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### Open Access Journals

DCDS

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.

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.

IPI

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.

DCDS

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.

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.

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