日時：2023 年 10月 28日 （土） 14:00 -- 16:30
場所： 学習院大学　中央教育研究棟4階 405号室
（以前の南4号館205号室ではありません。ご注意ください。）

1. 14:00--15:00
講演者: Frederic Klopp氏（パリ・ソルボンヌ大学）

Title：On the spatial extent of localized eigenfunctions for random Schrödinger operators

Abstract: On $\mathbb{Z}^d$, consider $\varphi$, an $\ell^2-$normalized function that decays exponentially at $\infty$ at a rate at least $\mu$. One can define the \emph{onset length} (of the exponential decay) of $\varphi$ as the radius of the smallest ball, say, $B$, such that one has the following global bound $\D|\varphi(x)|\leq\|\varphi\|_\infty e^{-\mu\,\text{dist}(x,B)}$. The present talk will describe the onset lengths of the localized eigenfunctions of random Schrdinger operators. Under suitable assumptions, we prove that, with probability one, the number of eigenfunctions in the localization regime having onset length larger than $\ell$ and localization center in a ball of radius $L$ is smaller than $C L^d\exp(-c \ell)$, for $\ell>0$ large (for some constants $C,c>0$). Thus, most eigenfunctions localize on small size balls independent of the system size which is the physicists understanding of localization; to our knowledge, this did not result from existing mathematical estimates. The talk is mainly based on joint work with Jeff Schenker.

2. 15:30--16:30
講演者: Pavel Exner氏 (Doppler Institute for Mathematical Physics and Applied Mathematics, Prague)

Title: Quantum graphs non-invariant with respect to time reversal

Abstract： The topic of this talk are quantum graphs with the vertex coupling which does not preserve the time-reversal invariance. As a case study we analyze the simplest example in which the asymmetry is maximal at a fixed energy. In this situation the high-energy scattering depends crucially on the vertex parity; we will demonstrate implications of this fact for spectral and transport properties in several classes of graphs, both finite and infinite periodic ones. In particular, we prove the Band-Berkolaiko universality for kagome lattices with this coupling. Furthermore, we discuss other graphs violating the invariance and identify a class of such couplings which exhibits a non-trivial PT-symmetry despite being self-adjoint; we also illustrate the role of the Dirichlet component in the vertex coupling. Finally, we show how a square lattice with such a coupling behaves in the presence of a magnetic field when the two time-asymmetry effects compete. The results come from a common work with Marzieh Baradaran, Jir Lipovsky , and Milos Tater.