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Characterization of spacetime singularities for the Schrödinger equation by initial state

Takeru Fujii, Kenichi Ito

Abstract

We discuss spacetime singularities of a solution to the Schrödinger equation with a metric perturbation and a sublinear potential. The quasi-homogeneous wave front set, due to Lascar (1977), of a solution is characterized by that of the free solution, and a classical high-energy scattering data. In the one-dimensional case, it further reduces to the homogeneous wave front set, due to Nakamura (2005), of the initial time-slice. For the proof of the former result we implement an idea inspired by Nakamura (2009), which was originally devised for spatial singularities of the Schrödinger equation. As for the latter result, we use an exact Egorov-type formula for the free propagator, and a special partition of unity conforming with the classical flow.

Characterization of spacetime singularities for the Schrödinger equation by initial state

Abstract

We discuss spacetime singularities of a solution to the Schrödinger equation with a metric perturbation and a sublinear potential. The quasi-homogeneous wave front set, due to Lascar (1977), of a solution is characterized by that of the free solution, and a classical high-energy scattering data. In the one-dimensional case, it further reduces to the homogeneous wave front set, due to Nakamura (2005), of the initial time-slice. For the proof of the former result we implement an idea inspired by Nakamura (2009), which was originally devised for spatial singularities of the Schrödinger equation. As for the latter result, we use an exact Egorov-type formula for the free propagator, and a special partition of unity conforming with the classical flow.

Paper Structure

This paper contains 22 sections, 14 theorems, 127 equations.

Table of Contents

  1. Settings and results
  2. Introduction
  3. Assumptions
  4. Quasi-homogeneous wave front set
  5. Classical high-energy scattering data
  6. The first main result
  7. Homogeneous wave front set
  8. The second main result
  9. Classical mechanics in high-energy regime
  10. Classical Hamiltonian flow
  11. Reduction to ordinary Hamiltonians
  12. Time-dependent Hamiltonian
  13. Time-independent Hamiltonian without potential
  14. Proof of the main proposition
  15. Singularities in general dimensional space
  16. ...and 7 more sections

Key Result

Theorem 1.3

Suppose Assumption 250124. For any $k\in\mathbb N_0$ set and denote its dual space by $\Sigma(-k)=\Sigma(k)^*$. Then there exists a unique family $\{U(t,s)\}_{t,s\in\mathbb R}$ of isomorphisms on $\mathcal{S}'(\mathbb R^d)$ such that the following holds. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (39)

  • Remark 1.2
  • Theorem 1.3: MR1243098
  • Remark 1.4
  • Definition 1.5
  • Proposition 1.7: MR461592
  • Remark 1.8
  • Definition 1.9
  • Remark 1.10
  • Proposition 1.11: MR2488342
  • Theorem 1.12
  • ...and 29 more