论文标题
海森堡集团的谐波振荡器
The Harmonic Oscillator on the Heisenberg Group
论文作者
论文摘要
在本说明中，我们提出了Heisenberg Group $ \ Mathbf {H} _n $的谐波振荡器的概念$ \ mathbf {h} _n $上的sub-laplacian，本质上是纯粹具有离散频谱的自我偶会，其特征向量应该是光滑的函数，并形成$ l^2（\ mathbf {h} _n）$的正顺序基础。这种方法导致在$ \ mathbf {h} _n $上的差分运算符，该操作员由（分层）dynin-folland lie代数确定。我们为操作员提供明确的表达以及其特征值的渐近估计。
In this note we present a notion of harmonic oscillator on the Heisenberg group $\mathbf{H}_n$ which forms the natural analogue of the harmonic oscillator on $\mathbb{R}^n$ under a few reasonable assumptions: the harmonic oscillator on $\mathbf{H}_n$ should be a negative sum of squares of operators related to the sub-Laplacian on $\mathbf{H}_n$, essentially self-adjoint with purely discrete spectrum, and its eigenvectors should be smooth functions and form an orthonormal basis of $L^2(\mathbf{H}_n)$. This approach leads to a differential operator on $\mathbf{H}_n$ which is determined by the (stratified) Dynin-Folland Lie algebra. We provide an explicit expression for the operator as well as an asymptotic estimate for its eigenvalues.
