Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Construction and spectrum of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$

Yueh-Sheng Hsu, Cyril Labbé

TL;DR

This work provides a rigorous construction of the Anderson Hamiltonian $\mathcal{H}=-\Delta+\xi$ with white-noise potential on $\mathbb{R}^2$ and $\mathbb{R}^3$ by leveraging the parabolic Anderson model and a Klein–Landau semigroup framework. It renormalizes the ill-defined distributional potential, builds the PAM semigroup, and obtains a self-adjoint generator as the strong resolvent limit of renormalized operators; in 2D and 3D, it is shown that the spectrum is almost surely the entire real line. The analysis hinges on enhanced noise spaces for PAM in 2D, and regularity-structures techniques for PAM in 3D, together with a Kotani-type argument that transfers Weyl-sequence reasoning to the singular setting. The results offer a robust, broadly applicable methodology for singular random differential operators beyond the present model, with implications for spectral theory and stochastic PDE renormalization.

Abstract

We propose a simple construction of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$ based on the solution theory of the parabolic Anderson model. It relies on a theorem of Klein and Landau [KL81] that associates a unique self-adjoint generator to a symmetric semigroup satisfying some mild assumptions. Then, we show that almost surely the spectrum of this random Schrödinger operator is $\mathbf{R}$. To prove this result, we extend the method of Kotani [Kot85] to our setting of singular random operators.

Construction and spectrum of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$

TL;DR

This work provides a rigorous construction of the Anderson Hamiltonian with white-noise potential on and by leveraging the parabolic Anderson model and a Klein–Landau semigroup framework. It renormalizes the ill-defined distributional potential, builds the PAM semigroup, and obtains a self-adjoint generator as the strong resolvent limit of renormalized operators; in 2D and 3D, it is shown that the spectrum is almost surely the entire real line. The analysis hinges on enhanced noise spaces for PAM in 2D, and regularity-structures techniques for PAM in 3D, together with a Kotani-type argument that transfers Weyl-sequence reasoning to the singular setting. The results offer a robust, broadly applicable methodology for singular random differential operators beyond the present model, with implications for spectral theory and stochastic PDE renormalization.

Abstract

We propose a simple construction of the Anderson Hamiltonian with white noise potential on and based on the solution theory of the parabolic Anderson model. It relies on a theorem of Klein and Landau [KL81] that associates a unique self-adjoint generator to a symmetric semigroup satisfying some mild assumptions. Then, we show that almost surely the spectrum of this random Schrödinger operator is . To prove this result, we extend the method of Kotani [Kot85] to our setting of singular random operators.
Paper Structure (14 sections, 21 theorems, 146 equations, 1 figure)

This paper contains 14 sections, 21 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Self-adjoint generator of unbounded symmetric semigroup
  4. The Anderson Hamiltonian in dimension two
  5. The space of enhanced noise and the white noise case
  6. The parabolic evolution
  7. The semigroup
  8. Characterization of the support
  9. The Anderson Hamiltonian in dimension three
  10. The space of enhanced noises and the white noise case
  11. The parabolic evolution
  12. The self-adjoint operator
  13. Inclusion of the supports
  14. Identification of the spectrum

Key Result

Theorem 1

In dimensions $2$ and $3$, there exists a random operator $\mathcal{H}$ which is self-adjoint on (a dense subset of) $L^2({{\hbox{\bfR}}}^d,dx)$ and which is the limit in probability of $\mathcal{H}_{\epsilon}$ as $\epsilon\downarrow 0$ in the strong resolvent sense. For any $t\ge 0$, the domain of

Figures (1)

  • Figure 1: Basis vectors and homogeneities in the regularity structure.

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Theorem 3: KL81
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 4
  • proof
  • Definition 3.3
  • Remark 3.4
  • ...and 43 more