Regularized potentials of Schrödinger operators and a local landscape function

Abstract

We study localization properties of low-lying eigenfunctions $(-\Delta +V)\varphi =\lambda \varphi \mathrm{\hspace{0.5em}}\text{in} \Omega$ 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 $V_t(x)=V*(\frac{1}{t}{\int }_0^t\frac{\hspace{0.17em}\text{exp}\hspace{0.17em}(-\left|\right|·|{|}^2/(4s))}{{(4\pi s)}^{d/2}}ds).$ 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 $f:\Omega \to {\mathbb{R}}_{>0}.$

Keywords:

• Eigenfunction
• localization
• regularization
• Schrödinger operator

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 35J10
• 65N25 (primary)
• 82B44 (secondary)

Additional information

Funding

The author is by the NSF (Division of Mathematical Sciences, DMS-1763179) and the Alfred P. Sloan Foundation.