Proof of the fundamental gap conjecture
Proof of the fundamental gap conjecture
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DOI：
10.1090/s0894-0347-2011-00699-1
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发表时间：
2010-06
影响因子：
3.9
通讯作者：
B. Andrews;J. Clutterbuck
中科院分区：
文献类型：
--
作者：
B. Andrews;J. Clutterbuck
我们证明了基本间隙猜想，该猜想表明，具有凸势的薛定谔算子的前两个狄利克雷特征值（谱间隙）与凸域上狄利克雷边界数据之间的差由相同直径的零势区间上的谱间隙所限定。更一般地说，对于高维的任意光滑势，我们的证明给出了谱间隙的一个明显的下界和第一个特征函数的对数的一个明显的凹模，根据域的直径和势的凸模。
We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spec- tral gap) of a Schrodinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero potential. More generally, for an arbitrary smooth potential in higher dimensions, our proof gives both a sharp lower bound for the spectral gap and a sharp modulus of concavity for the logarithm of the first eigenfunction, in terms of the diameter of the domain and a modulus of convexity for the potential.