Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Conservativeness of time changed processes and Liouville property for Schrödinger operators

Yuichi Shiozawa, Masayoshi Takeda

Abstract

We establish a criterion for the Liouville property for Schrödinger operators via the conservativeness of time changed processes. Using this criterion, we obtain necessary and sufficient conditions for the Liouville property for some Schrödinger operators in terms of the decay rates of the potentials at infinity/boundary.

Conservativeness of time changed processes and Liouville property for Schrödinger operators

Abstract

We establish a criterion for the Liouville property for Schrödinger operators via the conservativeness of time changed processes. Using this criterion, we obtain necessary and sufficient conditions for the Liouville property for some Schrödinger operators in terms of the decay rates of the potentials at infinity/boundary.
Paper Structure (11 sections, 12 theorems, 76 equations)

This paper contains 11 sections, 12 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Divergence of positive continuous additive functionals
  4. Resurrected Dirichlet forms and Liouville property for Schrödinger operators
  5. Examples
  6. Potentials of positive functions
  7. Potentials of non-fully supported functions and measures
  8. Rotation invariance
  9. Rotation invariance of Feller processes
  10. Rotation invariance of Dirichlet forms
  11. Distributions of positive continuous additive functionals under rotation invariance

Key Result

Lemma 2.2

Let $\mu\in S_1$ and $\{A_t^{\mu}\}_{t\ge 0}\in {\mathbf A}_{c,1}^+$ be in the Revuz correspondence. Then for any $f\in {\cal B}^+(E)$,

Theorems & Definitions (32)

  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.3
  • ...and 22 more