黎曼流形上非半有界协变薛定谔算子的自伴性
Self-adjointness of non-semibounded covariant Schrödinger operators on Riemannian manifolds
摘要：
在测地完备黎曼流形$M$的背景下，我们研究了$\nabla^{\dagger}\nabla+V$的自伴性，其中$\nabla$是度量协变导数（具有形式伴随$\nabla^{\dagger}$），并且$V$是$\textrm{End }\mathcal{V}$的局部平方可积部分，使得“负”部分PDMATHAMT的（纤维方向）范数属于局部Kato类（或者更一般地，局部收缩Dynkin类）。而不是下半有界性假设，我们假设存在一个数$\varepsilon \in [0,1]$和一个正函数$q$的$M$满足一定的增长条件，使$\varepsilon \nabla^{\dagger}\nabla+V\geq -q$，在二次型意义上的不等式被理解为$C_{c}^{\infty}(\mathcal{V})$。在第一个结果中，它属于情况$ε\in [0,1)$，我们使用椭圆方程方法。在第二个结果中，它属于情况$\varepsilon=1$，我们使用双曲方程方法。
英文摘要：
In the context of a geodesically complete Riemannian manifold $M$, we study the self-adjointness of $\nabla^{\dagger}\nabla+V$ where $\nabla$ is a metric covariant derivative (with formal adjoint $\nabla^{\dagger}$) on a Hermitian vector bundle $\mathcal{V}$ over $M$, and $V$ is a locally square integrable section of $\textrm{End }\mathcal{V}$ such that the (fiberwise) norm of the "negative" part $V^{-}$ belongs to the local Kato class (or, more generally, local contractive Dynkin class). Instead of the lower semiboundedness hypothesis, we assume that there exists a number $\varepsilon \in [0,1]$ and a positive function $q$ on $M$ satisfying certain growth conditions, such that $\varepsilon \nabla^{\dagger}\nabla+V\geq -q$, the inequality being understood in the quadratic form sense over $C_{c}^{\infty}(\mathcal{V})$. In the first result, which pertains to the case $ε\in [0,1)$, we use the elliptic equation method. In the second result, which pertains to the case $\varepsilon=1$, we use the hyperbolic equation method.
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