薛定谔算子本征值和eigenvalue间隙的临界势
Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators
摘要：
设$M$是一个有边界或无边界的紧致黎曼流形，$-Δ$是它的Laplace-Beltrami算子.对于任何有界标量势$q$，我们用$λ\_i(q)$表示作用在具有狄利克雷或诺依曼边界条件的函数上的薛定谔型算子$-Δ+ q$的第$i$个本征值，在$\partial M \neq \emptyset$情形下。我们调查的本征值$λ\_i$和本征值差距$G\_{ij}=λ\_j -λ\_i$被认为是泛函上的一组有界的潜力具有给定的平均值$M$的临界潜力。我们给出了一个潜在的$q$是关键的，或者是这些泛函的局部极小或局部极大的必要和充分条件。例如，我们证明了一个潜在的$q \in L^\infty (M)$是关键的功能$λ\_2$当且仅当，$q$是光滑的，$λ\_2(q)=λ\_3(q)$和存在第二本征函数$f\_1 ,...,f\_k$的$-Δ+ q$，使$Σ\_j f\_j^2 = 1$。特别地，$λ\_2$（以及任何$λ\_i$）在Dirichlet边界条件下不允许临界势。此外，功能$λ\_2$从来没有承认本地最小化的潜力。
英文摘要：
Let $M$ be a compact Riemannian manifold with or without boundary, and let $-Δ$ be its Laplace-Beltrami operator. For any bounded scalar potential $q$, we denote by $λ\_i(q)$ the $i$-th eigenvalue of the Schrödinger type operator $-Δ+ q$ acting on functions with Dirichlet or Neumann boundary conditions in case $\partial M \neq \emptyset$. We investigate critical potentials of the eigenvalues $λ\_i$ and the eigenvalue gaps $G\_{ij}=λ\_j -λ\_i$ considered as functionals on the set of bounded potentials having a given mean value on $M$. We give necessary and sufficient conditions for a potential $q$ to be critical or to be a local minimizer or a local maximizer of these functionals. For instance, we prove that a potential $q \in L^\infty (M)$ is critical for the functional $λ\_2$ if and only if, $q$ is smooth, $λ\_2(q)=λ\_3(q)$ and there exist second eigenfunctions $f\_1 ,...,f\_k$ of $-Δ+ q$ such that $Σ\_j f\_j^2 = 1$. In particular, $λ\_2$ (as well as any $λ\_i$) admits no critical potentials under Dirichlet Boundary conditions. Moreover, the functional $λ\_2$ never admits locally minimizing potentials.
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