Random Hamiltonians with Arbitrary Point Interactions

2019/07/22 by David Damanik, Damanik, David, Jake Fillman +7
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1907.09530

openalex publication_date 2019/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Schrödinger operators with Bernoulli-type random singular potential and singular density.

Citations

• Holder Continuity of the Integrated Density of States for Quasi-Periodic Schrodinger Equations and Averages of Shifts of Subharmonic Functions
• Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

Related

• Random Schrödinger Operators on discrete structures
• Anderson Hamiltonians with singular potentials
• Anderson Localization for Time Periodic Random Schrödinger operators
• An introduction to multiscale techniques in the theory of Anderson localization. Part I
• Localization for Random Schrödinger Operators Defined by Block Factors