Nevanlinna Function Research Articles

Inverse square singularities and eigenparameter-dependent boundary conditions are two sides of the same coin

Abstract We show that inverse square singularities can be treated as boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter with ‘a negative number of poles’. More precisely, we treat in a unified manner one-dimensional Schrödinger operators with either an inverse square singularity or a boundary condition containing a rational Herglotz–Nevanlinna function of the eigenvalue parameter at each endpoint and define Darboux-type transformations between such operators. These transformations allow one, in particular, to transfer almost any spectral result from boundary value problems with eigenparameter-dependent boundary conditions to those with inverse square singularities and vice versa.

Reducibility of Self-Adjoint Linear Relations and Application to Generalized Nevanlinna Functions

Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function $Q$, represented by a self-adjoint linear relation $A$, is decomposed by means of the reducing subspaces of $A$. The sum of two functions $Q_{i}{\in N}_{\kappa_{i}}\left( \mathcal{H} \right),\thinspace i=1,\thinspace 2$, minimally represented by the triplets $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$, is also studied. For that purpose, a model $( \tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma } )$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$ is created. By means of that model, necessary and sufficient conditions for $\kappa =\kappa_{1}+\kappa_{2}$ are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation $A$ affect reducing subspaces of $A$ and decomposition of the corresponding function $Q$.

Model dielectric functions for fluctuation potential calculations in electron gas: A critical assessment

In this article, we report a critical assessment of dielectric function calculations in electron gas through the comparison of different modelling methods. This work is motivated by the fact that the dielectric function is a key quantity in the multiple scattering description of plasmon features in various electron-based spectroscopies. Starting from the standard random phase approximation (RPA) expression, we move on to correlation-augmented RPA, then damped RPA models. Finally, we study the reconstruction of the dielectric function from its moments, using the Nevanlinna and memory function approaches. We find the memory function method to be the most effective, being highly flexible and customizable.

Nonlinear free L\'evy--Khinchine formula and conformal mapping

There are two natural notions of L\'evy processes in free probability: the first one has free increments with homogeneous distributions and the other has homogeneous transition probabilities (P.~Biane, \textit{Math. Z.} {\bf 227}(1998), 143--174). In the two cases one can associate a Nevanlinna function to a free L\'evy process. The Nevanlinna functions appearing in the first notion were characterized by Bercovici and Voiculescu, \textit{Pacific J. Math.} {\bf 153}(1992), 217--248. I give an explicit parametrization for the Nevanlinna functions associated with the second kind of free L\'evy processes. This gives a nonlinear free L\'evy--Khinchine formula.

Generalized Schur\u2013Nevanlinna functions and their realizations

Pontryagin space operator valued generalized Schur functions and generalized Nevanlinna functions are investigated by using discrete-time systems, or operator colligations, and state space realizations. It is shown that generalized Schur functions have strong radial limit values almost everywhere on the unit circle. These limit values are contractive with respect to the indefinite inner product, which allows one to generalize the notion of an inner function to Pontryagin space operator valued setting. Transfer functions of self-adjoint systems such that their state spaces are Pontryagin spaces, are generalized Nevanlinna functions, and symmetric generalized Schur functions can be realized as transfer functions of self-adjoint systems with Kreĭn spaces as state spaces. A criterion when a symmetric generalized Schur function is also a generalized Nevanlinna function is given. The criterion involves the negative index of the weak similarity mapping between an optimal minimal realization and its dual. In the special case corresponding to the generalization of an inner function, a concrete model for the weak similarity mapping can be obtained by using the canonical realizations.

Uniqueness of solution to inverse Dirac spectral problems associated with incomplete spectral data

In this paper, we study the inverse Dirac spectral problem defined on [0, a] with the boundary condition at a involving a Nevanlinna function f(λ). When f(λ) is known a priori, we prove that the potentials and the other boundary condition are uniquely determined in terms of appropriate partial information on the spectral data consisting of eigenvalues and normalizing constants.

Spectral Analysis of the Spin-Boson Hamiltonian with Two Photons for Arbitrary Coupling

We study the spectrum of the spin-boson model with two photons in {mathbb {R}}^d for arbitrary coupling alpha >0. It is shown that the discrete spectrum is finite and the essential spectrum consists of a half-line, the bottom of which is a unique zero of a simple Nevanlinna function. Besides the simplicity and more abstract nature of our approach, the main novelty is the achievement of these results under minimal regularity conditions on the photon dispersion and the coupling function.

A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

Recent work on return properties of quantum walks (the quantum analogue of random walks) has identified their generating functions of first returns as Schur functions. This is connected with a representation of Schur functions in terms of the operators governing the evolution of quantum walks, i.e. the unitary operators on Hilbert spaces.In this paper we propose a generalization of Schur functions by extending the above operator representation to arbitrary closed operators on Banach spaces. Such generalized ‘Schur functions’ meet the formal structure of first return generating functions, thus we call them FR-functions. We derive some general properties of FR-functions, among them a simple relation with an operator version of Stieltjes functions which generalizes the renewal equation already known for random and quantum walks. We also prove that FR-functions satisfy splitting properties which extend useful factorizations of Schur functions.When specialized to self-adjoint operators on Hilbert spaces, we show that FR-functions become Nevanlinna functions. This allows us to obtain properties of Nevanlinna functions which, as far as we know, seem to be new. The FR-function structure leads to a new operator representation of Nevanlinna functions in terms of self-adjoint operators, whose spectral measures provide also new integral representations of such functions. This allows us to characterize each Nevanlinna function by a measure on the real line, which we refer to as ‘the measure of the Nevanlinna function’. In contrast to standard operator and integral representations of Nevanlinna functions, these new ones are exact analogues of those already known for Schur functions. The above results are also the source of a very simple ‘Schur algorithm’ for Nevanlinna functions based on interpolations at points on the real line, which we refer to as the ‘Schur algorithm on the real line’.The paper is completed with several applications of FR-functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest: an analogue for orthogonal polynomials on the real line of the Khrushchev formula for orthogonal polynomials on the unit circle, and the use of FR-functions to study recurrence in random walks, quantum walks and open quantum walks. These applications provide numerous explicit examples of FR-functions, clarifying the meaning of these functions –as first return generating functions– and their splittings –which become recurrence splitting rules. They also show that these new tools, despite being extensions of very classical ones, play an important role in the study of physical problems of a highly topical nature.

Distributional representations of Nκ(∞)‐functions

The subclasses of the classes of generalized Nevanlinna functions appear in the context of Pontryagin space models, where they correspond to model relations having a particular spectral behaviour. Applications are found, for instance, in the investigation of differential expressions with singular coefficients. We study representations of ‐functions as Cauchy‐type integrals in a distributional sense and characterize the class of distributions occurring in such representations. We make explicit how the Pontryagin space model of an ‐function is related to the multiplication operator in the L2‐space of the measure which describes the action of the representing distribution away from infinity. Moreover, we determine the distributional representations of a pair of functions associated with a symmetric generalized Nevanlinna function.

Products of generalized Nevanlinna functions with symmetric rational functions

New classes of generalized Nevanlinna functions, which under multiplication with an arbitrary fixed symmetric rational function remain generalized Nevanlinna functions, are introduced. Characterizations for these classes of functions are established by connecting the canonical factorizations of the product function and the original generalized Nevanlinna function in a constructive manner. Also, a detailed functional analytic treatment of these classes of functions is carried out by investigating the connection between the realizations of the product function and the original function. The operator theoretic treatment of these realizations is based on the notions of rigged spaces, boundary triplets, and associated Weyl functions.

Global and local behavior of zeros of nonpositive type

A generalized Nevanlinna function Q(z) with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by Qτ(z)=(Q(z)−τ)/(1+τQ(z)), τ∈R∪{∞}, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ) as a function of τ is being studied. In particular, it is shown that it is continuous and its behavior in the points where the function extends through the real line is investigated.

Some Propositions on Generalized Nevanlinna Functions of the ClassNk

Some propositions on the generalized Nevanlinna functions are derived. We indicate mainly that (1) the negative inertia index of a Hermitian generalized Loewner matrix generated by a generalized Nevanlinna function in the classNκdoes not exceedκ. This leads to an equivalent definition of a generalized Nevanlinna function; (2) if a generalized Nevanlinna function in the classNκhas a uniform asymptotic expansion at a real pointαor at infinity, then the negative inertia index of the Hankel matrix constructed with the partial coefficients of that asymptotic expansion does not exceedκ. Also, an explicit formula for the negative index of a real rational function is given by using relations among Loewner, Bézout, and Hankel matrices. These results will provide first tools for the solution of the indefinite truncated moment problems together with the multiple Nevanlinna-Pick interpolation problems in the classNκbased on the so-called Hankel vector approach.

Hierarchies of difference boundary value problems continued

This paper continues the work done in [7] in which, for eigenparameter-dependent boundary conditions, it was shown how, by making use of a Crum-type transformation, it is possible to go up and down a hierarchy of difference boundary value problems. In particular, in [7], the terminal end boundary condition is assumed to be of the formwhere is the eigenparameter and is a negative Nevanlinna function with g < 0 and for . In this work the case of g = 0 and consequently of h = 0 or is considered. It is shown how many eigenvalues such a boundary value problem will have and in addition the resulting transformed boundary value problem together with its spectrum is given.

Operator monotone functions, Jacobi operators and orthogonal polynomials

We reveal a connection between operator monotone functions and orthogonal polynomials. Especially, we express an operator monotone function with a Jacobi operator, and show that it is a limit of rational operator monotone functions. Further we prove that the ‘principal inverse’ of an orthogonal polynomial is operator monotone and hence it has a holomorphic extension to the open upper half plane, namely a Pick function (or Nevanlinna function).

Weyl-Titchmarsh Theory for Schrodinger Operators with Strongly Singular Potentials

We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a generalized Nevanlinna function. Moreover, we will show how essential supports for the Lebesgue decomposition of the spectral measure can be obtained from the boundary behavior of the singular Weyl function. Finally, we will prove a local Borg-Marchenko type uniqueness result. Our criteria will in particular cover the aforementioned case of perturbed spherical Schroedinger operators.

Zeros of nonpositive type of generalized Nevanlinna functions with one negative square

A generalized Nevanlinna function Q ( z ) with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by Q τ ( z ) = ( Q ( z ) − τ ) / ( 1 + τ Q ( z ) ) , τ ∈ R ∪ { ∞ } , is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α ( τ ) as a function of τ defines a path in the closed upper halfplane. Various properties of this path are studied in detail.

Existence of zerofree functions N-associated to a de Branges Pontryagin space

In the theory of de Branges Hilbert spaces of entire functions, so-called ‘functions associated to a space’ play an important role. In the present paper we deal with a generalization of this notion in two directions, namely with functions N-associated \(({N \in\mathbb {Z}})\) to a de Branges Pontryagin space. Let a de Branges Pontryagin space \({\mathcal {P}}\) and \({N \in \mathbb {Z}}\) be given. Our aim is to characterize whether there exists a real and zerofree function N-associated to \({\mathcal {P}}\) in terms of Kreĭn’s Q-function associated with the multiplication operator in \({\mathcal {P}}\) . The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (indefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g., dealing with admissible majorants in de Branges spaces or the continuation problem for hermitian indefinite functions.

Realization of nonstrict matrix Nevanlinna functions as Weyl functions of symmetric operators in Pontryagin spaces

Matrix-valued Nevanlinna functions with possibly noninvertible imaginary part are realized as Q-functions or Weyl functions of symmetric operators in Pontryagin spaces. The functions are decomposed into a constant part, which gives rise to a realization in a finite dimensional Pontryagin space K, and a strict or uniformly strict part, which gives rise to a realization in a Hilbert space H. A coupling procedure then leads to a symmetric operator in the product space H x K and to the realization of the given Nevanlinna function.

Characterization of the unbounded bicommutant of C_0 (N) contractions

Recent results have shown that any closed operator A commuting with the backwards shift S∗ restricted to K2 u := H2 uH2 , where u is an inner function, can be realized as a Nevanlinna function of S∗ u := S|K2 u , A = φ(S∗ u) , where φ belongs to a certain class of Nevanlinna functions which depend on u . In this paper this is generalized to show that given any contraction T of class C0(N) , that any closed (and not necessarily bounded) operator A commuting with the commutant of T is equal to φ(T ) where φ belongs to a certain class of Nevanlinna functions which depend on the minimal inner function mT of T . Mathematics subject classification (2000): 47A45, 47A50..

Interpolating sequences, Carleson measures and Wirtinger inequality

Let S be a sequence of points in the unit ball B of C n which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure S := P a2S (1j aj 2 ) n a is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of B such that any -separated sequence S has its associated measure S bounded by C= n ; then X is the zero set of a function in the Nevanlinna class of B: As an easy consequence, we prove that if S is a dual bounded sequence in H p (B); then S is a Carleson measure, which gives a short proof in one variable of a theorem of L. Carleson and in several variables of a theorem of P. Thomas.