Absolutely continuous spectrum for the Anderson model on a product of a tree with a finite graph

We prove the almost sure existence of absolutely continuous spectrum at low disorder for the Anderson model on the simplest example of a product of a regular tree with a finite graph. This graph contains loops of unbounded size.Comment: 30 pages, 2 figure

Absolutely continuous spectrum for a random potential on a tree with strong transverse correlations and large weighted loops

We consider random Schr\"odinger operators on tree graphs and prove absolutely continuous spectrum at small disorder for two models. The first model is the usual binary tree with certain strongly correlated random potentials. These potentials are of interest since for complete correlation they exhibit localization at all disorders. In the second model we change the tree graph by adding all possible edges to the graph inside each sphere, with weights proportional to the number of points in the sphere.Comment: 25 pages, 4 figure

Counting eigenvalues of Schr\"odinger operators using the landscape function

We prove an upper and a lower bound on the rank of the spectral projections of the Schr\"odinger operator $-\Delta + V$ in terms of the volume of the sublevel sets of an effective potential $\frac{1}{u}$. Here, $u$ is the `landscape function' of [(David, G., Filoche, M., & Mayboroda, S. (2021) Advances in Mathematics, 390, 107946)], namely a solution of $(-\Delta + V)u = 1$ in \bbR^d. We prove the result for non-negative potentials satisfying a Kato-type and a doubling condition, in all spatial dimensions, in infinite volume, and show that no coarse graining is required. Our result yields in particular a necessary and sufficient condition for discreteness of the spectrum. In the case of polynomial potentials, we prove that the spectrum is discrete if and only if no directional derivative vanishes identically.Comment: To appear in Journal of Spectral Theor