Conjecture on semiclassical transport of localized data along curved interfaces

Let $y_0\in\Gamma$, let $\alpha_1,\alpha_2\in\mathbb{C}$, and let $\lambda_1,\lambda_2$ be defined by the decomposition

$$\begin{bmatrix}\alpha_1\alpha_2\end{bmatrix}=\lambda_1\begin{bmatrix}e^{-i\theta_0/2}\\-e^{i\theta_0/2}\end{bmatrix}+\lambda_2\begin{bmatrix}e^{-i\theta_0/2}\e^{i\theta_0/2}\end{bmatrix}.$$

Semiclassical transport conjecture. There exists $\beta<3/4$ such that, under the assumptions in, the solution $\Psi_t$ of

$$(\varepsilon D_t+H)\Psi_t=0,\qquad \Psi_0(x)=\frac{1}{\sqrt{\varepsilon}}\exp\left(-\frac{|x-y_0|^2}{2\varepsilon}\right)\begin{bmatrix}\alpha_1\alpha_2\end{bmatrix}$$

satisfies, uniformly for $\varepsilon\in(0,1]$ and $t>0$,

$$\Psi_t(x)=\frac{\lambda_1}{\sqrt{\varepsilon}}\exp\left(-\frac{|x-y_t|^2}{2\varepsilon}\right)\begin{bmatrix}e^{-i\theta_t/2}\\-e^{i\theta_t/2}\end{bmatrix}+\mathcal{O}_{L^2}\bigl(\varepsilon^{1/2}\langle t\rangle\bigr)+\mathcal{O}_{L^\infty}\bigl(\varepsilon^{-\beta}\langle t\rangle^{-1/2}\bigr).$$

This extends the preceding result from initial data aligned with the interface state to general semiclassically localized data. The component proportional to $\lambda_2$ is predicted to disperse along the interface, with rate $\varepsilon^{-1/4}t^{-1/2}$ in the linear case, while the surviving localized component follows the state represented by $\lambda_1$.