# 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.

## 12 Citations

The inverse resonance problem for CMV operators

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We consider the class of CMV operators with super-exponentially decaying Verblunsky coefficients. For these we define the concept of a resonance. Then we prove the existence of Jost solutions and a…

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Weyl theory for a non-classical system depending rationally on the spectral parameter is treated. Borg–Marchenko-type uniqueness theorem is proved. The solution of the inverse problem is constructed.…

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Minimal Rank Decoupling of Full-Lattice CMV Operators with Scalar- and Matrix-Valued Verblunsky Coefficients

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Relations between half- and full-lattice CMV operators with scalar- and matrix-valued Verblunsky coefficients are investigated. In particular, the decoupling of full-lattice CMV operators into a…

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- 2012

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Self-adjoint Dirac systems and subclasses of canonical systems, which generalize Dirac systems are studied. Explicit and global solutions of direct and inverse problems are obtained. A local…

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Skew-Self-Adjoint Dirac System with a Rectangular Matrix Potential: Weyl Theory, Direct and Inverse Problems

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A non-classical Weyl theory is developed for skew-self-adjoint Dirac systems with rectangular matrix potentials. The notion of the Weyl function is introduced and direct and inverse problems are…

Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy's 60th Birthday

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Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schrodinger operators with matrix-valued potentials, with…

## References

SHOWING 1-10 OF 92 REFERENCES

Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle

- MathematicsJ. Approx. Theory
- 2006

Weyl-Titchmarsh M-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators

- Mathematics
- 2001

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

- Mathematics
- 2000

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

- Mathematics
- 2004

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

- Mathematics
- 2004

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

- Mathematics
- 2001

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

- Mathematics
- 2005

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

- Mathematics
- 2005

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

- Mathematics
- 2010

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…