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Dept. of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute (RKMVERI) has organised an colloquium on “The landscape function and the exponential decay of solutions to Schrödinger operators with electric or magnetic potentials” by Professor B. Poggi, University of Minnesota, USA on 16 July 2021.

Title: The landscape function and the exponential decay of solutions to Schrödinger operators with electric or magnetic potentials

Abstract

In the first half of the last decade, Marcel Filoche and Svitlana Mayboroda observed that a certain solution to an elliptic PDE problem (the landscape function) can be effectively used to numerically predict the localization of eigenfunctions to Schrödinger operators with electric potentials exhibiting randomness. This localization phenomenon through disorder is known as Anderson localization, for which Phillip Anderson won the Nobel prize in physics in 1977. In this talk, we will introduce recent mathematical results showing the exponential decay given by the landscape function, for fundamental solutions, Lax-Milgram solutions, and eigenfunctions for Schrödinger operators; in particular, we will see that the landscape function generalizes previous tools used to study the exponential decay of solutions to Schrödinger operators when the potentials are regular.