Let $H$ be a Schrödinger operator defined on a noncompact Riemannian manifold $\Omega$, and let $W\in L^\infty(\Omega;\mathbb{R})$. Suppose that the operator $H+W$ is critical in $\Omega$, and let $\varphi$ be the corresponding Agmon ground state. We prove that if $u$ is a generalized eigenfunction of $H$ satisfying $|u|\leq \varphi$ in $\Omega$, then the corresponding eigenvalue is in the spectrum of $H$. The conclusion also holds true if for some $K\Subset \Omega$ the operator $H$ admits a positive solution in $\tilde{\Omega}=\Omega\setminus K$, and $|u|\leq \psi$ in $\tilde{\Omega}$, where $\psi$ is a positive solution of minimal growth in a neighborhood of infinity in $\Omega$. Under natural assumptions, this result holds true also in the context of infinite graphs, and Dirichlet forms.

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Agmon基态的Shnol型定理

令$ H $是在非紧黎曼流形$ \ Omega $上定义的Schrödinger运算符，并令$ W \ in L ^ \ infty（\ Omega; \ mathbb {R}）$。假设运算符$ H + W $在$ \ Omega $中很关键，并且让$ \ varphi $为对应的Agmon基态。我们证明如果$ u $是满足$ \ omega $中的$ | u | \ leq \ varphi $的$ H $的广义特征函数，则对应的特征值在$ H $的频谱中。如果对于某些$ K \ Subset \ Omega $，运算符$ H $接受$ \ tilde {\ Omega} = \ Omega \ setminus K $和$ | u | \ leq \ psi $的正解，则该结论也成立。在$ \ tilde {\ Omega} $中，其中$ \ psi $是在$ \ Omega $中无穷大附近最小增长的正解。在自然假设下，该结果在无限图和Dirichlet形式的情况下也适用。