Seminar

Mini course on Topics in quantum chaos

Spectral Methods in Mathematical Physics

22 January 10:00 - 12:00

Nalini Anantharaman - University of Strasbourg

I. This minicourse will start with a survey talk about the three main conjectures in quantum chaos. If we consider a chaotic classical hamiltonian system, and consider it from the point of view of quantum mechanics, we have : - the Bohigas-Gianonni-Schmit conjecture, according to which the spacing of eigenvalues should obey the Wigner statistics; - the quantum unique ergodicity conjecture, according to which the stationary wave functions should occupy the phase space uniformly; - the Berry conjecture, according to which the stationary wave functions should resemble, on a certain scale, a random gaussian process.
II. A detailed proof of the quantum ergodicity theorem (or Shnirelman theorem) will be given : when an ergodic classical system is quantized, then the eigenfunctions of the corresponding Schrödinger operator occupy the phase space uniformly in the semiclassical (small wavelength) limit -- except possibly for a scarce family.
III. We will then describe recent work of Anantharaman & Sabri, extending the scope of quantum ergodicity to graphs. Here we deal with the discrete laplacian (or more general Schrödinger operators) on a finite graph, and the semiclassical limit is to be understood as the size of the graph going to infinity. Under certain geometric and spectral assumptions, we prove that the eigenfunctions -- except possibly for a scarce family -- occupy the vertices uniformly.
IV. Finally, we discuss recent work of Backhausz & Szegedy, proving that eigenfunctions of the laplacian on random regular graphs are gaussian -- this proves a variant of the Berry conjecture on random regular graphs.
Organizers
Søren Fournais
Aarhus University
Rupert Frank
LMU Munich
Benjamin Schlein
University of Zurich, UZH
Simone Warzel
TU Munich

Program
Contact

Rupert Frank

frank@math.lmu.de

Other
information

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