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Semiclassical limit of cubic nonlinear Schrödinger equations for mixed states

Daniel Han-Kwan, Frédéric Rousset

TL;DR

The paper proves a rigorous semiclassical limit for the cubic nonlinear Schrödinger equation with mixed states, showing convergence to a singular Vlasov equation with a force term proportional to $\nabla_x\rho_f$. The authors develop a robust framework based on the Wigner transform, introducing an extended Wigner system to manage high-order derivatives, and construct a parametrix that is a matrix-valued Fourier integral operator. A key innovation is the quantum averaging lemma, which provides a derivative gain in the averaging process and yields uniform-in-$\varepsilon$ bounds under a quantum Penrose stability condition. Under uniform weighted Sobolev bounds and Penrose stability (which holds for small data or certain Maxwellian-like perturbations), the Wigner solutions exist on a fixed time interval, remain controlled as $\varepsilon\to 0$, and converge to the unique solution of the singular Vlasov-Benney equation, thereby justifying the quantum-to-classical transition for mixed states in this setting.

Abstract

In this work, we study the semiclassical limit of cubic Nonlinear Schrödinger equations for mixed states. We justify the limit to a singular Vlasov equation (in which the force field is proportional to the gradient of the density), for data with finite Sobolev regularity whose velocity profiles satisfy a quantum Penrose stability condition. This latter condition is always satisfied for small data (with a smallness condition independent of the semiclassical parameter) both in the focusing and the defocusing case, and for small perturbations of a large class of physically relevant examples in the defocusing case, such as local Maxwellian-like profiles.

Semiclassical limit of cubic nonlinear Schrödinger equations for mixed states

TL;DR

The paper proves a rigorous semiclassical limit for the cubic nonlinear Schrödinger equation with mixed states, showing convergence to a singular Vlasov equation with a force term proportional to . The authors develop a robust framework based on the Wigner transform, introducing an extended Wigner system to manage high-order derivatives, and construct a parametrix that is a matrix-valued Fourier integral operator. A key innovation is the quantum averaging lemma, which provides a derivative gain in the averaging process and yields uniform-in- bounds under a quantum Penrose stability condition. Under uniform weighted Sobolev bounds and Penrose stability (which holds for small data or certain Maxwellian-like perturbations), the Wigner solutions exist on a fixed time interval, remain controlled as , and converge to the unique solution of the singular Vlasov-Benney equation, thereby justifying the quantum-to-classical transition for mixed states in this setting.

Abstract

In this work, we study the semiclassical limit of cubic Nonlinear Schrödinger equations for mixed states. We justify the limit to a singular Vlasov equation (in which the force field is proportional to the gradient of the density), for data with finite Sobolev regularity whose velocity profiles satisfy a quantum Penrose stability condition. This latter condition is always satisfied for small data (with a smallness condition independent of the semiclassical parameter) both in the focusing and the defocusing case, and for small perturbations of a large class of physically relevant examples in the defocusing case, such as local Maxwellian-like profiles.
Paper Structure (45 sections, 44 theorems, 615 equations)

This paper contains 45 sections, 44 theorems, 615 equations.

Table of Contents

  1. Introduction
  2. The semiclassical nonlinear Schrödinger and Hartree equations
  3. The semiclassical Wigner equation
  4. Previous justifications of semiclassical limits of the Hartree equations
  5. The Vlasov--Benney equation and the Penrose stability condition
  6. Main result
  7. The quantum Penrose stability condition
  8. Notations
  9. Strategy and organization of the paper
  10. Strategy
  11. Organization of the paper
  12. Preliminaries for the Wigner equation
  13. Functional inequalities in weighted Sobolev spaces
  14. Local well-posedness of the Wigner equation
  15. The bootstrap argument
  16. ...and 30 more sections

Key Result

Theorem 1.4

Let $r\geq 2d + 2 \lfloor d/2 \rfloor + 8$ and $m \geq \min \left(10d + d/2+ 14 + r , 3d + 6 + 2r\right)$. Let $(f^0_{\varepsilon})_{\varepsilon \in (0,1]}$ a real-valued family of initial data for eq:wigner-intro that satisfies the following assumptions. A1. Uniform weighted Sobolev regularit A2. Uniform quantum Penrose stability. The family $(f^0_\varepsilon)_{\varepsilon \in (0,1]}$ sati

Theorems & Definitions (99)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 3.1
  • Lemma 3.2
  • Remark 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • ...and 89 more