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Uniqueness for the Schrödinger equation with an inverse square potential and application to controllability and inverse problems

S. E. Chorfi

Abstract

In this paper, we prove a sharp uniqueness result for the singular Schrödinger equation with an inverse square potential. This will be done without assuming geometrical restrictions on the observation region. The proof relies on a recent technique transforming the Schrödinger equation into an elliptic equation. We show that this technique is still applicable for singular equations. In our case, substantial difficulties arise when dealing with singular potentials of cylindrical type. Using the uniqueness result, we show the approximate controllability of the equation using a distributed control. The uniqueness result is also applied to prove the uniqueness for an inverse source problem.

Uniqueness for the Schrödinger equation with an inverse square potential and application to controllability and inverse problems

Abstract

In this paper, we prove a sharp uniqueness result for the singular Schrödinger equation with an inverse square potential. This will be done without assuming geometrical restrictions on the observation region. The proof relies on a recent technique transforming the Schrödinger equation into an elliptic equation. We show that this technique is still applicable for singular equations. In our case, substantial difficulties arise when dealing with singular potentials of cylindrical type. Using the uniqueness result, we show the approximate controllability of the equation using a distributed control. The uniqueness result is also applied to prove the uniqueness for an inverse source problem.

Paper Structure

This paper contains 7 sections, 8 theorems, 71 equations.

Table of Contents

  1. Introduction and main results
  2. Uniqueness for an elliptic equation with a singular potential
  3. Sharp uniqueness for the singular Schrödinger equation
  4. Uniqueness for the singular Schrödinger equation
  5. Approximate controllability
  6. Uniqueness for an inverse source problem
  7. Conclusion

Key Result

Theorem 1.1

Let $\omega \subset \Omega$ be an arbitrary nonempty open set. We assume that $u \in C([0,T];L^2(\Omega))$ satisfies (1.2). Then $u = 0$ in $(0,T) \times \omega$ yields $u=0$ in $Q$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Remark 1
  • Proposition 2.3
  • proof
  • Remark 2
  • Remark 3
  • proof : Proof of Theorem \ref{['thm1']}
  • Theorem 3.1
  • ...and 8 more