• Corpus ID: 15649726

Borg-Marchenko-type Uniqueness Results for CMV Operators

@article{Clark2008BorgMarchenkotypeUR,
  title={Borg-Marchenko-type Uniqueness Results for CMV Operators},
  author={Stephen Clark and Fritz Gesztesy and Maxim Zinchenko},
  journal={arXiv: Spectral Theory},
  year={2008}
}
We prove local and global versions of Borg-Marchenko-type uniqueness theorems for half-lattice and full-lattice CMV operators (CMV for Cantero, Moral, and Velazquez \cite{CMV03}). While our half-lattice results are formulated in terms of Weyl-Titchmarsh functions, our full-lattice results involve the diagonal and main off-diagonal Green's functions. 
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References

SHOWING 1-10 OF 92 REFERENCES
A local borg-marchenko theorem for complex potentials
Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle
Weyl-Titchmarsh M-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators
We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on R. We also prove new local uniqueness results for
On Local Borg–Marchenko Uniqueness Results
Abstract:We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl–Titchmarsh m-functions, mj(z), of two Schrödinger operators , j≡ 1,2 in L2((0,R)), 0<R≤∞,
A local Borg-Marchenko theorem for difference equations with complex coefficients
We investigate the asymptotic behavior of the Titchmarsh-Weyl m-function for a difference equation with complex coefficients and prove a local Borg-Marchenko theorem. The proofs are based on a
Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle
We investigate the existence and properties of an integrable system related to orthogonal polynomials on the unit circle. We prove that the main evolution of the system is defocusing Ablowitz-Ladik
On a local uniqueness result for the inverse Sturm-Liouville problem
A new and fairly elementary proof is given of the result by B. Simon [S], that the potential in a Sturm-Liouville operator is determined by the asymptotics of the associatedm-function near −∞. The
A Borg‐Type Theorem Associated with Orthogonal Polynomials on the Unit Circle
We prove a general Borg‐type result for reflectionless unitary CMV operators U associated with orthogonal polynomials on the unit circle. The spectrum of U is assumed to be a connected arc on the
Some remarks on CMV matrices and dressing orbits
The CMV matrices are the unitary analogs of Jacobi matrices. In the finite case, it is well-known that the set of Jacobi matrices with a fixed trace is nothing but a coadjoint orbit of the lower
Algebro-Geometric Finite-Band Solutions of the Ablowitz–Ladik Hierarchy
We provide a detailed derivation of all complex-valued algebro-geometric finite-band solutions of the Ablowitz-Ladik hierarchy. In addition, we survey a recursive construction of the Ablowitz-Ladik
...
1
2
3
4
5
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