Localization of the 1D Non-Stationary Anderson Model

2025/12/20 by Karl Zieber, Zieber, Karl
Mathematics · Physics and Astronomy · #35J10 #47B80 (Secondary) #81Q10 (Primary) 37H99 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.2512.18520

openalex publication_date 2025/12/20 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28

Abstract

This paper considers the family of Schrödinger operators on ℓ²(ℤ) given by independent but not necessarily identically distributed and possibly unbounded potentials. We assume a finite exponential moment and allow the choice of distributions to come from any compact set away from deterministic distributions. With these assumptions we prove spectral localization with exponentially decaying eigenfunctions as well as dynamical localization. One of the main tools is a Furstenberg-type theorem for non-stationary matrix products.

Citations

• Localization for heavy-tailed Anderson models
• Localization and unique continuation for non-stationary Schrödinger operators on the 2D lattice
• Non-stationary Anderson Localization
• Non-stationary version of Furstenberg Theorem on random matrix products
• Hölder regularity of stationary measures
• Parametric Furstenberg Theorem on random products of S L ( 2 , R ) matrices
• Localization for Anderson Models on Metric and Discrete Tree Graphs
• Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
• Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
• Shnol's theorem and the spectrum of long range operators
• Schrödinger operators with dynamically defined potentials
• A Short Course on One-Dimensional Random Schrödinger Operators
• On localization in the continuous Anderson-Bernoulli model in higher dimension
• Anderson localization for Bernoulli and other singular potentials

Related

• Non-stationary Anderson Localization
• Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
• Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
• Localization for random operators with non-monotone potentials with exponentially decaying correlations
• On the Kunz-Souillard approach to localization for the discrete one dimensional generalized Anderson model