We apply a particular form of the inverse scattering theory to turbulent magnetic fluctuations in a plasma. In the present note we develop the theory, formulate the magnetic fluctuation problem in terms of its electrodynamic turbulent response function, and reduce it to the solution of a special form of the famous Gelfand–Levitan–Marchenko equation of quantum mechanical scattering theory. The l...

Motivated by the study of certain nonlinear wave equations (in particular, the Camassa–Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form −u′′ = z uω + zuυ on an interval [0, L), where ω is a real-valued distribution in H−1 loc [0, L), υ is a non-negative Borel measure on [0, L) and z is a complex spectral parameter. Apa...

Recent results on localization, both exponential and dynamical, for various models of one-dimensional, continuum, random Schrödinger operators are reviewed. This includes Anderson models with indefinite single site potentials, the Bernoulli– Anderson model, the Poisson model, and the random displacement model. Among the tools which are used to analyse these models are generalized spectral avera...

We show that the Nonlinear Schrödinger Equation and the related Lax pair in 1+1 dimensions can be derived from 2+1 dimensional Chern-Simons Topological Gauge Theory. The spectral parameter, a main object for the Loop algebra structure and the Inverse Spectral Transform, has appear as a homogeneous part (condensate) of the statistical gauge field, connected with the compactified extra space coor...

We prove a substantial extension of an inverse spectral theorem of Ambarzumyan, and show that it can be applied to arbitrary compact Riemannian manifolds, compact quantum graphs and finite combinatorial graphs, subject to the imposition of Neumann (or Kirchhoff) boundary conditions.

1 KdV equation and Schrödinger operator 2 1.1 Integrability of Korteweg – de Vries equation . . . . . . . . . . . . . . . . . . 2 1.2 Elements of scattering theory for the Schrödinger operator . . . . . . . . . . . 5 1.3 Inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Dressing operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

1 Introduction The inverse spectral problem on a Riemannian manifold (M, g), possibly with boundary, is to determine as much as possible of the geometry of (M, g) from the spectrum of its Laplacian ∆ g (with some given boundary conditions). The special inverse problem of Kac is to determine a Euclidean domain Ω ⊂ R n up to isometry from the spectrum Spec B (Ω) of its Laplacian ∆ B with Dirichle...

In this paper, we would like to sketch a picture aimed at giving a comprehensive answer to the question: how does one go about reconstructing a manifold M from the spectrum of its Laplace operator ? It is understood that, in general, there is no unique way of reconstructing M, because a manifold is not in general uniquely determined from its spectrum. So let us make the following deenition: Dee...

In the first edition of this book the main attention was focused on the methods of solving the inverse problem of spectral analysis and on the conditions (necessary and sufficient) which the spectral data must satisfy in order to make it possible to reconstruct the potential of the corresponding Sturm-Liouville operator. These conditions imply that the spectral data (e.g. spectral function or s...