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Inverse Problems and Index Formulae for Dirac Operators

Yaroslav Kurylev, Matti Lassas

TL;DR

The paper develops a boundary-control framework for inverse problems for Dirac-type operators on manifolds with boundary, including nonlocal boundary conditions. It derives an index formula linking the Fredholm index of the Dirac operator to a boundary-time operator and proves that the manifold, bundle, and operator can be reconstructed from boundary data (Cauchy data or boundary spectral data) up to natural symmetries. Key contributions include a constructive procedure to determine $(M,g)$, $V$, and $D$ from dynamic data, a boundary-data–based index formula, focusing sequences that recover local bundle structure, and a spectral-data approach that recovers the full setup with gauge/topological caveats. The Euclidean specialization reveals Aharonov–Bohm-type nonuniqueness in the magnetic potential up to exact and harmonic-topology terms, illustrating the interplay between geometry, topology, and boundary measurements.

Abstract

We consider a Dirac-type operator $D_P$ on a vector bundle $V$ over a compact Riemannian manifold $(M,g)$ with a nonempty boundary. The operator $D_P$ is specified by a boundary condition $P(u|_{\p M})=0$ where $P$ is a projector which may be a non-local, i.e. a pseudodifferential operator. We assume the existence of a chirality operator which decomposes $L^2(M, V)$ into two orthogonal subspaces $X_+ \oplus X_-$. Under certain conditions, the operator $D_P$ restricted to $X_+$ and $ X_-$ defines a pair of Fredholm operators which maps $X_+\to X_-$ and $X_-\to X_+$ correspondingly, giving rise to a superstructure on $V$. In this paper we consider the questions of determining the index of $D_P$ and the reconstruction of $(M, g), V$ and $D_P$ from the boundary data on $\p M$. The data used is either the Cauchy data, i.e. the restrictions to $\p M \times \R_+$ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e. the set of the eigenvalues and the boundary values of the eigenfunctions of $D_P$. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in $M\times \C^4$, $M \subset \R^3$.

Inverse Problems and Index Formulae for Dirac Operators

TL;DR

The paper develops a boundary-control framework for inverse problems for Dirac-type operators on manifolds with boundary, including nonlocal boundary conditions. It derives an index formula linking the Fredholm index of the Dirac operator to a boundary-time operator and proves that the manifold, bundle, and operator can be reconstructed from boundary data (Cauchy data or boundary spectral data) up to natural symmetries. Key contributions include a constructive procedure to determine , , and from dynamic data, a boundary-data–based index formula, focusing sequences that recover local bundle structure, and a spectral-data approach that recovers the full setup with gauge/topological caveats. The Euclidean specialization reveals Aharonov–Bohm-type nonuniqueness in the magnetic potential up to exact and harmonic-topology terms, illustrating the interplay between geometry, topology, and boundary measurements.

Abstract

We consider a Dirac-type operator on a vector bundle over a compact Riemannian manifold with a nonempty boundary. The operator is specified by a boundary condition where is a projector which may be a non-local, i.e. a pseudodifferential operator. We assume the existence of a chirality operator which decomposes into two orthogonal subspaces . Under certain conditions, the operator restricted to and defines a pair of Fredholm operators which maps and correspondingly, giving rise to a superstructure on . In this paper we consider the questions of determining the index of and the reconstruction of and from the boundary data on . The data used is either the Cauchy data, i.e. the restrictions to of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e. the set of the eigenvalues and the boundary values of the eigenfunctions of . We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in , .

Paper Structure

This paper contains 10 sections, 26 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Definitions and main results.
  3. Hyperbolic Dirac equation and unique continuation
  4. Inner products and controllability.
  5. Generalized boundary sources
  6. Index formulae using boundary data
  7. Reconstruction of the manifold
  8. Focusing sequences. Proof of Theorem \ref{['main1']}.
  9. Inverse problem spectral problem.
  10. Dirac-type operator in ${\mathbb R}^3$

Key Result

Theorem 2.1

Let P be a zero-order classical pseudo-differential operator defining an orthoprojector which satisfies conditions (anticommutation) and (Fredholm). Let D_{P} be the Dirac-type operator (dirac-type). Then D_{P} is self-adjoint, the spectrum of D_{P} is discrete, and all eigenspaces are finite dimens

Theorems & Definitions (36)

  • Theorem 2.1: Brüning-Lesch
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 26 more