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Unique continuation properties for the continuous Anderson operator in dimension 2

Nicolas Moench

TL;DR

The paper studies unique continuation properties for singular continuous Anderson operators $H=\Delta+\xi$ on 1D and 2D closed manifolds. It develops a two-pronged approach: in dimension 1, a Carleman-based conjugation by the ground state yields strong unique continuation and a control-theoretic spectral inequality; in dimension 2, a Beltrami/quasiconformal framework via the ground state conjugation establishes strong unique continuation and shows that eigenfunction nodal sets are locally quasi-conformal to Laplace nodal sets, yielding a Courant-type nodal bound. The work combines enhanced-noise renormalization, divergence-form reformulations, and complex-analytic tools (Ahlfors–Bers, Mori) to transfer elliptic-regularity results to the rough setting, and proves exact null-controllability in 1D via spectral inequalities. These results extend deterministic PDE continuation theory to rough stochastic operators, with implications for spectral geometry and control in singular settings.

Abstract

We consider singular continuous Anderson operators $H=Δ+ξ$ on closed manifolds of dimension 1 and 2, and prove a unique continuation property for its eigenfunctions using the theory of quasi-conformal mappings. We investigate its nodal set by proving that it is quasi-conformal to the nodal set of a Laplace eigenfunction and prove a Courant nodal theorem. We also present an application to control for singular operator in dimension 1.

Unique continuation properties for the continuous Anderson operator in dimension 2

TL;DR

The paper studies unique continuation properties for singular continuous Anderson operators on 1D and 2D closed manifolds. It develops a two-pronged approach: in dimension 1, a Carleman-based conjugation by the ground state yields strong unique continuation and a control-theoretic spectral inequality; in dimension 2, a Beltrami/quasiconformal framework via the ground state conjugation establishes strong unique continuation and shows that eigenfunction nodal sets are locally quasi-conformal to Laplace nodal sets, yielding a Courant-type nodal bound. The work combines enhanced-noise renormalization, divergence-form reformulations, and complex-analytic tools (Ahlfors–Bers, Mori) to transfer elliptic-regularity results to the rough setting, and proves exact null-controllability in 1D via spectral inequalities. These results extend deterministic PDE continuation theory to rough stochastic operators, with implications for spectral geometry and control in singular settings.

Abstract

We consider singular continuous Anderson operators on closed manifolds of dimension 1 and 2, and prove a unique continuation property for its eigenfunctions using the theory of quasi-conformal mappings. We investigate its nodal set by proving that it is quasi-conformal to the nodal set of a Laplace eigenfunction and prove a Courant nodal theorem. We also present an application to control for singular operator in dimension 1.
Paper Structure (9 sections, 56 equations)

This paper contains 9 sections, 56 equations.

Table of Contents

  1. Introduction
  2. Continuous Anderson operators.
  3. Unique continution.
  4. Quasiregular mappings on the plane.
  5. An application to control theory.
  6. Strong unique continuation in dimension 1.
  7. The case of dimension 2.
  8. Proof of theorems \ref{['thm_cont']} and \ref{['thm_quasiconformal']}.
  9. A spectral inequality for the Anderson operator in dimension 1.

Theorems & Definitions (5)

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