I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here.
The question is mostly related to homogenization theory in mathematical physics.
$\textbf{Background}$
I will describe the periodic case first and then generalize it to the random case later.
Let's fix a domain $\Omega \subset \mathbb{R}^d$. Define $\Omega_\varepsilon$ to be a domain with holes of radius $a_\varepsilon$ cut out periodically with period $\varepsilon$. Take a look at the picture for reference.
Now consider the laplacian on this cut out domain as follows: \begin{equation*} \begin{split} -\Delta u_\varepsilon&=f \hspace{1em} \text{ in } \Omega_\varepsilon\\ u_\varepsilon&=0 \hspace{1em} \text{ on } \partial \Omega_\varepsilon \end{split} \end{equation*} Not that the boundary conditions also imply that the solution is zero on the holes.
Now as we take $\varepsilon \to 0$ and if we have
$$ a_\varepsilon= \begin{cases} exp(\frac{-C_0}{\varepsilon^2}), \text{ if } d=2\\ C_0\varepsilon^{\frac{d}{d-2}}, \text{ if } d>2 \end{cases} $$
Then we have that $u_\varepsilon$ converges weakly to $u$ in $H^1_0(\Omega)$ where $u$ satisfies the following equation
\begin{equation*} \begin{split} -\Delta u+\mu u&=f \hspace{1em} \text{ in } \Omega\\ u&=0 \hspace{1em} \text{ on } \partial \Omega \end{split} \end{equation*}
Here $\mu$ is a specific constant which can be worked out explicitly. The interesting thing is that there is an extra term which appears to be coming out of nowhere. More on this can be found in the paper https://link.springer.com/chapter/10.1007/978-1-4612-2032-9_4.
Or here https://www.math.u-bordeaux.fr/~cprange/documents/coursEDMI2016_lecture2.pdf
What is also interesting is that if we play around with $a_\varepsilon$, and consider $a_\varepsilon$ to be smaller than the scaling provided above then the limiting function solves the usual laplacian equation without the $\mu$. If $a_\varepsilon$ is larger then the limiting function is zero.
Since this stuff is mostly qualitative it's an interesting question to see what are the quantitative rates for the convergence and what is the best Sobolev space in which we get convergence.
We recently considered this problem of quantifying this periodic case and a random case with appropriate assumptions which roughly guarantee us that we would not have a lot of holes in a specific region via a large deviation bound assumption(think of poisson point cloud as a representative example).
We were able to prove that even in the random case, under the same scaling we get the strange term. Though in this case, it will be a random variable. And if the scaling is changed then we get the Laplace's equation or the zero function a.s.
The same question can be seen as a random walk and the holes to be traps, i.e the random walker dies when it hits the boundary of the hole.
Question
Now let's consider the Schrodinger's equation with random potential on $\Omega$. The random potentials are supported on the random holes and are such that if an electron gets in, it essentially stays trapped(I'm thinking of an infinite potential well).
Now intuitively, a similar trichotomy for the limiting function(if it exists!) should hold ie: diffusion, Brinkmann type equation and localization.
I was wondering:
If this is at all related to the mighty Anderson Localization problem?
If so, would it be interesting to see if such a limiting object exists and if so, how fast does the convergence occur?
