薛定谔算子理论中的交换子、eigenvalue间隙和中曲率
Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators
摘要：
利用交换子关系研究了作用于光滑紧致$d$维流形$M$浸入$\bbr^ν, ν\geq d+1$的薛定谔哈密顿量$H = -Δ+ V({x}),$的谱.这里$Δ$表示拉普拉斯-贝尔特拉米算子，实值势能函数$V(x)$通过乘法起作用。流形$M$可以是完备的，也可以有边界，在这种情况下，施加狄利克雷边界条件。结果表明，流形的中曲率对$H$的谱有严格的约束。此外，一个特殊的代数作用被发现是由薛定谔算子与潜在的比例的平方的中曲率：$$H_{g} := -Δ+ g h^2,$$其中$ν= d+1$，$g$是一个真正的参数，和$$h := \sum\limits_{j = 1}^{d} {κ_j},$$与$\{κ_j\}$，$j = 1, ..., d$表示的主曲率$M$。例如，通过定理\ref{thm3.1}和推论\ref{cor4.5}，任意薛定谔算子的每个本征值间隙由使用$H_{1/4}$的表达式上界。这些定理的“等周”部分指出，对于基本本征值间隙和无穷多个其他本征值间隙，这些边界是尖锐的。
英文摘要：
Commutator relations are used to investigate the spectra of Schrödinger Hamiltonians, $H = -Δ+ V({x}),$ acting on functions of a smooth, compact $d$-dimensional manifold $M$ immersed in $\bbr^ν, ν\geq d+1$. Here $Δ$ denotes the Laplace-Beltrami operator, and the real-valued potential--energy function $V(x)$ acts by multiplication. The manifold $M$ may be complete or it may have a boundary, in which case Dirichlet boundary conditions are imposed. It is found that the mean curvature of a manifold poses tight constraints on the spectrum of $H$. Further, a special algebraic rôle is found to be played by a Schrödinger operator with potential proportional to the square of the mean curvature: $$H_{g} := -Δ+ g h^2,$$ where $ν= d+1$, $g$ is a real parameter, and $$h := \sum\limits_{j = 1}^{d} {κ_j},$$ with $\{κ_j\}$, $j = 1, ..., d$ denoting the principal curvatures of $M$. For instance, by Theorem \ref{thm3.1} and Corollary \ref{cor4.5}, each eigenvalue gap of an arbitrary Schrödinger operator is bounded above by an expression using $H_{1/4}$. The "isoperimetric" parts of these theorems state that these bounds are sharp for the fundamental eigenvalue gap and for infinitely many other eigenvalue gaps.
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