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Frozen Gaussian approximation for the fractional Schrödinger equation

Lihui Chai, Hengzhun Chen, Xu Yang

TL;DR

This work develops a momentum-space Frozen Gaussian Approximation (FGA) for the fractional Schrödinger equation in the semi-classical limit $\varepsilon\to0$, addressing the low regularity of the kinetic symbol by introducing a regularization parameter $\delta$ and a phase-space cutoff. The authors formulate the FGA using a dual phase and Fourier integral operators, derive remainder estimates, and prove convergence to the true FSE solution under precise $\delta$- and $\varepsilon$-dependent bounds. They provide rigorous remainder analyses (R1, R2) and show how to balance $\delta$ and $\varepsilon$ to achieve a quantified convergence rate, with improved results in the linear-potential case. Numerical tests in 1D and 2D validate the method, demonstrating near-linear convergence in $\varepsilon$ for $\delta=\varepsilon$ and confirming the theoretical insights, indicating that the momentum-space FGA is a robust, efficient tool for high-frequency FSE dynamics.

Abstract

We develop a refined Frozen Gaussian approximation (FGA) for the fractional Schrödinger equation in the semi-classical regime, where the solution exhibits rapid oscillations as the scaled Planck constant $\varepsilon$ becomes small. Our approach utilizes an integral representation based on asymptotic analysis, offering a highly efficient computational framework for high-frequency wave function evolution. Crucially, we introduce the momentum space representation of the FGA and a regularization parameter $δ$ to address singularities in the higher-order derivatives of the Hamiltonian flow coefficients, which are typically assumed to be second-order differentiable or smooth in conventional analysis. We rigorously prove convergence of the method to the true solution and provide numerical experiments that demonstrate its precision and robust convergence behavior.

Frozen Gaussian approximation for the fractional Schrödinger equation

TL;DR

This work develops a momentum-space Frozen Gaussian Approximation (FGA) for the fractional Schrödinger equation in the semi-classical limit , addressing the low regularity of the kinetic symbol by introducing a regularization parameter and a phase-space cutoff. The authors formulate the FGA using a dual phase and Fourier integral operators, derive remainder estimates, and prove convergence to the true FSE solution under precise - and -dependent bounds. They provide rigorous remainder analyses (R1, R2) and show how to balance and to achieve a quantified convergence rate, with improved results in the linear-potential case. Numerical tests in 1D and 2D validate the method, demonstrating near-linear convergence in for and confirming the theoretical insights, indicating that the momentum-space FGA is a robust, efficient tool for high-frequency FSE dynamics.

Abstract

We develop a refined Frozen Gaussian approximation (FGA) for the fractional Schrödinger equation in the semi-classical regime, where the solution exhibits rapid oscillations as the scaled Planck constant becomes small. Our approach utilizes an integral representation based on asymptotic analysis, offering a highly efficient computational framework for high-frequency wave function evolution. Crucially, we introduce the momentum space representation of the FGA and a regularization parameter to address singularities in the higher-order derivatives of the Hamiltonian flow coefficients, which are typically assumed to be second-order differentiable or smooth in conventional analysis. We rigorously prove convergence of the method to the true solution and provide numerical experiments that demonstrate its precision and robust convergence behavior.
Paper Structure (20 sections, 17 theorems, 134 equations, 2 figures)

This paper contains 20 sections, 17 theorems, 134 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The frozen Gaussian approximation
  3. Hamiltonian flow and action
  4. Fourier integral operator
  5. Formulation of the frozen Gaussian approximation
  6. The FGA for the modified FSE
  7. Cutoff strategy
  8. The modified Hamiltonian flow
  9. The modified FGA and the associated remainder
  10. Estimate for R1
  11. Estimate for R2
  12. Convergence of FGA to the FSE
  13. Numerical tests
  14. One dimension
  15. Two dimension
  16. ...and 5 more sections

Key Result

Lemma 2

$Z(t,q,p)$ is invertible for $(q,p)\in \mathbb{R}^{2d}$ with $|\det(Z(t,q,p)| \geq 2^{d/2}$.

Figures (2)

  • Figure 1: $L^2$ error decay behavior of FGA solution with different $\delta$ values in Example \ref{['example:one_dim']}.
  • Figure 2: $L^2$ error decay behavior of FGA solution in Example \ref{['example:two_dim']}.

Theorems & Definitions (38)

  • Definition 1: Canonical Transformation
  • Lemma 2
  • Definition 3: Action
  • Definition 4: Fourier Integral Operator
  • Definition 5
  • Lemma 6
  • Lemma 7
  • Proposition 8
  • Remark 1
  • Proposition 1
  • ...and 28 more