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Short time asymptotics of the fundamental solutions for Schrödinger equations with non-smooth potentials

Shun Takizawa

Abstract

This paper deals with Schrödinger equations with potentials which are time-dependent non-smooth and at most quadratic growth. In the case where potentials are smooth with respect to spatial variables, fundamental solutions have explicit formulas in short time by D. Fujiwara. On the otherhand in the case where ones are non-smooth, we cannot expect that fundamental solutions have similar formula as above because dispersive estimates fail to hold in general. We show that a principal part of an asymptotic form of the fundamental solution has similar form as above even in the case where a potential is in $C^2$ with respect to spatial variables.

Short time asymptotics of the fundamental solutions for Schrödinger equations with non-smooth potentials

Abstract

This paper deals with Schrödinger equations with potentials which are time-dependent non-smooth and at most quadratic growth. In the case where potentials are smooth with respect to spatial variables, fundamental solutions have explicit formulas in short time by D. Fujiwara. On the otherhand in the case where ones are non-smooth, we cannot expect that fundamental solutions have similar formula as above because dispersive estimates fail to hold in general. We show that a principal part of an asymptotic form of the fundamental solution has similar form as above even in the case where a potential is in with respect to spatial variables.
Paper Structure (9 sections, 67 equations)

This paper contains 9 sections, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wave packet transform
  4. Classical orbits
  5. Key estimate
  6. Proof of Theorem \ref{['main thm']}
  7. Transformation of equation via wave packet transform
  8. Estimate for principal part
  9. Estimate for remainder term

Theorems & Definitions (7)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Lemma \ref{['Lem 3.1']}
  • proof