Stephen Clark, Fritz Gesztesy, Maxim Zinchenko
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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We prove a general Borg-type inverse spectral result for a reflectionless unitary CMV operator (CMV for Cantero, Moral, and Velázquez) associated with matrix-valued Verblunsky coefficients. More precisely, we find an explicit formula for the Verblunsky coefficients of a reflectionless CMV matrix whose spectrum consists of a connected arc on the unit circle. This extends a recent result on CMV operators with scalar-valued coefficients. In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory matrix-valued functions to prove an infinite sequ…
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) with matrix-valued Verblunsky coefficients. While our half-lattice results are formulated in terms of matrix-valued Weyl-Titchmarsh functions, our full-lattice results involve the diagonal and main off-diagonal Green's matrices. We also develop the basics of Weyl-Titchmarsh theory for CMV operators with matrix-valued Verblunsky coefficients as this is of independent interest and an essential ingredient in proving the corresponding…
B. Simon proved the existence of the wave operators for the CMV matrices with Szego class Verblunsky coefficients, and therefore the existence of the scattering function. Generally, there is no hope to restore a CMV matrix when we start from the scattering function, in particular, because it does not contain any information about the (possible) singular measure. Our main point of interest is the solution of the inverse scattering problem (the heart of the Faddeev--Marchenko theory), that is, to give necessary and sufficient conditions on a certain class of CMV matrices such that the restrictio…
We prove a general Borg-type result for reflectionless unitary Cantero-Moral-Velazquez (CMV) operators U associated with orthogonal polynomials on the unit circle. The spectrum of U is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator with spectrum the whole unit circle. In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory functions to prove an infinite sequence of trace formulas connected with the CMV operator U.
New formulas on the inverse problem for the continuous skew-self-adjoint Dirac type system are obtained. For the discrete skew-self-adjoint Dirac type system the solution of a general type inverse spectral problem is also derived in terms of the Weyl functions. The description of the Weyl functions on the interval is given. Borg-Marchenko type uniqueness theorems are derived for both discrete and continuous non-self-adjoint systems too.
We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl-Titchmarsh m-functions, $m_j(z)$, of two Schrödinger operators $H_j = -\f{d^2}{dx^2} + q_j$, j=1,2 in $L^2 ((0,R))$, $0<R\leq \infty$, are exponentially close, that is, $|m_1(z)- m_2(z)| \underset{|z|\to\infty}{=} O(e^{-2\Ima (z^{1/2})a})$, 0<a<R, then $q_1 = q_2$ a.e.~on $[0,a]$. The result applies to any boundary conditions at x=0 and x=R and should be considered a local version of the celebrated Borg-Marchenko uniqueness result (which is quickly recovered as a corollary to our proof). Moreover…
The Borg-Marchenko uniqueness theorem for complex potentials
本文针对具有有界复值势的 Schrödinger 算子引入谱对概念，将经典谱测度推广到非自伴情形，证明了谱对唯一确定算子的 Borg-Marchenko 型唯一性定理，并推导了谱对的渐近公式及其与谱测度、单特征值的关系。
We develop a scattering theory for CMV matrices, similar to the Faddeev--Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for the uniqueness, which are connected with the Helson--Szeg\H o and the Strong Szeg\H o theorems. The first condition is given in terms of the boundedness of a transformation operator associated to the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, correspo…
Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schrödinger operators with matrix-valued potentials, with special emphasis on distributional potential coefficients. Our principal method relies on a supersymmetric (factorization) formalism underlying Miura's transformation, which intimately connects the triple of operators $(D, H_1, H_2)$ of the form [D= (0 & A^*, A & 0) \text{in} L^2(\mathbb{R})^{2m} \text{and} H_1 = A^* A, H_2 = A A^* \text{in} L^2(\mathbb{R})^m.] Here $A= I_m (d/dx) + ϕ$ in $L^2(\mathbb…
Discrete Dirac type self-adjoint system is equivalent to the block Szegö recurrence. Representation of the fundamental solution is obtained, inverse problems on the interval and semi-axis are solved. A Borg-Marchenko type result is obtained too. Connections with the block Toeplitz matrices are treated.
We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a genera…
We discuss inverse spectral theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[τf = \frac{1}{r} \left(- \big(p[f' + s f]\big)' + s p[f' + s f] + qf\right), \] where the coefficients $p$, $q$, $r$, $s$ are Lebesgue measurable on $(a,b)$ with $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$ and real-valued with $p\not=0$ and $r>0$ a.e.\ on $(a,b)$. In particular, we explicitly permit certain distributional potential coefficients. The inverse spectral theory results derived in …
A transfer matrix function representation of the fundamental solution of the general-type discrete Dirac system, corresponding to rectangular Schur coefficients and Weyl functions, is obtained. Connections with Szegö recurrence, Schur coefficients and structured matrices are treated. Borg-Marchenko-type uniqueness theorem is derived. Inverse problems on the interval and semiaxis are solved.
A Weyl Matrix Perspective on Unbounded Non-Self-Adjoint Jacobi Matrices
针对Pushnitski--\u0160Stampach的编码方法，提出基于外尔函数的新视角，简化证明并推广至无界非自伴雅可比矩阵情形，建立正非对角元真雅可比矩阵与谱数据的双射且证明其连续性，还给出具独立意义的局部Borg--Marchenko定理。
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