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The boundary control approach to the Titchmarsh-Weyl $m-$function

S. A. Avdonin, V. S. Mikhaylov, A. V. Rybkin

TL;DR

The paper develops a Boundary Control framework that connects the dynamic Dirichlet-to-Neumann (response) data to the spectral Titchmarsh-Weyl m-function for half-line Schrödinger operators, providing a physical interpretation of Simon's A-amplitude via the boundary kernel. It derives a linear Volterra integral equation for the kernel A(x,y) whose diagonal gives the A-amplitude and proves convergence with explicit exponential-type bounds, addressing an open question about A. A practical algorithm is then proposed to compute m(z) by solving the integral equation for A and using a Laplace-type representation, yielding an absolutely convergent series in terms of A_n with A_0 = q. This approach enables efficient computation of m(z) from boundary data, offers robustness to non-smooth potentials, and suggests extensions to matrix-valued problems within the Boundary Control framework.

Abstract

We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the $A-$amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl $m-$function associated with the Schrödinger operator $H=-\partial _{x}^{2}+q\left( x\right) $ on $L_{2}\left( 0,\infty \right) $ with Dirichlet boundary condition at $x=0.$

The boundary control approach to the Titchmarsh-Weyl $m-$function

TL;DR

The paper develops a Boundary Control framework that connects the dynamic Dirichlet-to-Neumann (response) data to the spectral Titchmarsh-Weyl m-function for half-line Schrödinger operators, providing a physical interpretation of Simon's A-amplitude via the boundary kernel. It derives a linear Volterra integral equation for the kernel A(x,y) whose diagonal gives the A-amplitude and proves convergence with explicit exponential-type bounds, addressing an open question about A. A practical algorithm is then proposed to compute m(z) by solving the integral equation for A and using a Laplace-type representation, yielding an absolutely convergent series in terms of A_n with A_0 = q. This approach enables efficient computation of m(z) from boundary data, offers robustness to non-smooth potentials, and suggests extensions to matrix-valued problems within the Boundary Control framework.

Abstract

We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl function associated with the Schrödinger operator on with Dirichlet boundary condition at

Paper Structure

This paper contains 6 sections, 3 theorems, 79 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. The response operator and the $A$-amplitude
  3. An integral equation for the $A$-amplitude
  4. Analysis of iterations
  5. A new procedure for evaluating the Titchmarsh-Weyl $m-$function
  6. Acknowledgments

Key Result

Theorem 1

Let $q\in L_{\mathop{\rm loc}}^{1}\left( \mathbb{R}_{+}\right) .$ Then for a. e. ${ \if@compatibility \mathchar"010B {} \mathchar"010B } >0$ where $A\left( x,y\right)$ is the solution to the integral equation

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Remark 3
  • Remark 4
  • ...and 1 more