Herglotz Function Research Articles

Extended higher Herglotz functions I. Functional equations

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F(x) which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, F(x) satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study Fk,N(x), an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series as well as with a generalized cotangent Dirichlet series inspired from Krätzel's work. We derive two different kinds of functional equations satisfied by Fk,N(x). Radchenko and Zagier gave a beautiful relation between the integral ∫01log⁡(1+tx)1+tdt and F(x) and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between Fk,N(x) and a generalization of the above integral involving polylogarithm. The asymptotic expansions of Fk,N(x) and some generalized Lambert series are also obtained along with other supplementary results.

Arithmetic properties of the Herglotz-Zagier-Novikov function

In this article, we undertake the study of the function F(x;u,v), which we refer to as the Herglotz-Zagier-Novikov function. This function appears in Novikov's work on the Kronecker limit formula, which was motivated by Zagier's paper where he obtained the Kronecker limit formula in terms of the Herglotz function F(x). Two, three, and six-term functional equations satisfied by F(x;u,v) are exhibited. These are cohomological relations coming from the action of an involution and SL2(Z) on C×D12 (the unit circle D1). We also provide the special values of F(x;u,v) at rational arguments of x. Importantly, F(x;u,v) serves as a unified generalization of three other interesting functions, namely F(x), J(x), and T(x), which also appear in various Kronecker limit formulas and are previously studied by Cohen, Herglotz, Muzaffar and Williams, and Radchenko and Zagier. Consequently, our study not only reveals the numerous elegant properties of F(x;u,v) but also helps us to further develop the theories of functions related to its special cases such as J(x) and T(x).

Extended higher Herglotz function II

Very recently, Radchenko and Zagier revived the theory of Herglotz functions. The main goal of the article is to show that one of the formulas on page 220 of Ramanujan's Lost Notebook actually lives in the realms of this theory. As a consequence of our general theorem, we derive an interesting identity analogous to Ramanujan's formula for ζ(2m+1). We also introduce a character analogue of the Herglotz function and initiate its theory by obtaining an elegant functional equation governed by it.

Fundamental constraints on broadband passive acoustic treatments in unidimensional scattering problems

In a passive lossy acoustical system, sum rules derived from passivity explicitly relate the broadband response to the spatial dimension of the system, which provide important design criteria as well. In this work, the theory of Herglotz function is applied systematically to derive sum rules for unidimensional scattering problems relying on passive acoustic treatments which are generally made of rigid, motionless and subwavelength structures saturated by air. The rigid-boundary reflection, soft-boundary reflection and transmission problems are analysed. The derived sum rules are applied for guiding the designs of passive absorbers and mufflers: the required minimum space is directly predicted from the target (i.e. the desired absorption or transmission-loss spectra) without any specific design. Besides, it is possible to break this type of sum rules and fundamental constraints in particular cases. This property, if well used, could result in ultra-compact absorbers working at long wavelength up to infinity.

Parameterized families of polylog integrals

It is commonly known that integrals containing log-polylog integrands admit representations in terms of special functions such as the Dirichlet eta and Dirichlet beta functions. We investigate two parameterized families of such integrals and in a particular case demonstrate a connection with the Herglotz function. In the process of the investigation we recover some known Euler sum equalities and discover some new identities.

Inverse of the flow and moments of the free Jacobi process associated with one projection

This paper is a companion to a series of papers devoted to the study of the spectral distribution of the free Jacobi process associated with a single projection. Actually, we note that the flow derived in [N. Demni and T. Hmidi, Spectral distribution of the free Jacobi process associated with one projection, Colloq. Math. 137(2) (2014) 271–296] solves a radial Löwner equation and as such, the general theory of Löwner equations implies that it is univalent in some connected region in the open unit disc. We also prove that its inverse defines the Aleksandrov–Clark measure at [Formula: see text] of some Herglotz function which is absolutely-continuous with an essentially bounded density. As a by-product, we deduce that [Formula: see text] does not belong to the continuous singular spectrum of the unitary operator whose spectral dynamics are governed by the flow. Moreover, we use a previous result due to the first author in order to derive an explicit, yet complicated, expression of the moments of both the unitary and the free Jacobi processes. The paper is closed with some remarks on the boundary behavior of the flow’s inverse.

Representation of free Herglotz functions

A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free analytic analogue of the Herglotz representation and describe how our representations specialize to the free probabilistic case. We also show that the set of representable Herglotz functions arising from noncommutative conditional expectations must be closed in a natural topology.

Inverse Scattering Problems for Sturm–Liouville Operators with Spectral Parameter Dependent on Boundary Conditions

In this paper, we consider the inverse scattering problem for the Sturm–Liouville operator on the half-line [0,∞) with Herglotz function of spectral parameter in the boundary condition. The scattering data of the problem is defined, and its properties are investigated. The main equation is obtained for the solution of the inverse problem and it is shown that the potential is uniquely recovered in terms of the scattering data.

Meromorphic Continuations of Finite Gap Herglotz Functions and Periodic Jacobi Matrices

We find a necessary and sufficient condition for a Herglotz function m to be the Borel transform of the spectral measure of an exponential decaying perturbation of a periodic Jacobi matrix. The condition is in terms of meromorphic continuation of m to a natural Riemann surface and the structure of its zeros and poles.The analogous result is also established for the Borel transform of the spectral measure of eventually periodic Jacobi matrices.This paper generalizes the corresponding result from the author’s (Constr Approx 36(2):267–309, 2012) for exponential perturbations of the free Jacobi matrix.

Optimal Realizations of Passive Structures

This paper presents a convex optimization approach to study optimal realizations of passive electromagnetic structures. The optimization approach complements recently developed theory and techniques to derive sum rules and physical limitations for passive systems operating over a given bandwidth. The sum rules are based solely on the analytical properties of the corresponding Herglotz functions. However, the application of sum rules is limited by certain assumptions regarding the low- and high-frequency asymptotic behavior of the system, and the sum rules typically do not give much information towards an optimal realization of the passive system at hand. In contrast, the corresponding convex optimization problem is formulated to explicitly generate a Herglotz function as an optimal realization of the passive structure. The procedure does not require any additional assumptions on the low- and high frequency asymptotic behavior, but additional convex constraints can straightforwardly be incorporated in the formulation. Typical application areas are concerned with antennas, periodic structures, material responses, scattering, absorption, reflection, and extinction. In this paper, we consider three concrete examples regarding dispersion compensation for waveguides, passive metamaterials and passive radar absorbers.

Time-domain approach to the forward scattering sum rule

The forward scattering sum rule relates the extinction cross section integrated over all wavelengths with the polarizability dyadics. It is useful for deriving bounds on the interaction between scatterers and electromagnetic fields, antenna bandwidth and directivity and energy transmission through sub-wavelength apertures. The sum rule is valid for linearly polarized plane waves impinging on linear, passive and time translational invariant scattering objects in free space. Here, a time-domain approach is used to clarify the derivation and the used assumptions. The time-domain forward scattered field defines an impulse response. Energy conservation shows that this impulse response is the kernel of a passive convolution operator, which implies that the Fourier transform of the impulse response is a Herglotz function. The forward scattering sum rule is finally constructed from integral identities for Herglotz functions.

M-matrix asymptotics for Sturm–Liouville problems on graphs

We consider a system formulation for Sturm–Liouville operators with formally self-adjoint boundary conditions on a graph. An M-matrix associated with the boundary value problem is defined and related to the matrix Prüfer angle associated with the system boundary value problem, and consequently with the boundary value problem on the graph. Asymptotics for the M-matrix are obtained as the eigenparameter tends to negative infinity. We show that the boundary conditions may be recovered, up to a unitary equivalence, from the M-matrix and that the M-matrix is a Herglotz function. This is the first in a series of papers devoted to the reconstruction of the Sturm–Liouville problem on a graph from its M-matrix.

On functional equations associated with the renormalization of non-commuting circle mappings

We investigate the functional equations associated with critical non-commuting circle homeomorphisms. Using Epstein's Herglotz function methods, we show that there are analytic solutions of a class of functional equations closely related to period-two points of the circle-map renormalization transformations for rotation number with continued fraction .

A method to solve an acoustic inverse scattering problem involving smart obstacles

A time-harmonic acoustic inverse scattering problem involving smart obstacles is formulated and a method to solve it is proposed. A smart obstacle is an obstacle that, when hit by an incoming acoustic wave, tries to pursue a given goal circulating a suitable pressure current on its boundary. A pressure current is a quantity whose physical dimension is pressure divided by time. The goals pursued by the smart obstacles that we have considered are the following ones: to be undetectable or to appear with a shape and/or acoustic boundary impedance different from its actual ones eventually in a location in space different from the actual location. The following time-harmonic inverse scattering problem is considered: from the knowledge of several far fields generated by the smart obstacle when hit by known time-harmonic waves, the knowledge of the goal pursued by the smart obstacle and of its acoustic boundary impedance reconstruct the boundary of the obstacle. A method to solve this inverse problem that generalizes the so-called Herglotz function method is proposed. Some numerical experiments that validate the method proposed are presented. The website http://www.econ.univpm.it/recchioni/w13 contains some auxiliary material that helps the understanding of the current paper.

Alternative approach to the derivation of dispersion relations for optical constants

The derivation of dispersion relations for linear optical constants is considered starting from the representation of an optical property as a Herglotz function. One form of the Kramers–Kronig relations is determined directly and the second is obtained using elementary properties of the Hilbert transform. Application to the complex refractive index is considered.

On spectral theory for Schrödinger operators with strongly singular potentials

AbstractWe examine two kinds of spectral theoretic situations: First, we recall the case of self‐adjoint half‐line Schrödinger operators on [a , ∞), a ∈ ℝ, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix‐valued Herglotz functions representing the associated Weyl–Titchmarsh coefficients. Second, we contrast this with the case of self‐adjoint half‐line Schrödinger operators on (a , ∞) with a potential strongly singular at the end point a . We focus on situations where the potential is so singular that the associated maximally defined Schrödinger operator is self‐adjoint (equivalently, the associated minimally defined Schrödinger operator is essentially selfadjoint) and hence no boundary condition is required at the finite end point a . For this case we show that the Weyl–Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrödinger operators. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Sturm–Liouville problems with boundary conditions depending quadratically on the eigenparameter

We study Sturm–Liouville problems with right-hand boundary conditions depending on the spectral parameter in a quadratic manner. A modified Crum–Darboux transformation is used to produce chains of problems almost isospectral with the given one. The problems in the chain have boundary conditions which in various cases are affine or bilinear in the spectral parameter, and in all cases culminate in a problem with constant boundary conditions. This extends recent work of Binding, Browne, Code and Watson when the right-hand condition is either an affine function of the spectral parameter with negative leading coefficient or a Herglotz function.

Geometrical Aspects of Spectral Theory and Value Distribution for Herglotz Functions

In this paper we show how spectral theory for Herglotz functions and differential operators is related to and dependent on the geometrical properties of the complex upper half-plane, viewed as a hyperbolic space. We establish a theory of value distribution for Lebesgue measurable functions f: R→R and introduce the value distribution function associated with any given Herglotz function F. We relate the theory of value distribution for boundary values of Herglotz functions to the description of asymptotics for solutions of the Schrodinger equation on the half-line. We establish two results which play a key role in understanding asymptotic value distribution for Schrodinger operators with sparse potentials, and its implications for spectral theory.

THE USE OF THE HERGLOTZ FUNCTION METHOD TO RECONSTRUCT OBSTACLES FROM REAL AND FROM SYNTHETIC SCATTERING DATA

We consider the problem of the reconstruction of the shape of an obstacle from some knowledge of the scattered waves generated from the interaction of the obstacle with known incident waves. More precisely we study this inverse scattering problem considering acoustic waves or electromagnetic waves. In both cases the waves are assumed harmonic in time. The obstacle is assumed cylindrically symmetric and some special incident waves are considered. This allows us to formulate the two scattering problems, i.e. the acoustic scattering problem and the electromagnetic scattering problem, as a boundary value problem for the scalar Helmholtz equation in two independent variables. The numerical algorithms proposed are based on the Herglotz Function Method, which has been introduced by Colton and Monk.1 We report the results obtained with these algorithms in the reconstruction of simple obstacles with Lipschitz boundary using experimental electromagnetic scattering data, that is the Ipswich Data2,3 and in the reconstruction of "multiscale obstacles" using synthetic acoustic scattering data.

Asymptotic Value Distribution for Solutions of the Schrödinger Equation

We consider the Dirichlet Schrodinger operator T=−(d2/d x2)+V, acting in L2(0,∞), where Vis an arbitrary locally integrable potential which gives rise to absolutely continuous spectrum. Without any other restrictive assumptions on the potential V, the description of asymptotics for solutions of the Schrodinger equation is carried out within the context of the theory of value distribution for boundary values of analytic functions. The large xasymptotic behaviour of the solution v(x,λ) of the equation Tf(x,λ)=λf(x,λ), for λ in the support of the absolutely continuous part μa.c. of the spectral measure μ, is linked to the spectral properties of this measure which are determined by the boundary value of the Weyl–Titchmarsh m-function. Our main result (Theorem 1) shows that the value distribution for v′(N,λ)/v(N,λ) approaches the associated value distribution of the Herglotz function mN(z) in the limit N→∞, where mN(z) is the Weyl–Titchmarsh m-function for the Schrodinger operator −(d2/d x2)+Vacting in L2(N,∞), with Dirichlet boundary condition at x=N. We will relate the analysis of spectral asymptotics for the absolutely continuous component of Schrodinger operators to geometrical properties of the upper half-plane, viewed as a hyperbolic space.