Construction of Solutions with Extraordinary Gradient Amplification and Localization for Schr\"odinger Equations

TL;DR

This paper constructs solutions to Schr"odinger equations that exhibit extreme, localized gradient amplification near prescribed points, illustrating a deterministic analogue of quantum localization phenomena.

Contribution

It demonstrates how to design initial and boundary data to produce solutions with highly localized and amplified gradients outside the domain, revealing a trade-off consistent with the uncertainty principle.

Findings

Solutions exhibit gradient magnitudes exceeding any threshold outside the domain.

The regions of high gradient are highly localized, with measure tending to zero as the threshold increases.

The results connect deterministic Schr"odinger dynamics with quantum localization phenomena.

Abstract

This paper constructs solutions to linear and nonlinear Schr\"odinger-type equations in two and three spatial dimensions that exhibit prescribed, extraordinary gradient amplification and localization. For any finite time interval $[0,T]$, any prescribed collection of $n\in\mathbb{N}$ distinct points on $\partial D$, where $D$ is the compact support of the anisotropic coefficients, lower-order terms, or nonlinearities, and any amplitude threshold $\mathcal{M}>0$, we show that one can design smooth initial and/or boundary data such that the spatial gradients of the resulting solutions exceed $\mathcal{M}$ in neighborhoods of these points outside $D$ for almost every $t\in[0,T]$. Moreover, the ratio between the local $C^{1,\frac12}$-norm of the solution near each prescribed point outside $D$ and the $C^{1,\frac12}$-norm inside $D$ is bounded from below by $\mathcal{M}/2$ for almost every $t\in[0,T]$. We further prove that the spatial measure of the regions where the gradient magnitude exceeds $\mathcal{M}$ tends to zero as $\mathcal{M}\to\infty$, demonstrating that the amplification phenomenon is highly localized. This effect arises from the structure of the Schr\"odinger-type equation combined with carefully designed input profiles. From a physical perspective, the results provide a deterministic analogue of localization phenomena observed in quantum scattering and Anderson localization. In addition, the observed trade-off between extreme spatial localization and large gradient amplification is fully consistent with the spirit of the Heisenberg uncertainty principle: while the latter is traditionally formulated in a global $L^2$ space--frequency framework, our results offer a complementary deterministic manifestation at the level of localized spatial gradients in Schr\"odinger dynamics.