The Harmonic Oscillator on the Heisenberg Group

$L^p$-estimates for the Schrödinger equation associated to the harmonic oscillator

In this paper we obtain some Strichartz estimates for the Schrödinger equation associated to the harmonic oscillator and the Laplacian. Our main tool will be some embeddings between Lebesgue spaces and suitable Triebel-Lizorkin spaces.

KAM for KG on $\mathbb S^2$ and for the quantum harmonic oscillator on $\mathbb R^2$

In this paper we prove an abstract KAM theorem adapted to the Klein Gordon equation on the sphere $\mathbb S^2$ and for the quantum harmonic oscillator on $\mathbb R^2$ with regularizing nonlinearity.

海森堡振荡器

The Heisenberg Oscillator

在这篇短文中，我们确定了海森堡振子的谱，它是定义在三维海森堡群$H_1=\mathbb C \times \mathbb R$上的算子$L + |z|^2$，其中$L$表示正的次拉普拉斯算子，$z$表示群中一点的复变数.

Quantum Limits for Harmonic Oscillator

In this note we consider high energy eigenfunctions of the harmonic oscillator in $\mathbb{R}^d$ and prove that any invariant measure on the energy surface can be written as a weak limit of eigenfunctions.

关于微扰谐振子Feynman Kac公式的一点注记

A remark concerning Feynman Kac formulas for the perturbed harmonic oscillator

我们给出了某些抛物型演化问题的解决方案（依赖于时间的扰动的热方程的谐振子）作为显式积分的维纳空间。

Sharp Strichartz type estimates for the Schrödinger equation associated with harmonic oscillator

In this article we study the Schrödinger equation associated with Harmonic oscillator in the form of Strichartz type inequality. We give simple proofs for Strichartz type inequalities using purely the $L^2 \to L^p$ operator norm estimates of the spectral projections associated harmonic oscillator proved in \cite{KT}. Our Strichartz type estimates are sharp in sense of regularity of initial data.

非线性谐振子解的正则性和衰减性

Regularity and decay of solutions of nonlinear harmonic oscillators

证明了变系数非线性谐振子解的解析正则性和无穷远衰减性。也就是说，我们根据R^d中Hermite函数的基本性质，给出了复域中扇形的全纯延拓，以及相应的高斯衰减。我们的结果特别适用于谐振子的非线性本征值问题，该谐振子与R^d中的实解析散射或渐近圆锥度规有关，也适用于经典谐振子的某些扰动。

自相似集上的奇异积分与Heisenberg群中Lipschitz调和函数的可移性

Singular integrals on self-similar sets and removability for Lipschitz harmonic functions in Heisenberg groups

本文研究度量群的小子集（即测度为零且小于满维的子集）上的奇异积分。我们想到的群的主要例子是欧几里得空间和海森堡群。除了获得结果在一个非常一般的设置，这项工作的目的是双重的，我们将延长一些结果在欧几里得空间更一般的内核比以前考虑，我们将获得在海森堡群的一些应用调和（在海森堡意义上）功能的一些结果较早在欧几里得空间。

On heat equations associated with fractional harmonic oscillators

We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator $e^{-tH^β}$, $t, β>0$, associated with the harmonic oscillator $H=-Δ+ |x|^2$. We then prove some local and global wellposedness results for nonlinear fractional heat equations.

Hörmander condition for pseudo-multipliers associated to the harmonic oscillator

In this paper we prove Hörmander-Mihlin multiplier theorems for pseudo-multipliers associated to the harmonic oscillator (also called the Hermite operator). Our approach can be extended to also obtain the $L^p$-boundedness results for multilinear pseudo-multipliers. By using the Littlewood-Paley theorem associated to the harmonic oscillator we also give $L^p$-boundedness and $L^p$-compactness properties for multipliers. $(L^p,L^q)$-estimates for spectral pseudo-multipliers also are investigated.

Semiclassical asymptotics for nonselfadjoint harmonic oscillators

We consider nonselfadjoint perturbations of semiclassical harmonic oscillators. Under appropriate dynamical assumptions, we establish some spectral estimates such as upper bounds on the resolvent near the real axis when no geometric control condition is satisfied.

一维谐振子的Fulling-Kuchment定理

A Fulling-Kuchment theorem for the 1D harmonic oscillator

证明了存在一对“非等谱”一维半经典薛定谔算子，其谱模h^ \infty一致.特别地，它们的所有半经典迹不变量都是相同的。我们的证明是基于一个想法的Fulling-Kuchment和阿达玛的变分公式适用于适当的扰动的谐振子。保留字：逆谱问题，半经典薛定谔算子，迹不变量，阿达玛变分公式，谐振子，彭罗斯蘑菇，斯特姆-刘维理论。