An inverse spectral problem for a nonnormal first order differential operator

We study the eigenvalues of two restrictions ofB _x +P whereB is the two-by-two matrix that is zero on the diagonal and one off the diagonal andP is a two-by-two matrix of Lipschitz functions on the unit interval. We establish asymptotic forms for their eigenvalues and associated root vectors and demonstrate that these root vectors constitute a Riesz basis inL ²(0, 1)². We show that our forward analysis makes rigorous the attack on the associated inverse problem by M. Yamamoto,Inverse spectral problem for systems of ordinary differential equations of first order, I, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math. 35, 1988, pp. 519–546. We apply these results to the recovery of the line resistance and leakage conductance of a nonuniform transmission line.Supported by NSF grant DMS-9258312.     

The bitangential inverse spectral problem for canonical systems

The theory of the direct and bitangential inverse input impedance problem is used to solve the direct and bitangential inverse spectral problem. The analysis of the direct spectral problem uses and extends a number of results that appear in the literature. Special attention is paid to the class of canonical integral systems with matrizants that are strongly regular J-inner matrix valued functions in the sense introduced in [7]. The bitangential inverse spectral problem is solved in this class. In our considerations, the data for this inverse problem is a given nondecreasing p×p matrix valued function σ(μ) on and a normalized monotonic continuous chain of pairs , of entire inner p×p matrix valued functions. Each such chain defines a class of canonical integral systems in which we find a solution of the inverse problem for the given spectral function σ(μ). A detailed comparison of our investigations of inverse problems with those of Sakhnovich is presented.     

Lyapunov-Razumikhin pairs in the instability problem for infinite delay equations

Some theorems on complete instability of the zero solution relative to a set for nonautonomous nonlinear equations with infinite delay are provided. The right-hand side of the equation is assumed to be defined in a fading memory space and to satisfy conditions that allow the construction of limiting equations. We use conceptions of Lyapunov-Razumikhin pairs and limiting equations to obtain new instability results, which are applicable, in particular, to autonomous, periodic and almost periodic in t delay differential equations.     

Inverse spectral problems for Sturm-Liouville operators in impedance form

The spectral properties of Sturm-Liouville operators with an impedance in , for some p∈[1,∞), are studied. In particular, a complete solution of the inverse spectral problem is provided.     

J-inner matrix functions, interpolation and inverse problems for canonical systems, V: The inverse input scattering problem for Wiener class and rational p×q input scattering matrices

The general formulas developed in the fourth paper in this series are applied to solve the inverse input scattering problem for canonical integral systems in the special cases that the input scattering matrix is ap×q matrix valued function in the Wiener class (and the associated pairs are homogeneous). These formulas are then further specialized to the rational case. Whenp=q, these formulas are connected to the earlier results of Alpay-Gohberg and Gohberg-Kaashoek-Sakhnovich, who studied inverse problems for a related system of differential equations.This research was partially supported by a Minerva Foundation grant that is acknowledged with thanks.     

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

A direct and inverse scattering theory on the full line is developed for a class of first-order selfadjoint 2n×2n systems of differential equations with integrable potential matrices. Various properties of the corresponding scattering matrices including unitarity and canonical Wiener-Hopf factorization are established. The Marchenko integral equations are derived and their unique solvability is proved. The unique recovery of the potential from the solutions of the Marchenko equations is shown. In the case of rational scattering matrices, state space methods are employed to construct the scattering matrix from a reflection coefficient and to recover the potential explicitly.Dedicated to Israel Gohberg on the Occasion of his 70^th Birthday     

Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ⁺ and such that ImQ(z 0 and ImzQ(z) 0 ifz ⁺. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in ⁺ and such that the kernel has negative squares of ⁺ and ImzQ(z) 0 ifz ⁺ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + ² fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832     

Reflectionless Sturm–Liouville equations

We consider compactly supported perturbations of periodic Sturm–Liouville equations. In this context, one can use the Floquet solutions of the periodic background to define scattering coefficients. We prove that if the reflection coefficient is identically zero, then the operators corresponding to the periodic and perturbed equations, respectively, are unitarily equivalent. In some appendices, we also provide the proofs of several basic estimates, e.g., bounds and asymptotics for the relevant m$m$-functions.     

Point initial value problem for linear functional differential equations in Banach spaces

Summary The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.     

Non-semibounded sesquilinear forms and left-indefinite Sturm-Liouville problems

By Kato's First and Second Representation Theorem a closed densely defined semibounded hermitian sesquilinear form [·,·] in a Hilbert spaceH can be represented by a selfadjoint operatorT inH and if, in particular, the formt[·,·] is nonnegative thenD(t)=D(T ^1/2 ). In the present note this result is generalized to non-semibounded forms by means of Krein space methods. An application to the form where the functionp changes its sign leads to an expansion theorem for the Sturm-Liouville problem –(pu)=u,u(a)=u(b)=0 with respect to the weighted Sobolev norm .     

An interpolation problem in the class of stieltjes functions and its connection with other problems

This paper contains the proofs and some development of results that were published without proofs in [KN2]. It is completed with comments added by the second author explaining how these results became the basis of statements and the solutions of problems in the theory of entire functions, in the moment problem, in direct and inverse problems of the spectral theory of nonhomogeneous strings and in other problems.     

Bound states of a canonical system with a pseudo-exponential potential

Explicit formulas are given for the bound states (theL ²-eigenfunctions) and the corresponding eigenvalues of a self-adjoint operator defined by a canonical system with a pseudo-exponential potential. The formulas are expressed in terms of three matrices determining the potential. Both the half line and the full line case are considered.     

Stability of axial motions of nonlinearly elastic loops

We establish the stability of axial motions (steady motions along the lengthwise direction) of nonlinearly elastic loops of string. A key observation here is that a linear combination of the total energy and the total circulation of the string, both of which are conserved quantities, yields an appropriate Liapunov function. From our previous work [5], we know that there are uncountably many shapes corresponding to a given axial speed. Accordingly, we establish orbitai stability (modulo this collection of relative equilibria). For a well-defined class of soft materials, there is an upper bound on the axial speed sufficient for stability; stiff materials are shown to be orbitally stable at any axial speed.     

Description of the algebro-geometric Sturm-Liouville coefficients

It was discovered recently that there is a class of Sturm-Liouville operators whose coefficients are related to algebro-geometric data via a construction analogous to that carried out in the 1970s for the Schrödinger operator by Dubrovin, Matveev, and Novikov. In this paper, we study the “algebro-geometric” Sturm-Liouville coefficients in detail. Emphasis is placed on their recurrence properties. We make systematic use of the theory of abelian group extensions.     

An initial value problem for a functional equation

We consider the functional equationf(A(x,y))=B(f(x),f(y)), whereA andB are averages. It is known that such a functional equation has exactly one continuous solution satisfying a given two-point condition. By analogy with the theory of differential equations, we may regard the functional equation, together with a two-point condition, as a boundary value problem. (Then each boundary value problem has a unique continuous solution.) If we replace the two-point condition with the specification of a value and derivative at just one point, we obtain an initial value problem.Consider the initial value problemsf(A(x,y))=B(f(x),f(y)),f(a)=s,f(a)=, obtained by fixinga ands and allowing to vary through the set of positive real numbers. The main result of this paper gives a necessary and sufficient condition for each of the initial value problems to have a unique continuous solution, under the hypothesis that at least one of the problems has a continuous solution. This is a partial answer to the problem of determining conditions which are sufficient for the existence of a unique continuous solution of a given initial value problem.     

Small perturbations of canonical systems

We consider a singular two-dimensional canonical systemJy=–zHy on [0, ) such that at Weyl's limit point case holds. HereH is a measurable, real and nonnegative definite matrix function, called Hamiltonian. From results of L. de Branges it follows that the correspondence between canonical systems and their Titchmarsh-Weyl coefficients is a bijection between the class of all Hamiltonians with trH=1 and the class of Nevanlinna functions. In this note we show how the HamiltonianH of a canonical system changes if its Titchmarsh-Weyl coefficient or the corresponding spectral measure undergoes certain small perturbations. This generalizes results of H. Dym and N. Kravitsky for so-called vibrating strings, in particular a generalization of a construction principle of I.M. Gelfand and B.M. Levitan can be shown.     

Principal functions of non-selfadjoint matrix Sturm-Liouville equations

In this paper we investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem

     

Sine-Gordon theory in a semi-strip

Initial-boundary value problems for sine-Gordon and complex sine-Gordon equations in a semi-strip are treated. The evolution of the Weyl function and a uniqueness result are obtained for the complex sine-Gordon equation. The evolution of the Weyl function as well as an existence result and a procedure to recover solution are given for the sine-Gordon equation. It is shown that for a wide class of examples the solutions of the sine-Gordon equation are unbounded in the quarter-plane.