论文标题
光谱通过“侧向”扰动移动
Spectral shift via "lateral" perturbation
论文作者
论文摘要
我们考虑一个紧凑的扰动$ h_0 = s + k_0^* k_0 $ k_0 $ sexhainexhighatixhoint operator $ s $，其特征值$λ^\ circ $以下其基本频谱和相应的特征函数$ f $。假定扰动是“沿”特征功能$ f $的“沿”，即$ k_0f = 0 $。特征值$λ^\ circ $属于$ h_0 $和$ s $的光谱。让$ s $具有低于$λ^\ circ $的$σ$比$ h_0 $多； $σ$被称为$λ^\ Circ $的光谱移动。 现在，我们允许扰动在合适的操作员空间中变化，并研究$ h（k）= s + k^* k $的特征值$λ^\ circ $的延续。我们表明，特征值作为$ k $的函数在$ k = k_0 $中的关键点，此关键点的摩尔斯索引是频谱偏移$σ$。该定理的一个版本也适用于某些非阳性扰动。
We consider a compact perturbation $H_0 = S + K_0^* K_0$ of a self-adjoint operator $S$ with an eigenvalue $λ^\circ$ below its essential spectrum and the corresponding eigenfunction $f$. The perturbation is assumed to be "along" the eigenfunction $f$, namely $K_0f=0$. The eigenvalue $λ^\circ$ belongs to the spectra of both $H_0$ and $S$. Let $S$ have $σ$ more eigenvalues below $λ^\circ$ than $H_0$; $σ$ is known as the spectral shift at $λ^\circ$. We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue $λ^\circ$ in the spectrum of $H(K)=S + K^* K$. We show that the eigenvalue as a function of $K$ has a critical point at $K=K_0$ and the Morse index of this critical point is the spectral shift $σ$. A version of this theorem also holds for some non-positive perturbations.
