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Hypercontractivity for a family of quantum Ornstein-Uhlenbeck semigroups

Longfa Sun, Zhendong Xu, Hao Zhang

Abstract

We show that a family of quantum Ornstein-Uhlenbeck semigroups is hypercontractive. We also obtain the optimal order of the optimal time up to a constant for those elements whose Gibbs state is zero. The main ingredient of our proof is Meixiner polynomials.

Hypercontractivity for a family of quantum Ornstein-Uhlenbeck semigroups

Abstract

We show that a family of quantum Ornstein-Uhlenbeck semigroups is hypercontractive. We also obtain the optimal order of the optimal time up to a constant for those elements whose Gibbs state is zero. The main ingredient of our proof is Meixiner polynomials.
Paper Structure (9 sections, 15 theorems, 132 equations)

This paper contains 9 sections, 15 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eigenvalues and eigenvectors
  4. Theorem \ref{['Main-Lem']} implies Theorem \ref{['MainThm']}
  5. Proof of Theorem \ref{['Main-Lem']}
  6. Orthogonal polynomials
  7. Proof of Theorem \ref{['Main-Lem']}
  8. Estimate of Optimal time $t_p$
  9. Proof of Corollary \ref{['MainThm2']}

Key Result

Theorem 1.1

$(T_t^{(2)})_{t\geq 0}$ is hypercontractive. Moreover, the optimal time $t_p$ satisfies where $\widetilde{c}(\beta)$ and $\widetilde{C}(\beta)$ are positive constants depending only on $\beta$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • ...and 19 more