Heisenberg群上的调和与非调和振子
Harmonic and Anharmonic Oscillators on the Heisenberg Group
AI总结：
本文利用Dynin–Folland群表示定义Heisenberg群上的调和振子，并推广到非调和振子，获得谱估计、谱乘子 $L^p$–$L^q$ 估计及热半群估计。
摘要：
尽管迄今为止在Heisenberg群 $\mathbf{H}_n$ 上还不存在标准版本的调和振子，但我们利用Dynin–Folland群 $\mathbf{H}_{n,2}$ 的表示理论，为一种特定算子选择提供了有力依据；$\mathbf{H}_{n,2}$ 是一个3步分层Lie群，其一般表示作用于 $L^2(\mathbf{H}_n)$。我们的方法受到 $\mathbb{R}^n$ 上调和振子与 $\mathbf{H}_n$ 第一分层中平方和之间联系的启发，即我们将 $\mathbf{H}_n$ 上的调和振子定义为次Laplace算子 $\mathcal{L}_{\mathbf{H}_{n,2}}$ 在Dynin–Folland群的一般酉不可约表示 $\pi$ 下的像，该表示的形式维数为 $d_\pi=1$。更一般地，该方法通过在 $\mathbf{H}_{n,2}$ 上采用正Rockland算子，使我们能够定义一大类所谓的非调和振子。利用ter Elst和Robinson [tERo] 发展的方法，我们得到了 $\mathbf{H}_n$ 上调和与非调和振子的谱估计。此外，我们证明该方法可推广到中心维数为1的分次 $SI/Z$-群，即具有模1维中心 $Z(G)$ 平方可积的酉不可约表示的分次群。本文后半部分涉及谱乘子。通过将ter Elst和Robinson的技术与[AkRu18]中的最新结果相结合，我们得到了关于 $\mathcal{L}_{\mathbf{H}_{n,2}}$ 次Laplace算子谱乘子的有用的 $L^\mathbf{p}$–$L^\mathbf{q}$ 估计，事实上更一般地，也得到了一般分次群上一般Rockland算子谱乘子的此类估计。作为副产品，我们恢复了[FiRu17]中建立的分次群上的Sobolev嵌入，并得到了显式的亚椭圆热半群估计。
英文摘要：
Although there is no canonical version of the harmonic oscillator on the Heisenberg group $\mathbf{H}_n$ so far, we make a strong case for a particular choice of operator by using the representation theory of the Dynin-Folland group $\mathbf{H}_{n, 2}$, a $3$-step stratified Lie group, whose generic representations act on $L^2(\mathbf{H}_n)$. Our approach is inspired by the connection between the harmonic oscillator on $\mathbb{R}^n$ and the sum of squares in the first stratum of $\mathbf{H}_n$ in the sense that we define the harmonic oscillator on $\mathbf{H}_n$ as the image of the sub-Laplacian $\mathcal{L}_{\mathbf{H}_{n, 2}}$ under the generic unitary irreducible representation $π$ of the Dynin-Folland group which has formal dimension $d_π= 1$. This approach, more generally, permits us to define a large class of so-called anharmonic oscillators by employing positive Rockland operators on $\mathbf{H}_{n, 2}$. By using the methods developed in ter Elst and Robinson [tERo], we obtain spectral estimates for the harmonic and anharmonic oscillators on $\mathbf{H}_n$. Moreover, we show that our approach extends to graded $SI/Z$-groups of central dimension $1$, i.e., graded groups which possess unitary irreducible representations which are square-integrable modulo the $1$-dimensional center $Z(G)$. The latter part of the article is concerned with spectral multipliers. By combining ter Elst and Robinson's techniques with recent results in [AkRu18], we obtain useful $L^\mathbf{p}$-$L^\mathbf{q}$-estimates for spectral multipliers of the sub-Laplacian $\mathcal{L}_{\mathbf{H}_{n, 2}}$ and, in fact more generally, of general Rockland operators on general graded groups. As a by-product, we recover the Sobolev embeddings on graded groups established in [FiRu17], and obtain explicit hypoelliptic heat semigroup estimates.
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