# Random Hamiltonians with Arbitrary Point Interactions

## TL;DR

This paper studies disordered Hamiltonians with random singular point interactions, establishing a dichotomy between purely absolutely continuous spectrum and localization, including Anderson localization for Bernoulli-type potentials.

## Contribution

It introduces a general framework for Hamiltonians with arbitrary singular point interactions and proves a spectral localization dichotomy under minimal assumptions.

## Key findings

- Spectral and dynamical localization can occur in these models.
- Purely absolutely continuous spectrum is also possible in the same setting.
- Anderson localization is established for Bernoulli-type singular potentials.

## Abstract

We consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the real line. Under minimal assumptions on the type of disorder, we prove the following dichotomy: Either every realization of the random operator has purely absolutely continuous spectrum or spectral and exponential dynamical localization hold. In particular, we establish Anderson localization for Schr\"odinger operators with Bernoulli-type random singular potential and singular density.