Inverse problems for Schrödinger equations with unbounded potentials

2019/09/24 by Mourad Choulli, Choulli, Mourad · 1 citation
Mathematics · #Numerical methods in inverse problems #Advanced Mathematical Physics Problems #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1909.11133

Abstract

We summarize in these notes the course given at the Summer School of AIP 2019 held in Grenoble from July 1st to July 5th. This course was mainly devoted to the determination of the unbounded potential in a Schrödinger equation from the associated Dirichlet-to-Neumann map (abbreviated to DN map in this text). We establish a stability inequality for potentials belonging to Lⁿ, where n≥ 3 is the dimension of the space. Next, we prove a uniqueness result for potentials in L^n/2, n≥ 3, and apply this uniqueness result to demonstrate a Borg-Levinson type theorem. We use a classical approach which is essentially based on the construction of the so-called complex geometric optic solutions (abbreviated to CGO solutions in this text).

Cited by

• Comments on the determination of the conductivity at the boundary from\n the Dirichlet-to-Neumann map

Related

• Determining an unbounded potential from Cauchy data in admissible geometries
• Inverse boundary value problem for linear Schrödinger equation in two dimensions
• Multidimensionnel Borg-Levinson uniqueness and stability results for the Robin Laplacian with unbounded potential
• Stability inequality for the problem of determining an unbounded potential from boundary measurements
• Increasing stability of the inverse boundary value problem for the Schrödinger equation