The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy
Fritz Gesztesy Helge Holden Johanna Michor Gerald Teschl
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 151-196 doi: 10.3934/dcds.2010.26.151
We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy.
   The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.
keywords: initial value problem. complex-valued solutions Ablowitz-Ladik hierarchy
Spectral estimation and inverse initial boundary value problems
Sergei Avdonin Fritz Gesztesy Konstantin A. Makarov
Inverse Problems & Imaging 2010, 4(1): 1-9 doi: 10.3934/ipi.2010.4.1
We extend the classical spectral estimation problem to the infinite-dimensional case and propose a new approach to this problem using the Boundary Control (BC) method. Several applications to inverse problems for partial differential equations are provided.
keywords: Spectral estimation inverse problems boundary control method.
On a class of model Hilbert spaces
Fritz Gesztesy Rudi Weikard Maxim Zinchenko
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): 5067-5088 doi: 10.3934/dcds.2013.33.5067
A detailed description of the model Hilbert space $L^2(\mathbb{R}; d\Sigma; K)$, where $K$ represents a complex, separable Hilbert space, and $\Sigma$ denotes a bounded operator-valued measure, is provided. In particular, we show that several alternative approaches to such a construction in the literature are equivalent.
    These spaces are of fundamental importance in the context of perturbation theory of self-adjoint extensions of symmetric operators, and the spectral theory of ordinary differential operators with operator-valued coefficients.
keywords: Direct integrals of Hilbert spaces model Hilbert spaces.

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