There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda at the International Congress of Mathematicians 2018 (towards the end): https://www.youtube.com/watch?v=FhPsWJL9eNQ

Namely, she considers the following eigenvalue problem (otherwise known as the Schrödinger equation): $$ [-\Delta + V(x)] \psi(x) = E\psi(x) $$ with $x\in \Omega \subset \mathbb{R}^d$ and $\psi({x})\Bigr|_{\partial \Omega}=0$,and where $V(x)$ is a random potential (in some sense, as defined in the paper). I.e., the potential has many valleys of random location and possibly random depth, but the exact form of randomness appears unimportant.

The statement seems to be that if we solve instead the following much simpler problem $$ [-\Delta + V(x)] u(x) = 1, \mbox{with } u(x)\Bigr|_{\partial \Omega}=0 $$ then the $n$-th consecutive minimum of the function, $u^{-1}(x)$, dubbed localization landscape will determine with great accuracy (there is no exact statement) the $n$-consecutive eigenvalue as follows $$ E_n \approx (1 + d/4) \inf_x u^{-1}(x)|_n $$

I wonder if there are experts here in ODE, etc, who could comment on the status of this statement/conjecture and in general this localization landscape perspective.

The conjecture seems suspicious to me, because diagonalizing and inverting operators are in different computational complexity classes (the latter - required for finding $u$ - is much simpler). But if it's actually true, it would have important implications for physics (I am a physicist).