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Landau Hamiltonian with Gaussian white noise potential and the asymptotic of its bottom of spectrum

Yueh-Sheng Hsu

TL;DR

This work constructs a rigorous, renormalized Landau-type Hamiltonian in 2D with a Gaussian white-noise potential under a broad class of magnetic fields by combining an exponential transformation with a semigroup (Hille–Yosida) approach, avoiding heavier SPDE machinery. It proves the operator exists on bounded domains with Dirichlet boundary conditions and has a compact resolvent, enabling a pure-point spectrum. The authors extend known bottom-of-spectrum asymptotics from the non-magnetic continuous Anderson model to the magnetic setting, showing the lowest eigenvalues on large boxes diverge like −$C_{ m GN}$ log L under mild growth conditions on the magnetic potential A; this relies on a detailed large deviation framework for the random models and a meticulous tail bound argument. The paper also discusses scaling and translation properties, and provides concrete instances such as a uniform magnetic field and a Gaussian free field-driven magnetic field, highlighting the robustness of the method for magnetic perturbations.

Abstract

We present a simple construction of a random Schrödinger operator subject to a magnetic field with a regularity as low as $0^-$-Hölder and a Gaussian white noise electric potential on a two-dimensional bounded box. This construction is based on the exponential Ansatz introduced in [HL15] and leverages the semigroup approach developed in [HL24]. The proposed construction enables us to generalise an asymptotic result for the bottom of the spectrum of the two-dimensional continuous Anderson Hamiltonian, first proved in [CvZ21], to the magnetic case. Our choice of potential not only covers the case of a uniform magnetic field, but also those which would break translational invariance.

Landau Hamiltonian with Gaussian white noise potential and the asymptotic of its bottom of spectrum

TL;DR

This work constructs a rigorous, renormalized Landau-type Hamiltonian in 2D with a Gaussian white-noise potential under a broad class of magnetic fields by combining an exponential transformation with a semigroup (Hille–Yosida) approach, avoiding heavier SPDE machinery. It proves the operator exists on bounded domains with Dirichlet boundary conditions and has a compact resolvent, enabling a pure-point spectrum. The authors extend known bottom-of-spectrum asymptotics from the non-magnetic continuous Anderson model to the magnetic setting, showing the lowest eigenvalues on large boxes diverge like − log L under mild growth conditions on the magnetic potential A; this relies on a detailed large deviation framework for the random models and a meticulous tail bound argument. The paper also discusses scaling and translation properties, and provides concrete instances such as a uniform magnetic field and a Gaussian free field-driven magnetic field, highlighting the robustness of the method for magnetic perturbations.

Abstract

We present a simple construction of a random Schrödinger operator subject to a magnetic field with a regularity as low as -Hölder and a Gaussian white noise electric potential on a two-dimensional bounded box. This construction is based on the exponential Ansatz introduced in [HL15] and leverages the semigroup approach developed in [HL24]. The proposed construction enables us to generalise an asymptotic result for the bottom of the spectrum of the two-dimensional continuous Anderson Hamiltonian, first proved in [CvZ21], to the magnetic case. Our choice of potential not only covers the case of a uniform magnetic field, but also those which would break translational invariance.

Paper Structure

This paper contains 11 sections, 14 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Construction of $H$ on bounded open box
  4. Construction of $R_t^B$
  5. Construction of $Q$
  6. Asymptotic of $\lambda^L_n$ as $L \to \infty$
  7. Intermediate properties
  8. Deterministic bounds on the eigenvalues
  9. Rescaling
  10. Large deviation principle
  11. Proof of the tail bound Theorem \ref{['thm:ev_tail_estimate']}

Key Result

Theorem 1

For $\kappa \in (0, 1/4)$, let $\mathbf{A} \in \mathcal{C}^{1-\kappa}_{\mathrm{loc}}({{\hbox{\bfR}}}^2; {{\hbox{\bfR}}}^2)$. Fix a bounded open box $B \subset {{\hbox{\bfR}}}^2$. Then, there exists a random operator $H$ on $L^2(B; {{\hbox{\bfC}}})$ which, almost surely, satisfies the zero Dirichlet This operator $H$ is therefore regarded as a realisation of the formal expression $(i\nabla + \math

Figures (1)

  • Figure 1: Scheme for the construction of $H$ and its eigenvalues $\lambda_n, n \ge 1$. Here, $\Omega$ denotes the canonical sample space for $\xi$, $\mathscr{A}$ is a function space accommodating the field $\mathbf{A}$ (in our case, $\mathscr{A} = \mathcal{C}^{1-\kappa}_{\mathrm{loc}}({{\hbox{\bfR}}}^2;{{\hbox{\bfR}}}^2)$), and $\mathscr{M}$ is an auxiliary space designed to accommodate the models$Q$ (in our case, $\mathscr{M} = \mathcal{C}^{1-\kappa}_{\mathrm{loc}}({{\hbox{\bfR}}}^2) \times \mathcal{C}^{-\kappa}_{\mathrm{loc}}({{\hbox{\bfR}}}^2))$.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Example 1.1: Uniform magnetic field
  • Example 1.2: Gaussian free field
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Definition 2.3
  • Corollary 2.4
  • ...and 19 more