Random Sturm–Liouville operators

Random Sturm-Liouville operators

Selfadjoint Sturm-Liouville operators Hω on L2(a, b) with random potentials are considered and it is proven, using positivity conditions, that for almost every ω the operator Hω does not share eigenvalues with a broad family of random operators and in particular with operators generated in the same way as Hω but in L2(ã, b̃) where (ã, b̃) ⊂ (a, b).

Random Rearrangements and Operators

Localization for Random Unitary Operators

We consider unitary analogs of 1−dimensional Anderson models on l2(Z) defined by the product Uω = DωS where S is a deterministic unitary and Dω is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of Uω is pure point almost surely for all val...

Lectures on Random Schrödinger Operators

These notes are based on lectures given in the IV Escuela de Verano en Análisis y Fisica Matemática at the Instituto de Matemáticas, Cuernavaca, Mexico, in May 2005. It is a complete description of Anderson localization for random Schrödinger operators on L2(IRd) and basic properties of the integrated density of states for these operators. A discussion of localization for randomly perturbed Lan...

Random Schrödinger Operators on Manifolds

We consider a random family of Schrödinger operators on a cover X of a compact Riemannian manifold M = X/Γ. We present several results on their spectral theory, in particular almost sure constancy of the spectral components and existence and non-randomness of an integrated density of states. We also sketch a groupoid based general framework which allows to treat basic features of random operato...