Regularized Potentials of Schrödinger Operators and a Local Landscape Function

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 $f:Ω\rightarrow \mathbb{R}_{>0}$.

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