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Inverse problems for semilinear Schrödinger equations at large frequency via polynomial resolvent estimates on manifolds

Katya Krupchyk, Shiqi Ma, Suman Kumar Sahoo, Mikko Salo, Simon St-Amant

Abstract

We study inverse boundary problems for semilinear Schrödinger equations on smooth compact Riemannian manifolds of dimensions $\ge 2$ with smooth boundary, at a large fixed frequency. We show that certain classes of cubic nonlinearities are determined uniquely from the knowledge of the nonlinear Dirichlet--to--Neumann map at a large fixed frequency on quite general Riemannian manifolds. In particular, in contrast to the previous results available, here the manifolds need not satisfy any product structure, may have trapped geodesics, and the geodesic ray transform need not be injective. Only a mild assumption about the geometry of intersecting geodesics is required. We also establish a polynomial resolvent estimate for the Laplacian on an arbitrary smooth compact Riemannian manifold without boundary, valid for most frequencies. This estimate, along with the invariant construction of Gaussian beam quasimodes with uniform bounds for underlying constants and a stationary phase lemma with explicit control over all involved constants, constitutes the key elements in proving the uniqueness results for the considered inverse problems.

Inverse problems for semilinear Schrödinger equations at large frequency via polynomial resolvent estimates on manifolds

Abstract

We study inverse boundary problems for semilinear Schrödinger equations on smooth compact Riemannian manifolds of dimensions with smooth boundary, at a large fixed frequency. We show that certain classes of cubic nonlinearities are determined uniquely from the knowledge of the nonlinear Dirichlet--to--Neumann map at a large fixed frequency on quite general Riemannian manifolds. In particular, in contrast to the previous results available, here the manifolds need not satisfy any product structure, may have trapped geodesics, and the geodesic ray transform need not be injective. Only a mild assumption about the geometry of intersecting geodesics is required. We also establish a polynomial resolvent estimate for the Laplacian on an arbitrary smooth compact Riemannian manifold without boundary, valid for most frequencies. This estimate, along with the invariant construction of Gaussian beam quasimodes with uniform bounds for underlying constants and a stationary phase lemma with explicit control over all involved constants, constitutes the key elements in proving the uniqueness results for the considered inverse problems.
Paper Structure (19 sections, 20 theorems, 200 equations)

This paper contains 19 sections, 20 theorems, 200 equations.

Table of Contents

  1. Introduction and statement of results
  2. Polynomial resolvent estimates for the Laplacian on a smooth compact Riemannian manifold
  3. Proof of Theorem \ref{['thm_main']}
  4. Gaussian beam quasimodes and solutions to Helmholtz equations
  5. Main part of the proof of Theorem \ref{['thm_main']}
  6. Proof of Theorem \ref{['thm_main_2']}
  7. Gaussian beam quasimodes and solutions to Helmholtz equations with uniform constants
  8. Main part of the proof of Theorem \ref{['thm_main_2']}
  9. Proof of Theorem \ref{['thm:intersection']}
  10. Existence of non-tangential geodesics between boundary points
  11. Perturbation of pairs of geodesics
  12. Proof of Theorem \ref{['thm:intersection']}
  13. A rough stationary phase argument
  14. Boundary determination
  15. Example of an infinite set of linearly independent admissible perturbations
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a smooth compact oriented Riemannian manifold of dimension $n\ge 2$ with smooth boundary, satisfying the condition (H). Let $0<\alpha<1$ and let $q_1,q_2\in C^{0,\alpha}(M)$. Then for any $\delta>0$, there exists a set $J\subset [1,\infty)$, $J=J(M,g,\delta)$, satisfying $|J|\le \delt

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Theorem 1.6
  • Example 1.7
  • Theorem 1.8
  • Example 1.9
  • Example 1.10
  • ...and 36 more