A Spectral Gap Estimate and Applications
A Spectral Gap Estimate and Applications
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谱间隙估计和应用
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
S. Steinerberger
中科院分区：
文献类型：
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作者：
B. Georgiev;Mayukh Mukherjee;S. Steinerberger
我们考虑Schrödinger运算符
−d2dx2+Von一个具有Dirichlet边界条件的区间[a，b], $$ ext{-} frac{d^{2}}{d x^{2}} + V { ext{on an interval}}~~[a,b]~{ ext{with Dirichlet boundary conditions}},$$其中V从下有界，并证明了第一个特征值λ1的下界，如果wV(y) = |{x∈[a,b]: V(x)≤y}|，则
λ1≥1250min >minV1wV(y)2+y。$$lambda_{1} geq frac{1}{250} minlimits_{y > min V}{left( frac{1}{w_{V}(y)^{2}} + y
ight)}.$$如果{x∈[a， b]，其结果是一个普遍常数：V(x)≤y}是求解最小化问题的y值的区间。一个直接的应用如下：设Ω∧∂2 ${Omega } subset mathbb {R}^{2}$是一个凸域，设u：Ω→∈$u:{Omega }
ightarrow mathbb {R}$是∂Ω上具有Dirichlet边界条件的Ω上的拉普拉斯算子−Δ的第一个特征函数。我们证明
∥u∥L∞（Ω）≤1inrad（Ω）inrad（Ω）diam（Ω）1/6∥u∥L2（Ω）， $$| u |_{L^{infty}({Omega})} lesssim frac{1}{ ext{inrad}({Omega})} left( frac{ ext{inrad}({Omega})}{ ext{diam}({Omega})}
ight)^{1/6} |u|_{L^{2}({Omega})},$$，它回答了二维特殊情况下的van den Berg问题。
AbstractWe consider the Schrödinger operator
−d2dx2+Von an interval[a,b]with Dirichlet boundary conditions,$$ ext{-} frac{d^{2}}{d x^{2}} + V { ext{on an interval}}~~[a,b]~{ ext{with Dirichlet boundary conditions}},$$where V is bounded from below and prove a lower bound on the first eigenvalue λ1 in terms of sublevel estimates: if wV(y) = |{x ∈ [a, b] : V (x) ≤ y}|, then
λ1≥1250miny>minV1wV(y)2+y.$$lambda_{1} geq frac{1}{250} minlimits_{y > min V}{left( frac{1}{w_{V}(y)^{2}} + y
ight)}.$$The result is sharp up to a universal constant if {x ∈ [a, b] : V(x) ≤ y} is an interval for the value of y solving the minimization problem. An immediate application is as follows: let Ω⊂ℝ2${Omega } subset mathbb {R}^{2}$ be a convex domain and let u:Ω→ℝ$u:{Omega }
ightarrow mathbb {R}$ be the first eigenfunction of the Laplacian − Δ on Ω with Dirichlet boundary conditions on ∂Ω. We prove
∥u∥L∞(Ω)≲1inrad(Ω)inrad(Ω)diam(Ω)1/6∥u∥L2(Ω),$$| u |_{L^{infty}({Omega})} lesssim frac{1}{ ext{inrad}({Omega})} left( frac{ ext{inrad}({Omega})}{ ext{diam}({Omega})}
ight)^{1/6} |u|_{L^{2}({Omega})},$$which answers a question of van den Berg in the special case of two dimensions.