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Inverse Problem for the Schrödinger Equation with Non-self-adjoint Matrix Potential

Sergei Avdonin, Alexander Mikhaylov, Victor Mikhaylov, Jeff Park

TL;DR

This work extends the Boundary Control method to the one-dimensional vector Schrödinger equation with a nonsymmetric matrix potential by deriving how to recover spectral data from dynamical boundary data and then solving the inverse problem for the potential. It develops a rigorous abstract BC framework for non-self-adjoint generators, establishes generalized spectral problems that link dynamic data to the spectrum and eigenvectors, and provides procedures for both simple and general (multiplicity) cases. A central achievement is the spectral-controllability-based reconstruction of the matrix potential $Q$ from the response operator $R^T$ via a connecting-operator formalism and a transmutation-based null controllability argument, culminating in a formula $Q(T)=M''(T)M^{-1}(T)$ derived from the wave-analytic representation. The results enable stable recovery of $Q$ on the interval by varying a parameter $T$ and aggregating the spectral data, with potential applications to multidimensional non-self-adjoint inverse problems through the boundary-control paradigm.

Abstract

We consider the dynamical system with boundary control for the vector Schrödinger equation on the interval with a non-self-adjoint matrix potential. For this system, we study the inverse problem of recovering the matrix potential from the dynamical Dirichlet--to--Neumann operator. We first provide a method to recover spectral data for an abstract system from dynamic data and apply it to the Schrödinger equation. We then develop a strategy for solving the inverse problem for the Schrödinger equation using this method with other techniques of the Boundary control method.

Inverse Problem for the Schrödinger Equation with Non-self-adjoint Matrix Potential

TL;DR

This work extends the Boundary Control method to the one-dimensional vector Schrödinger equation with a nonsymmetric matrix potential by deriving how to recover spectral data from dynamical boundary data and then solving the inverse problem for the potential. It develops a rigorous abstract BC framework for non-self-adjoint generators, establishes generalized spectral problems that link dynamic data to the spectrum and eigenvectors, and provides procedures for both simple and general (multiplicity) cases. A central achievement is the spectral-controllability-based reconstruction of the matrix potential from the response operator via a connecting-operator formalism and a transmutation-based null controllability argument, culminating in a formula derived from the wave-analytic representation. The results enable stable recovery of on the interval by varying a parameter and aggregating the spectral data, with potential applications to multidimensional non-self-adjoint inverse problems through the boundary-control paradigm.

Abstract

We consider the dynamical system with boundary control for the vector Schrödinger equation on the interval with a non-self-adjoint matrix potential. For this system, we study the inverse problem of recovering the matrix potential from the dynamical Dirichlet--to--Neumann operator. We first provide a method to recover spectral data for an abstract system from dynamic data and apply it to the Schrödinger equation. We then develop a strategy for solving the inverse problem for the Schrödinger equation using this method with other techniques of the Boundary control method.
Paper Structure (8 sections, 7 theorems, 86 equations, 2 algorithms)

This paper contains 8 sections, 7 theorems, 86 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Equations of the BC method
  3. The Spectral Problem for the Simple Case -- algebraically simple spectrum
  4. Recovery of the Spectral Data in the Simple Case
  5. The Spectral Problem and Recovery of the Spectral Data in the General Case
  6. Dynamical Inverse Problem for the nonsymmetric matrix Schrödinger Operator on an Interval
  7. Spectral Controllability of the Schrödinger Equation with nonsymmetric matrix potential
  8. Recovery of the Spectral Data and the Matrix Potential

Key Result

Lemma 2.1

The following identity holds.

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 1
  • Definition 4.1
  • Lemma 4.2
  • proof
  • ...and 6 more