Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral invariants of the Dirichlet-to-Neumann map associated to the Witten-Laplacian with potential

Xiaoming Tan

TL;DR

This work delivers an explicit, procedure-driven determination of the spectral invariants for the Dirichlet-to-Neumann map Λ associated with the Witten-Laplacian Δ_φ plus a potential V on a compact manifold with boundary. By flattening the boundary and applying full symbol calculus to factor Δ_φ+V, the authors construct the full symbol of Λ and relate the heat-trace Tr e^{-tΛ} to a local asymptotic expansion with coefficients a_k that decompose into a universal Laplace-Beltrami part and a φ,V-dependent part. They compute the first four coefficients a0, a1, a2, and a3, yielding explicit geometric and analytic expressions in terms of boundary curvature, the drifting function φ, and the potential V; a corollary under constant sectional curvature further simplifies these expressions. The results provide a concrete bridge between Steklov-type spectral data and boundary geometry, enabling reconstruction of geometric and analytic features from Λ. The methodology extends prior Steklov-related heat-trace results to the Witten-Laplacian setting and demonstrates an effective framework for extracting higher-order spectral invariants via pseudodifferential symbol calculus.

Abstract

For a compact connected Riemannian manifold with smooth boundary, we establish an effective procedure, by which we can calculate all the coefficients of the spectral asymptotic formula of the Dirichlet-to-Neumann map associated to the Witten-Laplacian with potential. In particular, by computing the full symbol of the Dirichlet-to-Neumann map we explicitly give the first four coefficients. They are spectral invariants, which provide precise information concerning the volume, curvatures, drifting function and potential.

Spectral invariants of the Dirichlet-to-Neumann map associated to the Witten-Laplacian with potential

TL;DR

This work delivers an explicit, procedure-driven determination of the spectral invariants for the Dirichlet-to-Neumann map Λ associated with the Witten-Laplacian Δ_φ plus a potential V on a compact manifold with boundary. By flattening the boundary and applying full symbol calculus to factor Δ_φ+V, the authors construct the full symbol of Λ and relate the heat-trace Tr e^{-tΛ} to a local asymptotic expansion with coefficients a_k that decompose into a universal Laplace-Beltrami part and a φ,V-dependent part. They compute the first four coefficients a0, a1, a2, and a3, yielding explicit geometric and analytic expressions in terms of boundary curvature, the drifting function φ, and the potential V; a corollary under constant sectional curvature further simplifies these expressions. The results provide a concrete bridge between Steklov-type spectral data and boundary geometry, enabling reconstruction of geometric and analytic features from Λ. The methodology extends prior Steklov-related heat-trace results to the Witten-Laplacian setting and demonstrates an effective framework for extracting higher-order spectral invariants via pseudodifferential symbol calculus.

Abstract

For a compact connected Riemannian manifold with smooth boundary, we establish an effective procedure, by which we can calculate all the coefficients of the spectral asymptotic formula of the Dirichlet-to-Neumann map associated to the Witten-Laplacian with potential. In particular, by computing the full symbol of the Dirichlet-to-Neumann map we explicitly give the first four coefficients. They are spectral invariants, which provide precise information concerning the volume, curvatures, drifting function and potential.
Paper Structure (15 sections, 8 theorems, 206 equations)

This paper contains 15 sections, 8 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Notations and main result
  3. Preliminaries
  4. Differential operators and curvatures
  5. Boundary normal coordinates
  6. Pseudodifferential operators and symbols
  7. Symbol calculus and the asymptotic expansion of heat trace
  8. Symbols of pseudodifferential operators
  9. Coefficients of the asymptotic expansion of heat trace
  10. Computations of
  11. Computation of
  12. Computation of
  13. Computation of
  14. Computation of
  15. Proof of Corollary \ref{['cor1.2']}

Key Result

Theorem 1.1

Suppose that $(M,g)$ is a compact connected Riemannian manifold of dimension $n$ with smooth boundary $\partial M$. Let $\{\lambda_k\}_{k \geqslant 1}$ be the eigenvalues of the Dirichlet-to-Neumann map $\Lambda$ associated to the Witten-Laplacian with potential $\Delta_{\phi}+V$. Then the heat trac where the coefficients $a_k = \int_{\partial M} a_k(x^{\prime})\,dS,\,x^{\prime}\in\partial M,\,0 \

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • ...and 6 more