Localization and eigenfunctions to 2nd - order elliptic PDEs (François Pagano, Unige)

26.02.2024 16:15 – 18:15

Abstract:
In the 70’s, Anderson studied the motion of electrons in materials. If the atomic structure is periodic, electrons can travel freely: the material conducts electricity. On the other hand, if the material has impurities or if the atomic structure is more random, electrons can get trapped: the material is now an insulator. Anderson received the Nobel Prize in Physics for this discovery in ’77.
Understanding this question mathematically amounts to understanding the nature of the spectrum for a periodic or random Schrödinger operator.

In this talk, we will first illustrate, using results from Kuchment (’12) and Bourgain-Kenig (’05), how this problem is related to the following (deterministic) question going back to Landis (late 60' s): given A elliptic, C^1 (or smoother) and V bounded, how rapidly can a non-trivial solution to −div(A∇u) + V u = 0 decay to zero at infinity?

We will discuss the construction of an operator on the cylinder T^2 × R with an eigenfunction div(A∇u) = −µu, which has double exponential decay at both ± ∞, where A is uniformly elliptic and uniformly C^1 smooth in the cylinder.
Joint work with S. Krymskii and A. Logunov.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-15, Séminaire "Maths-Physique"

entrée libre