Fredholm Eigenvalues Research Articles

Extremal properties of Sobolev's Beltrami coefficients and distortion of curvelinear functionals

An important problem in applications of quasiconformal analysis and in its numerical aspect is to establish algorithms for explicit or approximate determination of the basic quasiinvariant curvelinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficients of associated quasicircles. We prove two general theorems of new type answering this question for univalent functions in arbitrary quasiconformal domains and provide its applications. The results are strengthened in the case of maps of the disk and give rise to extremal Beltrami coefficients of a new type.

On the Extremality of Harmonic Beltrami Coefficients

We prove a general theorem, which provides a broad collection of univalent functions with equal Grunsky and Teichmüller norms and thereby the Fredholm eigenvalues and the reflection coefficients of associated quasicircles. It concerns an important problem to establish the exact or approximate values of basic quasiinvariant functionals of Jordan curves, which is crucial in applications and in the numerical aspect of quasiconformal analysis.

Fredholm Eigenvalues and Quasiconformal Geometry of Polygons

An important open problem in geometric complex analysis is to establish algorithms for the explicit determination of the basic curvilinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues, and the quasireflection coefficient. This is important also for the potential theory but has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory.This survey extends our previous survey of 2005 and presents the new approaches and recent essential progress in this field of geometric complex analysis and potential theory, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals (to which the notion of Fredholm eigenvalues also can be extended).

Fredholm eigenvalues and quasiconformal geometry of polygons

An important open problem in geometric complex analysis is to establish algorithms for the explicit determination of the basic curvilinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues, and the quasireflection coefficient. This is important also for the potential theory but has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory. This survey extends our previous survey of 2005 and presents the newapproaches and recent essential progress in this field of geometric complex analysis and potential theory, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals (to which the notion of Fredholm eigenvalues also can be extended).

On Fredholm Eigenvalues of Unbounded Polygons

An important open problem in geometric complex analysis is to establish some algorithms for explicit determination of the basic functionals intrinsically connected with conformal and quasiconformal mappings such as their Teichmuller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficient. This problem has not been solved even for generic quadrilaterals. We provide a restricted solution of the problem for unbounded rectilinear polygons.

Optimal Shape Design by Partial Spectral Data

In this paper, we are concerned with a shape design problem, in which our target is to design, up to rigid transformations and scaling, the shape of an object given either its polarization tensor at multiple contrasts or the partial eigenvalues of its Neumann--Poincaré operator, which are known as the Fredholm eigenvalues. We begin by proposing to recover the eigenvalues of the Neumann--Poincaré operator from the polarization tensor by means of the holomorphic functional calculus. Then we develop a regularized Gauss--Newton optimization method for the shape reconstruction process. We present numerical results to demonstrate the effectiveness of the proposed methods and to illustrate important properties of the Fredholm eigenvalues and their associated eigenfunctions. Our results are expected to have important applications in the design of plasmon resonances in nanoparticles as well as in the multifrequency or pulsed imaging of small anomalies.

Transformation of the quasiconformal reflection coefficient and of the Fredholm eigenvalue

Abstract. The behavior of the quasiconformal reflection coefficient and of the Fredholm eigenvalue of a plane analytic Jordan curve is studied under the square root transformation.

Fredholm's alternative breaks the confinement of electromagnetic waves

The recovery of information from the so called electromagnetic evanescent waves seems to be a very well explained item. Nevertheless, the travelling waves that becomes from the evanescent waves emerge from integral or differential equations that are very different to those describing the conventional ones. Indeed, we can say that the two kinds of solutions, the travelling and evanescent waves represent a mutually discriminating problem in which we cannot have simultaneous validity of both kinds of equations even they represents the physical evolution of a the same system. But if we can describe our system with a Fredholm's equation we can relate both situations through the properties of the Fredholm's eigenvalue. When the Fredholm's eigenvalue has its values into certain range then Fredholm's equation describes a normal travelling spectrum; otherwise, we are in the presence of another type of equation with abnormal or special behavior. In this work, we analyze the so-named Fredholm's alternative, which enables us to describe the change of positive refraction index-like conditions of broadcasting media to negative refraction index-like conditions. We also sketch some general conditions for the Fredholm's eigenvalue in order to establish general rules for the breaking of the waves’ confinement.

Generalized Fourier coefficients of a quasisymmetric homeomorphism and the Fredholm eigenvalue

Recently, Reiner Kuhnau gave some new characterizations of the Fredholm eigenvalue by means of generalized Fourier coefficients. In this note we give a fast approach to Kuhnau’s results and answer some questions posed by Kuhnau as well.

Fredholm eigenvalue for a quasi-circle and Grunsky functionals

We give several new formulas of the least positive Fredholm eigenvalue for a quasi- circle, answering a problem posed recently by Kuhnau. During the proof, we show that the Grunsky functionals corresponding to the two complementary domains of the quasi-circle are the same and equal to the reciprocal of the Fredholm eigenvalue.

Extremal problems for Fredholm eigenvalues

The study of extremal problems for Fredholm eigenvalues was initiated by Schiffer in the context of the existence of conformal maps onto canonical domains.

Gap probabilities in non-Hermitian random matrix theory

We compute the gap probability that a circle of radius r around the origin contains exactly k complex eigenvalues. Four different ensembles of random matrices are considered: the Ginibre ensembles and their chiral complex counterparts, with both complex (β=2) or quaternion real (β=4) matrix elements. For general non-Gaussian weights we give a Fredholm determinant or Pfaffian representation respectively, depending on the non-Hermiticity parameter. At maximal non-Hermiticity, that is, for rotationally invariant weights, the product of Fredholm eigenvalues for β=4 follows from the β=2 case by skipping every second factor, in contrast to the known relation for Hermitian ensembles. On additionally choosing Gaussian weights we give new explicit expressions for the Fredholm eigenvalues in the chiral case, in terms of Bessel-K and incomplete Bessel-I functions. This compares with known results for the Ginibre ensembles in terms of incomplete exponentials. Furthermore, we present an asymptotic expansion of the logarithm of the gap probability for large argument r at large N in all four ensembles, up to and including the third order linear term. We can provide strict upper and lower bounds and present numerical evidence for the conjectured values of the linear term, depending on the number of exact zero eigenvalues in the chiral ensembles. For the Ginibre ensemble at β=2, exact results were previously derived by Forrester [Phys. Lett. A 169, 21 (1992)].

Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues

The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to \(\widehat{\mathbb{C}}\) are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = {z: |z| > 1} can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.

A new matrix characterization of fredholm eigenvalues of quasicircles

We give a new characterization of the Fredholm eigenvalues of a quasicircle or of a quasisymmetric transformation. This leads to a matrix eigenvalue problem for a suitable Hermitian matrix. There are connections to extremal quasiconformal mappings and reflections.

Inversion of Ahlfors and Grunsky Inequalities

Abstract We solve the old Kühnau's problem on the exact lower bound in the inverse inequality estimating the dilatation of a univalent function by its Grunsky norm and in the related Ahlfors inequality for Fredholm eigenvalues.

Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on $\mathbb {R}^{N}$

We investigate the relationship between the decay at infinity of the right-hand side f and solutions u of an equation Lu = f when L is a second order elliptic operator on R N . It is shown that when L is Fredholm, u inherits the type of decay of f (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of L, isolated or not, decay exponentially. No use is made of spectral theory. The result is next extended when L is replaced by a Fredholm quasilinear operator. Various generalizations to other unbounded domains, higher order operators or elliptic systems are possible and briefly alluded to, but not discussed in detail.

Defect basis property of the eigenvectors of a selfadjoint operator-valued function and formula for defect

A continuous selfadjoint operator-valued functionA(x) is considered. Possible changes of dimension of the positive partA+(t) are investigated ast transverses through some Fredholm eigenvalue λ0 ∈ ℝ. Some results on defect basis property are obtained, and the formula for the defect is proved. Both holomorphic and non-holomorphic cases are considered.

The Grunsky coefficient conditions

Starting from [3], several authors have repeatedly posed the following problem (see [6, p. 126; 7] about this): Does the inequality (4) (and those equivalent to it) characterize the class E(k), i.e., is it also sufficient for the k-quasiconformal extendability of f? Pommerenke [2] has established that in each case the inequality (4) ensures the k'-quasiconformal extendability of f with k'~k. Kiihnau [6], using the connection between the Grunsky matrix and the least (nontrivial) Fredholm eigenvalue ~ of the curve f(Izl = I) (found in [8]), has shown that the posed problem Qhas a negative solution, namely: The function

The fredholm eigenvalues of plane domains

The problem of how the Fredholm eigenvalue δ(D) of a simply connected domain D varies under “small” perturbations of D has been much studied. The difficulty consists in determining the sense in which such a perturbation is small. Here we consider a metric which arises in the definition of universal Teichmuller space. We show that, with this metric, the Fredholm eigenvalue δ(D) is a locally Lipschitz function of D, if D is a quasiconformal disk.

Grunsky inequalities and Fredholm spectrum in general domains

We discuss the Fredholm spectrum for general domains and study its applications to conformal and quasi-conformal mappings. In particular, we establish an improvement of the Grunsky inequalities which is valid for general domains. This improvement constitutes an extension of a recent result of Schiffer concerning the sharpening of Grunsky inequalities for the unit disk by a factor smaller than 1, and which is the reciprocal of the least Fredholm eigenvalue of the smooth simply-connected image domain.