Uniqueness results for matrix-valued Schrödinger, Jacobi, and Dirac-type operators. (English) Zbl 1017.34020
The motivation of the paper is a close investigation of inverse spectral problems and the proof of few uniqueness theorems for isospectral sets of periodic matrix-valued Schrödinger operators. That is to determine spectral data (interpreted in a very broad sense) which uniquely determine the matrix-valued coefficients in Schrödinger, Jacobi and Dirac type systems. It begins with a brief review of the basic Weyl-Titchmarsh theory for matrix-valued Schrödinger- and Dirac-type operators as needed in the sequel.
The principal result proven is that, for a fixed \(x_0\in\mathbb{R}\) and all \(z\in\mathbb{C}_+\), \(g(z,x_0)\) and \(g'(z,x_0)\) uniquely determine the matrix-valued \(m\times m\) potential \(Q(z)\) for a.e. \(x\in\mathbb{R}\), where \(g(z,x)\) denotes the diagonal Green matrix of the selfadjoint \(m\times m\)-matrix-valued Schrödinger operator \[ H= -{d^2\over dx^2} I_m+ Q\quad\text{in }L^2(\mathbb{R})^m,\quad m\in\mathbb{N}. \] A local version of this result is also given as an adjoint. The last section is devoted to the investigation of the Weyl-Titchmarsh theory and uniqueness theorems for matrix-valued Jacobi (i.e., second-order finite differences) operators.
The paper not only presents the material as stated above, it also mentions in a synchronous manner almost all the previous work related to the topic discussed. This is evident from the fact that it is supplemented with 145 references.
The principal result proven is that, for a fixed \(x_0\in\mathbb{R}\) and all \(z\in\mathbb{C}_+\), \(g(z,x_0)\) and \(g'(z,x_0)\) uniquely determine the matrix-valued \(m\times m\) potential \(Q(z)\) for a.e. \(x\in\mathbb{R}\), where \(g(z,x)\) denotes the diagonal Green matrix of the selfadjoint \(m\times m\)-matrix-valued Schrödinger operator \[ H= -{d^2\over dx^2} I_m+ Q\quad\text{in }L^2(\mathbb{R})^m,\quad m\in\mathbb{N}. \] A local version of this result is also given as an adjoint. The last section is devoted to the investigation of the Weyl-Titchmarsh theory and uniqueness theorems for matrix-valued Jacobi (i.e., second-order finite differences) operators.
The paper not only presents the material as stated above, it also mentions in a synchronous manner almost all the previous work related to the topic discussed. This is evident from the fact that it is supplemented with 145 references.
Reviewer: N.D.Sengupta (Mumbai)
MSC:
34B20 | Weyl theory and its generalizations for ordinary differential equations |
34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
34E05 | Asymptotic expansions of solutions to ordinary differential equations |