Boundary observation of a Schrödinger field with slower surface dispersion
On the annulus $\Omega = \{x \in \mathbb{R}^2 : 1 < |x| < 2\}$ let $\Gamma_1$ and $\Gamma_0$ be the inner and outer circles, and $\nu$ the unit normal pointing out of $\Omega$. Consider complex solutions $Z = (z, z_\Gamma)$ of $$i z_t + \Delta z = 0 \ \text{ in } \Omega, \qquad z = 0 \ \text{ on } \Gamma_0,$$ $$z_\Gamma = z|_{\Gamma_1}, \qquad i z_{\Gamma,t} + \tfrac12 \Delta_\Gamma z_\Gamma - \partial_\nu z = 0 \ \text{ on } \Gamma_1,$$ where $\Delta_\Gamma$ is the Laplace–Beltrami operator of $\Gamma_1$, with the energy $$\|Z\|_{\mathcal{E}}^2 = \int_\Omega |\nabla z|^2 \, dx + \tfrac12 \int_{\Gamma_1} |\nabla_\Gamma z_\Gamma|^2 \, dS.$$ This is the case $d = 1$, $\delta = 1/2$ of the system $i z_t + d \Delta z = 0$, $i z_{\Gamma,t} + \delta \Delta_\Gamma z_\Gamma - d \partial_\nu z = 0$, with bulk coefficient $d$ and surface coefficient $\delta$. For domains of this shape (an outer domain minus a strongly convex inner one), Mercado and Morales proved a Carleman estimate and exact controllability by a control on the outer boundary when $\delta > d$, a hypothesis they use to absorb the surface gradient term.
Problem. Is it true that for every $T > 0$ a constant $C_T$ satisfies $$\|Z(0)\|_{\mathcal{E}}^2 \le C_T \int_0^T \int_{\Gamma_0} |\partial_\nu z|^2 \, dS \, dt$$ for every smooth compatible solution?
Source: Chorfi, Maniar and Morales (2025), §5.1, for exact controllability of the Schrödinger equation with dynamic boundary conditions when the surface coefficient is smaller than the bulk one; stated as an observability inequality for one annulus and coefficient 1/2 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, dynamic boundary conditions, observability, Schrödinger equation
References
@article{mercado2023,
author = {Mercado, Alberto and Morales, Roberto},
title = {Exact controllability for a {S}chr{\"o}dinger equation with dynamic boundary conditions},
journal = {SIAM J. Control Optim.},
volume = {61},
number = {6},
year = {2023},
pages = {3501--3525},
doi = {10.1137/21M1439407},
eprint = {2301.02573},
archivePrefix = {arXiv},
}
@misc{chorfi2025,
author = {Chorfi, S. E. and Maniar, L. and Morales, R.},
title = {Controllability and inverse problems for hyperbolic and dispersive equations with dynamic boundary conditions},
year = {2025},
eprint = {2505.14795},
archivePrefix = {arXiv},
}
@misc{aim-slow-surface-schrodinger-observability,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Boundary observation of a Schrödinger field with slower surface dispersion}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/386-slow-surface-schrodinger-observability.md},
note = {Entry 386 at commit a153f46, 2026-10-13},
}
Resolution claims
We consider a Schrödinger field on a planar annulus with a dynamic inner boundary and a Dirichlet outer boundary. When the surface dispersion coefficient is one half of the bulk coefficient, observation of the outer normal derivative fails to control the initial energy at every positive time. We construct smooth solutions with unit initial energy and exponentially small observation on …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/exterior_observability.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00772, "Boundary observation of a Schrödinger field with slower surface dispersion", clopen.solutions.