APDE seminar@BCAM: On the stability of the initial-to-final inverse problem for the Schrödinger equation

Abstract

The initial-to-final inverse problem consists in inferring the dynamics of an evolution PDE from measurements only on the initial and final state of the solutions of the equation in a fixed time interval. This is, by knowledge of the map ψ(0, ) 7→ ψ(T, ) for a fixed T > 0.
In particular, we deal with the evolution of a quantum particle under the influence of an electric potential, and we study the theoretical stability of the Hamiltonian with respect to errors in the measurements. Building on recent uniqueness results by P. Caro and A. Ruiz, and by the speaker, P. Caro, I. Parissis, and T. Zacharopoulos, we consider bounded time-dependent potentials V = V (t, x) with super-exponential decay, as well as time-independent potentials V = V (x) with super-linear decay.

In the first case, we find a logarithmic modulus of continuity, in line with other classical results in inverse problems. However, we find a significant improvement for the case of time-independent potentials, in the form of a Hölder type continuity.  

To this end, we introduce an ad-hoc quasi-distance that allows us to quantify these errors, and we prove a suitable Runge approximation in the relevant weighted L2 spaces.
This work is a collaboration with Thanasis Zacharopoulos, from Aarhus University.