Subordinate solutions and spectral measures of canonical systems

The theory of 2×2 trace-normed canonical systems of differential equations on ?⁺ can be put in the framework of abstract extension theory, cf. [9]. This includes the theory of strings as developed by I.S. Kac and M.G. Kre?n. In the present paper the spectral properties of such canonical systems are characterized by means of subordinate solutions. The theory of subordinacy for Schrödinger operators on the halfline ?⁺, was originally developed D.J. Gilbert and D.B. Pearson. Its extension to the framework of canonical systems makes it possible to describe the spectral measure of any Nevanlinna function in terms of subordinate solutions of the corresponding trace-normed canonical system, which is uniquely determined by a result of L. de Branges.     

Spectral theory of Hamiltonian systems with almost constant coefficients

We derive the spectral theory for general linear Hamiltonian systems. The coefficients are assumed to be asymptotically constant and satisfy certain smoothness and decay conditions. These latter constraints preclude the appearance of singular continuous spectra. The results are thus far reaching extensions of earlier theorems of the authors. Two-, three- and four-dimensional systems are studied in greater detail. The results also apply to the case of the Dirichlet index and Dirichlet spectrum.     

Schrödinger operators and de Branges spaces

We present an approach to de Branges's theory of Hilbert spaces of entire functions that emphasizes the connections to the spectral theory of differential operators. The theory is used to discuss the spectral representation of one-dimensional Schrödinger operators and to solve the inverse spectral problem.     

Bundle bispectrality for matrix differential equations

We consider the fundamental solutions of a wide class of first order systems with polynomial dependence on the spectral parameter and rational matrix potentials. Such matrix potentials are rational solutions of a large class of integrable nonlinear equations, which play an important role in different mathematical physics problems. The concept of bispectrality, which was originally introduced by Grünbaum, is extended in a natural way for the systems under consideration and their bispectrality is derived via the representation of the fundamental solutions. This bispectrality is preserved under the flows of the corresponding integrable nonlinear equations. For the case of Dirac type (canonical) systems the complete characterization of the bispectral potentials under consideration is obtained in terms of the system's spectral function.     

Direct and Inverse Spectral Theory of One-dimensional Schrödinger Operators with Measures

We present a direct and rather elementary method for defining and analyzing one-dimensional Schrödinger operators H = −d²/dx² + μ with measures as potentials. The basic idea is to let the (suitably interpreted) equation −f′′ + μ f = zf take center stage.We show that the basic results from direct and inverse spectral theory then carry over to Schrödinger operators with measures.     

Eigenvalues of an elliptic system

We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations of the operator. We show that the leading term in the spectral asymptotics is closely related to a certain convex polygon, and that the spectrum does not determine the operator up to similarity. Two elliptic systems which only differ in their boundary conditions may have entirely different spectral asymptotics. While our study makes no claim to generality, the results obtained will have to be incorporated into any future general theory. Received: 15 August 2001 / in final form: 11 February 2002 / Published online: 24 February 2003     

To the spectral theory of Krein systems

We consider the Krein systems. For the set of Stummel class coefficients, we establish the criterion in terms of these coefficients for the system to satisfy the Szegö-type estimate on the spectral measure.     

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.     

Far from equilibrium steady states of 1D-Schrödinger–Poisson systems with quantum wells I

We describe the asymptotics of the steady states of the out-of-equilibrium Schrödinger–Poisson system, in the regime of quantum wells in a semiclassical island. After establishing uniform estimates on the nonlinearity, we show that the nonlinear steady states lie asymptotically in a finite-dimensional subspace of functions and that the involved spectral quantities are reduced to a finite number of so-called asymptotic resonant energies. The asymptotic finite dimensional nonlinear system is written in a general setting with only a partial information on its coefficients. After this first part, a complete derivation of the asymptotic nonlinear system will be done for some specific cases in a forthcoming article [V. Bonnaillie–Noël, F. Nier, M. Patel, Far from equilibrium steady states of 1D-Schrödinger–Poisson systems with quantum wells II, Prépublications IRMAR, 2007].     

A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) system?s poles in the complex plane. We include several examples to show the utility of this theory.     

Inverse problem of scattering theory for a class one-dimensional Schrödinger equation

Abstract

In this work, direct and inverse scattering problem on the real axis for the Schrödinger equation with piecewise-constant coefficient are studied. Using the new integral representations for solutions, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proved.     

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.     

The singular value decomposition to approximate spectra of dynamical systems. Theoretical aspects

In this paper we consider the singular value decomposition (SVD) of a fundamental matrix solution in order to approximate the Lyapunov and exponential dichotomy spectra of a given system. One of our main results is to prove that SVD techniques are sound approaches for systems with stable and distinct Lyapunov exponents. We also show how the information which emerges with the SVD techniques can be used to obtain information on the growth directions associated to given spectral intervals.     

Generalized rational first integrals of analytic differential systems

In this paper we mainly study the necessary conditions for the existence of functionally independent generalized rational first integrals of ordinary differential systems via the resonances. The main results extend some of the previous related ones, for instance the classical Poincaré?s one (Poincaré, 1891, 1897 [16]), the Furta?s one (Furta, 1996 [8]), part of Chen et al.?s ones (Chen et al., 2008 [4]), and the Shi?s one (Shi, 2007 [18]). The key point in the proof of our main results is that functionally independence of generalized rational functions implies the functionally independence of their lowest order rational homogeneous terms.     

On the isospectral problem of the dispersionless Camassa–Holm equation

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.     

Automatic solution of regular and singular vector Sturm-Liouville problems

This paper describes the algorithms and theory behind a new code for vector Sturm-Liouville problems. A new spectral function is defined for vector Sturm-Liouville problems; this is an integer valued function of the eigenparameter which has discontinuities precisely at the eigenvalues. We describe numerical algorithms which may be used to compute the new spectral function, and its use as amiss-distance function in a new code which solves automatically a large class of regular and singular vector Sturm-Liouville problems. Vector Sturm-Liouville problems arise naturally in quantum mechanical applications. Usually they are singular. The advantages of the author's code lie in its ability to solve singular problems automatically, and in the fact that the user may specify the required eigenvalue by its index.     

Estimates for eigenvalues of quasilinear elliptic systems. Part II

In this paper we find explicit lower bounds for Dirichlet eigenvalues of a weighted quasilinear elliptic system of resonant type in terms of the eigenvalues of a single p-Laplace equation. Also we obtain asymptotic bounds by studying the spectral counting function which is defined as the number of eigenvalues smaller than a given value.     

Inverse problem for Sturm-Liouville operators with rational reflection coefficient

In this paper we give exact formulas for the potential associated to a Sturm-Liouville equation when the reflection coefficient function is rational. The solution is given in terms of a minimal realization of the reflection coefficient function.Dedicated to the memory of M.G. Kreîn     

Invariant curves of one-step methods

This paper considers the invariant sets of numerical one-step integration methods in a neighbourhood of a hyperbolic periodic solution of a nonlinear ODE. Using results from the dynamical systems theory it was possible to show that for the usual one-step methods the invariant sets areC ^k-circles (closed curves) for small enough stepsizeh. Here we give a direct proof for that and also show that they areO(h ^p)C ^k-close to the true periodic trajectory, wherep is the order of the method.Most of this work was done during the author's visit at the Mittag-Leffler Institute, Djursholm, Sweden, academic year 1985–86.