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Local time decay for fractional Schrödinger operators with slowly decaying potentials and a weaker Agmon type estimate in a classically forbidden region

Kouichi Taira

Abstract

A local time decay estimate of fractional Schrödinger operators with slowly decaying positive potentials are studied. It is shown that its resolvent is smooth near zero and the time propagator has fast local time decay which is very different from very short-range cases. The key element of the proof is to establish a weaker Agmon estimate for a classically forbidden region using exotic symbol calculus. As a byproduct, we prove that the Riesz operator is a pseudodifferential operator with an exotic symbol.

Local time decay for fractional Schrödinger operators with slowly decaying potentials and a weaker Agmon type estimate in a classically forbidden region

Abstract

A local time decay estimate of fractional Schrödinger operators with slowly decaying positive potentials are studied. It is shown that its resolvent is smooth near zero and the time propagator has fast local time decay which is very different from very short-range cases. The key element of the proof is to establish a weaker Agmon estimate for a classically forbidden region using exotic symbol calculus. As a byproduct, we prove that the Riesz operator is a pseudodifferential operator with an exotic symbol.
Paper Structure (22 sections, 27 theorems, 128 equations)

This paper contains 22 sections, 27 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Introduction
  3. Main theorem
  4. Limiting absorption principle and time decay estimates
  5. Key estimates
  6. Related problems
  7. Notation
  8. Preliminary
  9. Scattering symbol class
  10. A slowly varying metric
  11. Functional calculus
  12. Weaker Agmon estimate
  13. Analysis of the Riesz operator
  14. Disjoint support property
  15. Proof of the resolvent estimates at the low-frequency regimes
  16. ...and 7 more sections

Key Result

Theorem 1.4

Let $P=p_0(D_x)+V(x)$ satisfy Assumptions symbolass, potass and abevass. $(i)$$($Uniform estimates for powers of the resolvent$)$ Let $N\in\mathbb{Z}_{>0}$ and $\gamma>\max(N-1/2,N\left(\frac{1}{2}+\frac{2k-1}{4k}\mu\right) )$. Then Moreover, the map $\overline{\mathbb{C}}_{\pm}\setminus \{0\}\ni z\mapsto \langle x \rangle^{-\gamma}R_{\pm}(z)^N\langle x \rangle^{-\gamma}\in B(L^2)$ is Hölder cont

Theorems & Definitions (65)

  • Remark 1.1
  • Remark 1.2
  • Example 1
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Proposition 2.1
  • Definition 1
  • Remark 2.2
  • ...and 55 more