Anderson Hamiltonians with singular potentials

Abstract

We construct random Schrödinger operators, called Anderson Hamiltonians, with Dirichlet and Neumann boundary conditions for a fairly general class of singular random potentials on bounded domains. Furthermore, we construct the integrated density of states of these Anderson Hamiltonians, and we relate the Lifschitz tails (the asymptotics of the left tails of the integrated density of states) to the left tails of the principal eigenvalues.

The main text contains 50 pages. The appendices contain 32 pages. Reorganisation and relocation of some text: most notably on the main assumptions: see the introduction, beginning Section 1.4 and Section 1.5; and Section 3

References in corpus (13)

• A theory of regularity structures
• A Wong-Zakai theorem for stochastic PDEs
• A simple construction of the continuum parabolic Anderson model on ${\mathbf{R}}^2$
• Multiplicative stochastic heat equations on the whole space
• Global solutions to elliptic and parabolic ${\Phi }^4$ models in Euclidean space
• The continuous Anderson hamiltonian in $d\le 3$
• The reconstruction theorem in Besov spaces
• Rough differential equations driven by signals in Besov spaces
• Localization of the continuous Anderson Hamiltonian in $1$-d
• Integrated density of states of the Anderson Hamiltonian with two-dimensional white noise
• Boundary renormalisation of SPDEs
• Finite speed of propagation for the 2- and 3-dimensional multiplicative stochastic wave equation
• Anderson Hamiltonian and associated Nonlinear Stochastic Wave and Schrödinger equations in the full space