Regularized Potentials of Schrödinger Operators and a Local Landscape Function

Abstract

We study localization properties of low-lying eigenfunctions $$(-Δ+V) ϕ= λϕ\qquad \mbox{in}~Ω$$ for rapidly varying potentials $V$ in bounded domains $\Omega \subset {\mathbb{R}}^d$. Filoche & Mayboroda introduced the landscape function $(-\Delta +V)u=1$ 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 $$V_t(x)=V\ast \left(\frac{1}{t}{\int }_0^t\frac{\exp \left(-\Vert \cdot {\Vert }^2/(4s)\right)}{(4\pi s{)}^{d/2}}ds\right).$$ We prove that for eigenfunctions $(-\Delta +V)\varphi =\lambda \varphi$ 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 $(-\Delta +V)u=f$ for a general right-hand side $f:\Omega \to {\mathbb{R}}_{>0}$.