We consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set \({\Omega }\subset \mathbb {R}^{m}\) of finite Lebesgue measure \(0<|{\Omega }|<\infty \) with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.

中文翻译：

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扭转函数的效率与局部化

我们考虑狄利克雷拉普拉斯-Δ扭转功能，并为操作者薛定谔-Δ+ V上的开集\（{\欧米茄} \子集\ mathbb {R} ^ {米} \）有限Lebesgue测度的\（ 0 <| {\欧米茄} | <\ infty \）与实值的非负的，可测量的电势V。我们研究了扭转函数的效率和局部化现象，以及它们与第一个Dirichlet本征函数的几何形状的相互影响。