论文标题
Schrödinger操作员的正规化潜力和当地景观功能
Regularized Potentials of Schrödinger Operators and a Local Landscape Function
论文作者
论文摘要
我们研究了低洼特征函数的定位属性$$（ - δ+v）ϕ = λϕ \ qquad \ qquad \ mbox {in} 〜Ω $$，用于在有限域中快速变化的电位$ v $ v $ $ω\ subset \ subset \ subset \ mathbb {r}^d $。 Filoche＆Mayboroda引入了景观功能$（-δ+ V）u = 1 $，并表明该功能$ u $具有出色的属性：本地化特征函数更喜欢本地化$ u $的本地最大值。 Arnold，David，Filoche，Jerison \＆Mayboroda表明，$ 1/U $自然而然地是相关方程中的潜力。在这些问题的激励下，我们引入了一个单参数的正规化潜力$ v_t $的家族，该家族是由与radial kernel $$ $$ v_t（x）= v * \ left（\ frac {1} {t} {t} {t} \ int_0^t \ frac frac {\ frac {\ exp {\ exp { \ right）}} {（4πs）^{d/2}} ds \ right）。$$我们证明，对于eigenfunctions $（ - δ+v）ϕ = λϕ $ this正则化$ v_t $在精确的感觉上是，小规模上的规范有效性。景观功能$ u $尊重相同类型的正则化。这允许我们从方程$（ - δ+ v）u = f $的解决方案中得出景观型函数，对于一般的右侧$ f：ω\ rightarrow \ rightarrow \ mathbb {r} _ {> 0} $。
We study localization properties of low-lying eigenfunctions $$(-Δ+V) ϕ= λϕ\qquad \mbox{in}~Ω$$ for rapidly varying potentials $V$ in bounded domains $Ω\subset \mathbb{R}^d$. Filoche & Mayboroda introduced the landscape function $(-Δ+ V)u=1$ and showed that the function $u$ has remarkable properties: localized eigenfunctions prefer to localize in the local maxima of $u$. Arnold, David, Filoche, Jerison \& Mayboroda showed that $1/u$ arises naturally as the potential in a related equation. Motivated by these questions, we introduce a one-parameter family of regularized potentials $V_t$ that arise from convolving $V$ with the radial kernel $$ V_t(x) = V * \left( \frac{1}{t} \int_0^t \frac{ \exp\left( - \|\cdot\|^2/ (4s) \right)}{(4 πs )^{d/2}} ds \right).$$ We prove that for eigenfunctions $(-Δ+V) ϕ= λϕ$ this regularization $V_t$ is, in a precise sense, the canonical effective potential on small scales. The landscape function $u$ respects the same type of regularization. This allows allows us to derive landscape-type functions out of solutions of the equation $(-Δ+ V)u = f$ for a general right-hand side $f:Ω\rightarrow \mathbb{R}_{>0}$.
