A classical result by Agmon implies that an eigenfunction of a Schrodinger operator on R^d decays exponentially if the classical region associated with that eigenstate is compactly supported. A Schrodinger operator is the sum of the negative Laplacian and a multiplication operator by a real valued function, known as the potential. The classical region is defined as the subset of R^d for which the potential is less than the eigenvalue.
In this work, we extend the result to Schrodinger operators with an eigenvalue such that the classical region is not compactly supported. We show that assuming integrability of the classical region with respect to an increasing weight function implies L^2-decay of the eigenfunction with respect to the same weight. Here, decay is measured in the Agmon metric which takes into account anisotropies of the potential.
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Hi Hengrui, I really like your poster and find the example especially helpful! Do you have any next steps planned for your research project?
Thanks Lisa! And yes, we are now trying to prove the optimality of our theorem through Fiemann path integrals; by that I mean that the decay rate of the eigenfunction is exactly the same as the inverse of the weight function when approaching infinity. We will then submit a paper.
Best,
Hengrui
Thanks Lisa! And yes, we are now trying to prove the optimality of our theorem through Fiemann path integrals; by that I mean that the decay rate of the eigenfunction is exactly the same as the inverse of the weight function when approaching infinity. We will then submit a paper.
Best,
Hengrui