Title: Quantum Chaos and Arithmetic
Speaker: Ilya Khayutin
Speaker Info: Northwestern University
Brief Description: Faculty Series
In 1920 Neils Bohr postulated that quantum mechanics reproduces classical physics in the large energy limit. One aspect of this philosophy is that high energy eigenstates of a chaotic classical system should “spread around”. An example of a classical dynamical system is the geodesic flow on a compact manifold. In some cases, e.g. for a negatively curved manifold, this flow is ergodic with respect to the volume measure. That is, the empirical distribution of almost every geodesic orbit approaches the volume measure in the long term.Date: Friday, May 21, 2021
For these systems, Bohr’s correspondence principle has generated many conjectures describing the limiting behavior of Laplace eigenstates. For example, one might ask whether the probability density of a high energy eigenstate approaches the volume measure; or what is growth rate, in terms of the eigenvalue, of the Lp-norm for an L2-normalized eigenfunction. Several very strong and general results were proved on average, but understanding an individual eigenstate of large eigenvalue remains mostly elusive. Moreover, naive conjectures and Bohr’s postulate can fail.
Most of the progress has been in the arithmetic setting, where the system has additional symmetries arising from number theory. I will discuss these topics and show why one might paradoxically expect arithmetic quantum chaos even when the classical physics is definitely not chaotic!