Recovering All Coefficients in the Schrödinger Equation With Finite Sets of Boundary Measurements
Shitao Liu, Antonio Pierrottet
TL;DR
This work addresses recovering all spatially varying coefficients $q_0(x)$, $\mathbf{q}_1(x)$, and $q_2(x)$ in the time-dependent Schrödinger equation from finitely many boundary measurements. The authors implement a direct Carleman-estimate framework on a Riemannian manifold with metric $g=q_2^{-1}(x)dx^2$ and a pseudo-convex weight $\phi(t,x)=d(x)-ct^2$, enabling stability estimates without cut-off or compactness-uniqueness arguments. The core strategy reduces the problem to a finite-dimensional inverse source problem via $f_2,\mathbf{f}_1,f_0$ and a time-differentiated linear system $i\big(u_t^{(1)}(0,x),\dots,u_t^{(n+2)}(0,x)\big)^T = U_{\mathbf{R}}(x)(f_0,\mathbf{f}_1,f_2)^T$, where positivity of $\det U_{\mathbf{R}}(x)$ ensures invertibility. Combining Carleman estimates with energy bounds yields Lipschitz stability for the unknown coefficients from boundary Neumann data on $\Sigma_1^T$, and this leads to stability for the original inverse problem; the paper also provides concrete examples and discusses extensions to related boundary data formulations.
Abstract
We consider an inverse problem of recovering all spatial dependent coefficients in the time dependent Schrödinger equation defined on an open bounded domain in $\mathbb{R}^n$, $n\geq 2$, with smooth enough boundary. We show that by appropriately selecting a finite number of initial conditions and a fixed Dirichlet boundary condition, we may recover all the coefficients in a Lipschitz stable fashion from the corresponding finitely many boundary measurements made on a portion of the boundary. The proof is based on a direct approach, which was introduced in \cite{HIY2020}, to derive the stability estimate directly from the Carleman estimates without any cut-off procedure or compactness-uniqueness argument.