Page 395 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 395
ANALYSIS ON GRAPHS (MS-48)
Effects of time-reversal asymmetry in the vertex coupling of quantum
graphs
Pavel Exner, exner@ujf.cas.cz
Czech Academy of Sciences, Nuclear Physics Institute, Czech Republic
The talk concerns the effects coming from a violation of time-reversal symmetry in the vertex
coupling of quantum graph, focusing on the situation when the asymmetry is maximal at a fixed
energy. It is shown that such a coupling has a topological property: the transport behaviour of
such a vertex in the high-energy regime depend substantially on the vertex parity. We explore
consequences of this fact for several classes of graphs, both finite and infinite periodic ones. The
results come from a common work with Marzieh Baradaran, Jiˇrí Lipovský, and Miloš Tater.
Steklov eigenvalues on graphs
Bobo Hua, bobohua@fudan.edu.cn
Fudan University, China
The eigenvalues of the Dirichlet-to-Neumann operator are called Steklov eigenvalues, which
are well studied in spectral geometry. In this talk, we introduce Steklov eigenvalues on graphs,
and estimate them using geometric quantities, based on joint works with Wen Han, Zunwu He,
Yan Huang, and Zuoqin Wang.
Stochastic completeness and uniqueness class for graphs
Xueping Huang, hxp@nuist.edu.cn
Nanjing University of Information Science and Technology, China
Uniqueness class for the heat equation on a weighted graph is closely related to stochastic
completeness of the corresponding minimal continuous time random walk. For a class of so
called globally local graphs, we obtain essentially sharp criteria which are in the same form as
for manifolds. Sharp volume growth type criteria for stochastic completeness then follow, after
a reduction to the globally local case.
This talk is based on joint work with M. Keller and M. Schmidt.
Multifractal eigenfunctions in a singular quantum billiard
Jon Keating, keating@maths.ox.ac.uk
University of Oxford and London Mathematical Society, United Kingdom
Spectral statistics and questions relating to quantum ergodicity in star graphs are closely re-
lated to those of Šeba billiards (rectangular billiards with a singular point scatterer) and other
intermediate systems. In intermediate systems, it has been suggested in the Physics literature
that quantum eigenfunctions should exhibit mulifractal properties. I shall discuss the proof that
this is the case for Šeba billiards. The work I shall report on was done jointly with Henrik
Ueberschaer.
393
Effects of time-reversal asymmetry in the vertex coupling of quantum
graphs
Pavel Exner, exner@ujf.cas.cz
Czech Academy of Sciences, Nuclear Physics Institute, Czech Republic
The talk concerns the effects coming from a violation of time-reversal symmetry in the vertex
coupling of quantum graph, focusing on the situation when the asymmetry is maximal at a fixed
energy. It is shown that such a coupling has a topological property: the transport behaviour of
such a vertex in the high-energy regime depend substantially on the vertex parity. We explore
consequences of this fact for several classes of graphs, both finite and infinite periodic ones. The
results come from a common work with Marzieh Baradaran, Jiˇrí Lipovský, and Miloš Tater.
Steklov eigenvalues on graphs
Bobo Hua, bobohua@fudan.edu.cn
Fudan University, China
The eigenvalues of the Dirichlet-to-Neumann operator are called Steklov eigenvalues, which
are well studied in spectral geometry. In this talk, we introduce Steklov eigenvalues on graphs,
and estimate them using geometric quantities, based on joint works with Wen Han, Zunwu He,
Yan Huang, and Zuoqin Wang.
Stochastic completeness and uniqueness class for graphs
Xueping Huang, hxp@nuist.edu.cn
Nanjing University of Information Science and Technology, China
Uniqueness class for the heat equation on a weighted graph is closely related to stochastic
completeness of the corresponding minimal continuous time random walk. For a class of so
called globally local graphs, we obtain essentially sharp criteria which are in the same form as
for manifolds. Sharp volume growth type criteria for stochastic completeness then follow, after
a reduction to the globally local case.
This talk is based on joint work with M. Keller and M. Schmidt.
Multifractal eigenfunctions in a singular quantum billiard
Jon Keating, keating@maths.ox.ac.uk
University of Oxford and London Mathematical Society, United Kingdom
Spectral statistics and questions relating to quantum ergodicity in star graphs are closely re-
lated to those of Šeba billiards (rectangular billiards with a singular point scatterer) and other
intermediate systems. In intermediate systems, it has been suggested in the Physics literature
that quantum eigenfunctions should exhibit mulifractal properties. I shall discuss the proof that
this is the case for Šeba billiards. The work I shall report on was done jointly with Henrik
Ueberschaer.
393
