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Propagation of space-time singularities for perturbed harmonic oscillators

Kenichi Ito, Tomoya Tagawa

TL;DR

This work analyzes how space-time singularities propagate for the Schrödinger equation with a time-dependent, short-range perturbed harmonic oscillator. It employs a semi-classical microlocal framework based on the quasi-homogeneous wave front set of order $\theta$, identifying the critical case $\theta=2$ and relating perturbed dynamics to the unperturbed Mehler-type evolution through classical scattering data. The main result shows that the space-time singularities of the perturbed solution correspond exactly to those of the unperturbed solution evaluated along the classical scattering data $(x_+(s,y,\eta),\xi_+(s,y,\eta))$, thus reducing space-time microlocal propagation to classical mechanics plus a semiclassical transport construction. The proof fuses an interpolation via a parameter $\kappa$, a symbol-transport construction solving transport equations along a time-dependent flow, and classical Mourre-type estimates to bridge the perturbed and unperturbed propagations, extending Lascar’s framework to space-time singularities for time-dependent perturbations of the harmonic oscillator. This advances understanding of infinite propagation speed for Schrödinger dynamics and complements prior spatial-slice results in the literature on harmonic oscillators with decaying or short-range perturbations.

Abstract

We discuss propagation of space-time singularities for the quantum harmonic oscillator with time-dependent metric and potential perturbations. Reformulating the quasi-homogeneous wave front set according to Lascar (1977) in a semiclassical manner, we obtain a characterization of its appearance in comparison with the unperturbed system. The idea of our proof is based on the argument of Nakamura (2009), which was originally devised for the analysis of spatial singularities of the Schrödinger equation, however, the application is non-trivial since the time is no more a parameter, but takes a part in the base variables.

Propagation of space-time singularities for perturbed harmonic oscillators

TL;DR

This work analyzes how space-time singularities propagate for the Schrödinger equation with a time-dependent, short-range perturbed harmonic oscillator. It employs a semi-classical microlocal framework based on the quasi-homogeneous wave front set of order , identifying the critical case and relating perturbed dynamics to the unperturbed Mehler-type evolution through classical scattering data. The main result shows that the space-time singularities of the perturbed solution correspond exactly to those of the unperturbed solution evaluated along the classical scattering data , thus reducing space-time microlocal propagation to classical mechanics plus a semiclassical transport construction. The proof fuses an interpolation via a parameter , a symbol-transport construction solving transport equations along a time-dependent flow, and classical Mourre-type estimates to bridge the perturbed and unperturbed propagations, extending Lascar’s framework to space-time singularities for time-dependent perturbations of the harmonic oscillator. This advances understanding of infinite propagation speed for Schrödinger dynamics and complements prior spatial-slice results in the literature on harmonic oscillators with decaying or short-range perturbations.

Abstract

We discuss propagation of space-time singularities for the quantum harmonic oscillator with time-dependent metric and potential perturbations. Reformulating the quasi-homogeneous wave front set according to Lascar (1977) in a semiclassical manner, we obtain a characterization of its appearance in comparison with the unperturbed system. The idea of our proof is based on the argument of Nakamura (2009), which was originally devised for the analysis of spatial singularities of the Schrödinger equation, however, the application is non-trivial since the time is no more a parameter, but takes a part in the base variables.
Paper Structure (13 sections, 10 theorems, 94 equations)

This paper contains 13 sections, 10 theorems, 94 equations.

Table of Contents

  1. Settings and results
  2. Perturbed harmonic oscillator
  3. Quasi-homogeneous wave front set
  4. Classical high-energy asymptotics
  5. Main result
  6. Strategy of the proof
  7. Classical mechanics
  8. Reduction to simpler Hamilton equations
  9. Classical Mourre-type estimates
  10. Conclusions
  11. Proof
  12. Construction of symbol
  13. Proof of the main result

Key Result

Theorem 1.3

Suppose Assumption 250124, and set respectively. Then there exists a unique family $\{U(t,s)\}_{t,s\in\mathbb R}$ of unitary operators on $L^2(\mathbb R^d)$ with the following properties.

Theorems & Definitions (19)

  • Remark 1.2
  • Theorem 1.3: Y
  • Definition 1.4
  • Proposition 1.5
  • Theorem 1.7
  • Definition 1.8
  • Proposition 1.9: N2
  • Theorem 1.10
  • Proposition 2.1
  • proof
  • ...and 9 more