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On the Gross-Pitaevskii evolution linearized around the degree-one vortex

Jonas Luhrmann, Wilhelm Schlag, Sohrab Shahshahani

TL;DR

The paper analyzes the Gross-Pitaevskii equation linearized around the degree-one vortex under equivariant symmetry, focusing on the non-selfadjoint matrix operator i\uL. It establishes that the spectrum is the entire real line and derives a Stone-type formula for the evolution by carefully analyzing the resolvent in the complex plane near zero energy, where a zero-energy resonance induces a distinct L^2-norm growth. A complete construction of the distorted Fourier transform at small energies is carried out via a delicate combination of Weyl solutions near r=0 and r=\infty, Lyapunov-Perron fixed points, and a detailed control of the resolvent’s jump. The results yield both sharp L^2-growth bounds for frequency-localized evolution and a framework to read off dispersive behavior through the distorted Fourier representation, complementing concurrent dispersive analyses and enabling a path toward full asymptotic stability questions for the degree-one vortex. The methods hinge on handling strongly singular inverse-square-type potentials, embedding resonances, and implementing a bespoke Stone-type formula to accommodate non-selfadjoint spectral data. These contributions provide a rigorous spectral and scattering-theoretic foundation for the linearized GP dynamics around topological vortices and illuminate how zero-energy phenomena govern long-time evolution and potential stabilization strategies via projection away from L^2-growth modes.

Abstract

We study the evolution of the Gross-Pitaevskii equation linearized around the Ginzburg-Landau vortex of degree one under equivariant symmetry. Among the main results of this work, we determine the spectrum of the linearized operator, uncover a remarkable $L^2$-norm growth phenomenon related to a zero-energy resonance, and provide a complete construction of the distorted Fourier transform at small energies. The latter hinges upon a meticulous analysis of the behavior of the resolvent in the upper and lower half-planes in a small disk around zero-energy.

On the Gross-Pitaevskii evolution linearized around the degree-one vortex

TL;DR

The paper analyzes the Gross-Pitaevskii equation linearized around the degree-one vortex under equivariant symmetry, focusing on the non-selfadjoint matrix operator i\uL. It establishes that the spectrum is the entire real line and derives a Stone-type formula for the evolution by carefully analyzing the resolvent in the complex plane near zero energy, where a zero-energy resonance induces a distinct L^2-norm growth. A complete construction of the distorted Fourier transform at small energies is carried out via a delicate combination of Weyl solutions near r=0 and r=\infty, Lyapunov-Perron fixed points, and a detailed control of the resolvent’s jump. The results yield both sharp L^2-growth bounds for frequency-localized evolution and a framework to read off dispersive behavior through the distorted Fourier representation, complementing concurrent dispersive analyses and enabling a path toward full asymptotic stability questions for the degree-one vortex. The methods hinge on handling strongly singular inverse-square-type potentials, embedding resonances, and implementing a bespoke Stone-type formula to accommodate non-selfadjoint spectral data. These contributions provide a rigorous spectral and scattering-theoretic foundation for the linearized GP dynamics around topological vortices and illuminate how zero-energy phenomena govern long-time evolution and potential stabilization strategies via projection away from L^2-growth modes.

Abstract

We study the evolution of the Gross-Pitaevskii equation linearized around the Ginzburg-Landau vortex of degree one under equivariant symmetry. Among the main results of this work, we determine the spectrum of the linearized operator, uncover a remarkable -norm growth phenomenon related to a zero-energy resonance, and provide a complete construction of the distorted Fourier transform at small energies. The latter hinges upon a meticulous analysis of the behavior of the resolvent in the upper and lower half-planes in a small disk around zero-energy.

Paper Structure

This paper contains 35 sections, 31 theorems, 486 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Gross-Pitaevskii equation linearized around the degree-one vortex
  3. Main results
  4. Proof ideas and discussion
  5. Spectrum of the linearized operator $\mathcal{L}$
  6. The zero-energy resonance and growth of the $L^2$-norm
  7. Construction of the distorted Fourier transform
  8. Upper bound on the growth of the $L^2$ norm
  9. Projecting away the $L^2$ growth
  10. References
  11. Organization of the paper
  12. Notation and conventions
  13. Spectral Properties of the Linearized Operator
  14. Basics
  15. Determining the spectrum and applying Hille-Yosida
  16. ...and 20 more sections

Key Result

Theorem 1.1

The operator $\mathcal{L}$ on $L^2_r(\mathbb R_+) \times L^2_r(\mathbb R_+)$ with domain $\mathcal{D} \times \mathcal{D}$ satisfies:

Figures (1)

  • Figure 1: Spectral properties of the operator $i\mathcal{L}$: The orange band corresponds to the spectrum. The red dot at the origin indicates the zero-energy resonance.

Theorems & Definitions (71)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 61 more