Observability of a 1D Schrödinger equation with time-varying boundaries

Exact observability properties of subelliptic wave and Schr{ö}dinger equations

In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{ö}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.

平面上具有周期有界势的薛定谔方程从可测集出发的可观测性估计

Observability estimates for the Schr{ö}dinger equation in the plane with periodic bounded potentials from measurable sets

本文证明了平面上具有 2πZ²-周期有界势的薛定谔方程从任意周期可测集在任意小时间内可观测，并利用 Floquet-Bloch 变换、Strichartz 估计和半经典缺陷测度将 Taüffer 的二维结果推广到更一般的情形。

Global Optimization via Schr{ö}dinger-F{ö}llmer Diffusion

We study the problem of finding global minimizers of $V(x):\mathbb{R}^d\rightarrow\mathbb{R}$ approximately via sampling from a probability distribution $μ_σ$ with density $p_σ(x)=\dfrac{\exp(-V(x)/σ)}{\int_{\mathbb R^d} \exp(-V(y)/σ) dy }$ with respect to the Lebesgue measure for $σ\in (0,1]$ small enough. We analyze a sampler based on the Euler-Maruyama discretization of the Schr{ö}dinger-F{ö}llmer diffusion processes with stochastic approximation under appropriate assumptions on the step size $s$ and the potential $V$. We prove that the output of the proposed sampler is an approximate globa…

Observability and quantum limits for the Schrödinger equation on the sphere

In this note, we describe our recent results on semiclassical measures for the Schr{ö}dinger evolution on Zoll manifolds. We focus on the particular case of eigenmodes of the Schr{ö}dinger operator on the sphere endowed with its canonical metric. We also recall the relation of this problem with the observability question from control theory. In particular, we exhibit examples of open sets and potentials on the 2-sphere for which observability fails for the evolution problem while it holds for the stationary one. Finally, we give some new results in the case where the Radon transform of the pot…

Observability for generalized Schrödinger equations and quantum limits on product manifolds

Given a closed product Riemannian manifold N = M x M equipped with the product Riemannian metric g = h + h , we explore the observability properties for the generalized Schr{ö}dinger equation i$\partial$ t u = F (g)u, where g is the Laplace-Beltrami operator on N and F : [0, +$\infty$) $\rightarrow$ [0, +$\infty$) is an increasing function. In this note, we prove observability in finite time on any open subset $ω$ satisfying the so-called Vertical Geometric Control Condition, stipulating that any vertical geodesic meets $ω$, under the additional assumption that the spectrum of F (g) satisfies …

Approximate controllability via adiabatic techniques for the three-inputs controlled Schr{ö}dinger equation

We consider a system described by a controlled bilinear Schr{ö}dinger equation with three external inputs. We provide a constructive method to approximately steer the system from a given energy level to a superposition of energy levels corresponding to a given probability distribution. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian admits eigenvalue intersections, with respect to variations of the controls, and if the latter are conical. We provide sharp estimates of the relation between the error and the controllability time, and we show how to improv…