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Fritz Gesztesy

Publications and source records attributed to Fritz Gesztesy.

At least 145 records · Page 8Linked to original sources
arXiv2005

Evans Functions, Jost Functions, and Fredholm Determinants

The principal results of this paper consist of an intrinsic definition of the Evans function in terms of newly introduced generalized matrix-valued Jost solutions for general first-order matrix-valued differential equations on the real line, and a proof of the fact that the Evans function, a finite-dimensional determinant by construction, coincides with a modified Fredholm determinant associated with a Birman-Schwinger-type integral operator up to a nonvanishing factor.

math.DS↗
arXiv2005

A Borg-Type Theorem Associated with Orthogonal Polynomials on the Unit Circle

We prove a general Borg-type result for reflectionless unitary Cantero-Moral-Velazquez (CMV) operators U associated with orthogonal polynomials on the unit circle. The spectrum of U is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator with spectrum the whole unit circle. In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory functions to prove an infinite sequence of trace formulas connected with the CMV operator U.

math.SP↗
arXiv2004

Trace Formulas and Borg-Type Theorems for Matrix-Valued Jacobi and Dirac Finite Difference Operators

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A, B in the self-adjoint Jacobi operator H=AS^+ + A^-S^- + B (with S^\pm the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E_+], E_- < E_+, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by [-E_+^{1/2},-E_-^{1/2}] \cup [E_-^{1/2},E_+^{1/2}], 0 \leq E_- < E_+.

math.SP↗
arXiv2004

J-Self-Adjointness of a Class of Dirac-Type Operators

In this note we prove that the maximally defined operator associated with a class of Dirac-type differential expressions M(Q) is J-self-adjoint with respect to a proper antilinear conjugation J under the general hypothesis that the entries of the matrix potential coefficient Q are locally integrable on the real line. The Dirac-type differential expression M(Q) is of significance as it appears in the Lax formulation of the nonabelian (matrix-valued) focusing nonlinear Schrödinger hierarchy of evolution equations.

math.SP↗
arXiv2003

(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green's functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants. We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrödinger operators and the inverse transmission coefficient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant. Finally, we rederive the explicit formula for the 2-modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener-Hopf analog of Day's formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.

math.SP↗
arXiv2003

On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KdV potentials

We characterize the spectrum of one-dimensional Schrödinger operators H=-d^2/dx^2+V with quasi-periodic complex-valued algebro-geometric potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The corresponding problem appears to have been open since the mid-seventies. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green's function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well.

math.SP↗
arXiv2003

An Explicit Characterization of Calogero--Moser Systems

Combining theorems of Halphen, Floquet, and Picard and a Frobenius type analysis, we characterize rational, meromorphic simply periodic, and elliptic KdV potentials. In particular, we explicitly describe the proper extension of the Calogero--Moser locus associated with these three classes of algebro-geometric solutions of the KdV hierarchy with special emphasis on the case of multiple collisions between the poles of solutions.

nlin.SI↗
arXiv2003

On a Theorem of Halphen and its Application to Integrable Systems

We extend Halphen's theorem which characterizes the solutions of certain $n$th-order differential equations with rational coefficients and meromorphic fundamental systems to a first-order $n \times n$ system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the $\kdv$ hierarchy and illustrate some of the connections with completely integrable models of the Calogero-Moser-type. In particular, our treatment recovers the complete characterization of the isospectral class of such rational KdV solutions in terms of a precise description of the Airault-McKean-Moser locus of their poles.

nlin.SI↗
arXiv2002

Spectral Analysis of Darboux Transformations for the Focusing NLS Hierarchy

We study Darboux transformations associated with the focusing nonlinear Schrödinger equation (NLS_-) and their effect on spectral properties of the underlying Lax operator. The latter is a formally J-self-adjoint (but non-self-adjoint) one-dimensional Dirac-type operator with potential q. As one of our principal results we prove that under the most general hypothesis of local integrability on q, the maximally defined operator D(q) generated by this Dirac operator is actually J-self-adjoint. We also establish the existence of Weyl--Titchmarsh-type solutions associated with D(q). The Darboux transformations considered in this paper guarantee that the resulting potentials q are locally nonsingular. Moreover, we prove that the construction of N-soliton NLS_- potentials q^{(N)} with respect to a general NLS_- background potential q, associated with Dirac operators D(q^{(N)}) and D(q), amounts to the insertion of N complex conjugate pairs of eigenvalues into the spectrum of D(q), leaving the rest of the spectrum (especially, the essential spectrum) invariant. These results are obtained by establishing the existence of bounded transformation operators which intertwine the background Dirac operator D(q) and the Dirac operator D(q^{(N)}) obtained after N Darboux transformations.

nlin.SI↗
arXiv2002

$\mathbf {SL_2(\bbR)}$, Exponential Herglotz Representations, and Spectral Averaging

We revisit the concept of spectral averaging and point out its origin in connection with one-parameter subgroups of $SL_2(\bbR)$ and the corresponding Möbius transformations. In particular, we identify exponential Herglotz representations as the basic ingredient for the absolute continuity of average spectral measures with respect to Lebesgue measure and the associated spectral shift function as the corresponding density for the averaged measure. As a by-product of our investigations we unify the treatment of rank-one perturbations of self-adjoint operators and that of self-adjoint extensions of symmetric operators with deficiency indices $(1,1)$. Moreover, we derive separate averaging results for absolutely continuous, singularly continuous, and pure point measures and conclude with an averaging result of the $κ$-continuous part (with respect to the $κ$-dimensional Hausdorff measure) of singularly continuous measures.

math.SP↗
arXiv2001

A Class of Matrix-Valued Schrödinger Operators with Prescribed Finite-Band Spectra

We construct a class of matrix-valued Schrödinger operators with prescribed finite-band spectra of maximum spectral multiplicity. The corresponding matrix potentials are shown to be stationary solutions of the KdV hierarchy. The methods employed in this paper rely on matrix-valued Herglotz functions, Weyl--Titchmarsh theory, pencils of matrices, and basic inverse spectral theory for matrix-valued Schrödinger operators.

math.SP↗
arXiv2001

Algebro-Geometric Solutions of the Camassa--Holm hierarchy

We provide a detailed treatment of the Camassa--Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions. In analogy to other completely integrable hierarchies of soliton equations such as the KdV or AKNS hierarchies, the CH hierarchy is recursively constructed by means of basic polynomial formalism invoking a spectral parameter. Moreover, we study Dubrovin-type equations for auxiliary divisors and associated trace formulas, consider the corresponding algebro-geometric initial value problem, and derive the theta function representations of algebro-geometric solutions of the CH hierarchy.

nlin.SI↗
arXiv2001

Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators

Let $g(z,x)$ denote the diagonal Green's matrix of a self-adjoint $m\times m$ matrix-valued Schrödinger operator $H= -\f{d^2}{dx^2}I_m +Q(x)$ in $L^2 (\bbR)^{m}$, $m\in\bbN$. One of the principal results proven in this paper states that for a fixed $x_0\in\bbR$ and all $z\in\bbC_+$, $g(z,x_0)$ and $g^\prime (z,x_0)$ uniquely determine the matrix-valued $m\times m$ potential $Q(x)$ for a.e.~$x\in\bbR$. We also prove the following local version of this result. Let $g_j(z,x)$, $j=1,2$ be the diagonal Green's matrices of the self-adjoint Schrödinger operators $H_j=-\f{d^2}{dx^2}I_m +Q_j(x)$ in $L^2 (\bbR)^{m}$. Suppose that for fixed $a>0$ and $x_0\in\bbR$, $\|g_1(z,x_0)-g_2(z,x_0)\|_{\bbC^{m\times m}}+ \|g_1^\prime (z,x_0)-g_2^\prime (z,x_0)\|_{\bbC^{m\times m}} \underset{|z|\to\infty}{=}O\big(e^{-2\Im(z^{1/2})a}\big)$ for $z$ inside a cone along the imaginary axis with vertex zero and opening angle less than $π/2$, excluding the real axis. Then $Q_1(x)=Q_2(x)$ for a.e.~$x\in [x_0-a,x_0+a]$. Analogous results are proved for matrix-valued Jacobi and Dirac-type operators.

math.SP↗