论文标题
某些非自我支持操作员的光谱分解
Spectral decomposition of some non-self-adjoint operators
论文作者
论文摘要
我们考虑了$ h = h_0+cwc $的希尔伯特空间中的非自身选择运算符，其中$ h_0 $是自我偶像，$ w $是有限的，$ c $是公制运算符，$ c $有限，相对于$ h_0 $相对紧凑。我们假设$ c（h_0-z）^{ - 1} c $在\ mathbb {c} \ setMinus \ mathbb {r} $中均匀地限制在$ z \中。我们将$ h $的频谱奇异性定义为基本频谱的点$λ\inσ_{\ mathrm {ess}}}（h）$，以至于$ c（h \ pm i \ pm i \ varepsilon）^{ - 1} cw cw没有限制为$ \ varepsilon lavepsilon。我们证明，$ h $的频谱奇异性与与共鸣状态相关的特征值一对一的信件，向更大的希尔伯特空间扩展了$ h $。接下来，我们表明，$ h $的渐近消失状态，即$φ$ $ e^{\ pm ith}φ\ to0 $ as $ t \ to \ infty $，与$ h $相对于eigenvalues $ eigenvalues $λ\ n \ math \ in \ in \ in $ \ mp \ mathrm {im}（λ）> 0 $。最后，我们定义了$ h $的绝对连续频谱子空间，并表明它满足$ \ mathcal {h} _ {\ mathrm {ac}}}}（h）= \ mathcal {h} _ {\ mathrm {\ mathrm {p}}}（h^*） $ \ MATHCAL {H} _ {\ MATHRM {P}}}（H^*）$代表$ H^*$的点频谱。因此，我们从$ h $的光谱子空间方面获得了希尔伯特空间的直接总和分解。我们证明的主要成分之一是用于有限操作员$ r（h）$将频谱奇异性身份的频谱分辨率公式。我们的结果适用于具有复杂潜力的Schrödinger运营商。
We consider non-self-adjoint operators in Hilbert spaces of the form $H=H_0+CWC$, where $H_0$ is self-adjoint, $W$ is bounded and $C$ is a metric operator, $C$ bounded and relatively compact with respect to $H_0$. We suppose that $C(H_0-z)^{-1}C$ is uniformly bounded in $z\in\mathbb{C}\setminus\mathbb{R}$. We define the spectral singularities of $H$ as the points of the essential spectrum $λ\inσ_{\mathrm{ess}}(H)$ such that $C(H\pm i\varepsilon)^{-1}CW$ does not have a limit as $\varepsilon\to0^+$. We prove that the spectral singularities of $H$ are in one-to-one correspondence with the eigenvalues, associated to resonant states, of an extension of $H$ to a larger Hilbert space. Next, we show that the asymptotically disappearing states for $H$, i.e. the set of vectors $φ$ such that $e^{\pm itH}φ\to0$ as $t\to\infty$, coincide with the generalized eigenstates of $H$ corresponding to eigenvalues $λ\in\mathbb{C}$, $\mp\mathrm{Im}(λ)>0$. Finally, we define the absolutely continuous spectral subspace of $H$ and show that it satisfies $\mathcal{H}_{\mathrm{ac}}(H)=\mathcal{H}_{\mathrm{p}}(H^*)^\perp$, where $\mathcal{H}_{\mathrm{p}}(H^*)$ stands for the point spectrum of $H^*$. We thus obtain a direct sum decomposition of the Hilbert spaces in terms of spectral subspaces of $H$. One of the main ingredients of our proofs is a spectral resolution formula for a bounded operator $r(H)$ regularizing the identity at spectral singularities. Our results apply to Schrödinger operators with complex potentials.
