Regularized Potentials of Schrödinger Operators and a Local Landscape Function
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We study localization properties of low-lying eigenfunctions (-Δ +V) ϕ = λϕ Ω for rapidly varying potentials V in bounded domains Ω⊂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 * ( 1/t∫_0^t ( - ·^2/ (4s) )/(4 π s )^d/2 ds ). 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→R_>0.
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