Construction of Solutions with Extraordinary Gradient Amplification and Localization for Schrödinger Equations

TL;DR

The paper constructs solutions to linear and nonlinear Schrödinger-type equations in $d=2,3$ with coefficients supported in a bounded domain $D$ that develop prescribed, highly localized gradient amplification near selected points on $\partial D$ for almost every $t\in[0,T]$, while maintaining interior regularity. The core approach builds a exterior input $u_0(\mathbf{x},t)=H_g(\mathbf{x})e^{-it}$ from transmission eigenfunctions and Herglotz approximations, and corrects it via a small exterior term to realize gradient amplification near the chosen points; for the nonlinear case, a Banach fixed-point argument extends the construction to arbitrary times $T$. The main results (linear and nonlinear) guarantee that the amplified regions shrink in measure as the amplification threshold $\mathcal{M}$ grows, establishing a deterministic analogue of localization phenomena in quantum dynamics and illustrating a trade-off between extreme localization and large gradients in Schrödinger evolution. The work combines spectral analysis, interior transmission theory, and regularity theory to produce a flexible framework applicable to broad classes of Schrödinger-type equations with anisotropic interior structure.

Abstract

This paper constructs solutions to linear and nonlinear Schrödinger-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 Schrödinger-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 Schrödinger dynamics.

Figures (1)

• Figure 1: Geometric configuration where each external point $\mathbf{y}_i$ is placed at a uniform distance from its corresponding boundary point $\mathbf{x}_i \in \partial D$.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Lemma 2.1
• Lemma 2.2
• Remark 2.3
• Theorem 3.1