Fast localization of eigenfunctions via smoothed potentials

10/28/2020

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by   Jianfeng Lu, et al.

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We study the problem of predicting highly localized low-lying eigenfunctions (-Δ +V) ϕ = λϕ in bounded domains Ω⊂ℝ^d for rapidly varying potentials V. Filoche Mayboroda introduced the function 1/u, where (-Δ + V)u=1, as a suitable regularization of V from whose minima one can predict the location of eigenfunctions with high accuracy. We proposed a fast method that produces a landscapes that is exceedingly similar, can be used for the same purposes and can be computed very efficiently: the computation time on an n × n grid, for example, is merely 𝒪(n^2 logn), the cost of two FFTs.