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Standing waves of the Anderson-Gross-Pitaevskii equation

Samaël Mackowiak

TL;DR

This work analyzes standing waves for the Anderson-Gross-Pitaevskii equation in dimension 1 and 2, incorporating a confining Hermite potential and spatial white noise. The authors develop a rigorous operator-theoretic framework for the renormalized Anderson operator; in 1D this is achieved via a quadratic-form construction, while in 2D the approach uses an exponential transform and Wick renormalization to define a lower-bounded self-adjoint operator with compact resolvent. Standing waves are constructed through energy minimization at fixed mass and, in the focusing case, via action minimization on the Nehari manifold, yielding positive, orbitally stable ground states in the defocusing or subcritical regimes and precise descriptions of mass dependence and small-mass asymptotics. A key feature is the demonstration of exponential localization for the waves and a detailed treatment of the critical and supercritical regimes, including the introduction of a noisy critical mass in dimension two. These results advance understanding of nonlinear Schrödinger dynamics in random, confining environments and provide a foundation for Bose-Einstein condensates in inhomogeneous media.

Abstract

In this paper, we study standing waves for the Anderson-Gross-Pitaevskii equation in dimension 1 and 2. The Anderson-Gross-Pitaevskii equation is a nonlinear Schrödinger equation with a confining potential and a multiplicative spatial white noise. Standing waves are characterized by a profile which is invariant by the dynamic and solves a nonlinear elliptic equation with spatial white noise potential. We construct such solutions via variational methods and obtain some results on their regularity, localization and stability.

Standing waves of the Anderson-Gross-Pitaevskii equation

TL;DR

This work analyzes standing waves for the Anderson-Gross-Pitaevskii equation in dimension 1 and 2, incorporating a confining Hermite potential and spatial white noise. The authors develop a rigorous operator-theoretic framework for the renormalized Anderson operator; in 1D this is achieved via a quadratic-form construction, while in 2D the approach uses an exponential transform and Wick renormalization to define a lower-bounded self-adjoint operator with compact resolvent. Standing waves are constructed through energy minimization at fixed mass and, in the focusing case, via action minimization on the Nehari manifold, yielding positive, orbitally stable ground states in the defocusing or subcritical regimes and precise descriptions of mass dependence and small-mass asymptotics. A key feature is the demonstration of exponential localization for the waves and a detailed treatment of the critical and supercritical regimes, including the introduction of a noisy critical mass in dimension two. These results advance understanding of nonlinear Schrödinger dynamics in random, confining environments and provide a foundation for Bose-Einstein condensates in inhomogeneous media.

Abstract

In this paper, we study standing waves for the Anderson-Gross-Pitaevskii equation in dimension 1 and 2. The Anderson-Gross-Pitaevskii equation is a nonlinear Schrödinger equation with a confining potential and a multiplicative spatial white noise. Standing waves are characterized by a profile which is invariant by the dynamic and solves a nonlinear elliptic equation with spatial white noise potential. We construct such solutions via variational methods and obtain some results on their regularity, localization and stability.
Paper Structure (14 sections, 43 theorems, 188 equations)

This paper contains 14 sections, 43 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. Hermite-Sobolev spaces
  5. White noise
  6. Confining Anderson operator in dimension 1
  7. Confining Anderson operator in dimension 2
  8. A priori properties of standing waves
  9. Exponential localization
  10. Energy ground-states
  11. Existence of energy ground-states
  12. Dependence on the mass
  13. The (super-)critical focusing case
  14. Action ground-states

Key Result

Theorem 1.1

Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$ and $\gamma>0$. Denote by $\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of Eq-StationaryAGP in $\mathrm{D}\left(\sqrt{|A|}\right)$, In every other cases, there is no non-trivial non-negative solution in $\mathrm{D}\left(\sqrt{|A|}\right)$.

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Corollary 2.6
  • ...and 63 more