The continuous Anderson hamiltonian in $d\le 3$

Abstract

We construct the continuous Anderson hamiltonian on $(-L,L{)}^d$ driven by a white noise and endowed with either Dirichlet or periodic boundary conditions. Our construction holds in any dimension $d\le 3$ and relies on the theory of regularity structures: it yields a self-adjoint operator in $L^2((-L,L{)}^d)$ with pure point spectrum. In $d\ge 2$, a renormalisation of the operator by means of infinite constants is required to compensate for ill-defined products involving functionals of the white noise. We also obtain left tail estimates on the distributions of the eigenvalues: in particular, for $d=3$ these estimates show that the eigenvalues do not have exponential moments.

arXiv comment38 pages

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Cited by in corpus (17)

• Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions
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• Integrated density of states of the Anderson Hamiltonian with two-dimensional white noise
• Local and global well-posedness of 2d periodic multiplicative stochastic NLS
• Boundary renormalisation of SPDEs
• Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
• Hairer's multilevel Schauder estimates without Regularity Structures
• Phase Transitions in Asymptotically Singular Anderson Hamiltonian and Parabolic Model
• Finite speed of propagation for the 2- and 3-dimensional multiplicative stochastic wave equation
• Singular SPDEs on Homogeneous Lie Groups
• Global well-posedness for the nonlinear generalized parabolic Anderson model equation
• Asymptotic of the smallest eigenvalues of the continuous Anderson Hamiltonian in $d\le 3$
• Constraint Minimization Problem of the Nonlinear Schrödinger Equation with the Anderson Hamiltonian
• Introduction to SPDEs from Probability and PDE
• Anderson Hamiltonians with singular potentials
• Construction and spectrum of the Anderson Hamiltonian with white noise potential on ${\mathbf{R}}^2$ and ${\mathbf{R}}^3$
• The 1D Schrödinger equation with a spacetime white noise: the average wave function