论文标题
某些Schrödinger操作员在有理dunkl设置中的特征值的行为
Behavior of eigenvalues of certain Schrödinger operators in the rational Dunkl setting
论文作者
论文摘要
对于标准化的root System $ r $ IN $ \ MATHBB r^n $和一个多重功能$ k \ geq 0 $ let $ \ mathbf n = n+\ sum_ {α\ in r} k（α）$。我们用$ dw（\ Mathbf {x}）=π_{α\ in R} | \ langle \ Mathbf {x}，α\ rangle |^{k（α）} \，d \，d \ Mathbf {x} $ $ \ mathbb {rangbf {x} $。令$ l =-Δ+v $，$ v \ geq 0 $，是$ \ mathbb r^n $上的dunkl--schrödinger运营商。假设存在$ q> \ max（1，\ frac {\ mathbf {n}} {2}）$，这样$ v $属于反向Hölder类$ {\ rm {rm {rh}}}^Q}（q}（dw）$。对于$λ> 0 $，我们提供$ l $的特征值数量的上和下部估计值，该数量较小或等于$λ$。我们在Fefferman中的主要工具 - 在理性dunkl设置中的phong类型不平等。
For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{α\in R} k(α)$. We denote by $dw(\mathbf{x})=Π_{α\in R}|\langle \mathbf{x},α\rangle|^{k(α)}\,d\mathbf{x}$ the associated measure in $\mathbb{R}^N$. Let $L=-Δ+V$, $V\geq 0$, be the Dunkl--Schrödinger operator on $\mathbb R^N$. Assume that there exists $q >\max(1,\frac{\mathbf{N}}{2})$ such that $V$ belongs to the reverse Hölder class ${\rm{RH}}^{q}(dw)$. For $λ>0$ we provide upper and lower estimates for the number of eigenvalues of $L$ which are less or equal to $λ$. Our main tool in the Fefferman--Phong type inequality in the rational Dunkl setting.
