonunitary contraction operators, where the more general two-component de Branges-Rovnyak model space ${\mathcal D}(S)$ and associated overlapping spaces play key roles. Connections with other function theory problems and applications are also discussed. More recent applications to a variety of subsequent applications are given in a companion survey article