Timetable : PDF

Invited Speaker : 

・NAKANO, Fumihiko （Tohoku University）

     Title : Statistics for random Schroedinger operators

     Abstract : Theory of random Schroedinger operators(RSO) are motivated to describe the behavior of electrons in disordered media. Many developments, such as spectral properties, exponential localization, dynamical localization, and conducting properties, have been made during this several decades. One of the active topics in RSO is probably the statisticsal properties and relation to the random matrix theory(RMT), which I will focus on after introduction : 

(1) Introduction

(2) Eigenvalue and eigenfunction statistics

(3) 1d Schroedinger operators with decaying randomness and relation to RMT

(4) Recent topics 

・NIIKUNI, Hiroaki（Maebashi Institute of Technology）

     Title (First talk) ：An estimate of the resolvent difference between 1D Schrödinger operators with δ-interactions and their Shortley--Weller approximations

   Abstract：In the first talk, we consider the one-dimensional continuous Schrödinger operators with δ-point interactions supported on a slightly perturbed lattice of ℓZ with a small parameter ℓ>0. Unlike the equidistance lattice ℓZ, it is natural that the corresponding discrete Schrödinger operator is constructed by the Shortley--Weller method. In spirit to the method constructed by P. Exner, S. Nakamura and Y. Tadano in 2022 for the equilateral lattice quantum graph Hamiltonian, we give an estimate for resolvent difference between the continuous and discrete Hamiltonian. The slide of the talk will be uploaded on September 1st on the following webpage:

http://www.maebashi-it.ac.jp/˜niikuni/slide/20230904.pdf

   Title (Second talk)： An explicit formulae for the number of the negative eigenvalues for carbon nanotubes with a ring of δ-impurities

   Abstract： In the second talk, we consider quantum graphs corresponding to zigzag carbon nanotubes with a ring of impurities. Recall that the quantum graph for a zigzag carbon nanotube without impurities is the triplet of the metric graph corresponding to the carbon nanotube with N-zigzags, the Schrödinger operator on the graph and the Kirchhoff--Neumann vertex conditions. To express impurities, we use the δ-type vertex condition. In this study, we assume that our impurities are located on a ring of carbon nanotubes and all the strength of impurities are common real constant β. Then, the number of negative eigenvalue can be explicitly given in terms of the strength β. The slide of the talk will be uploaded on September 1st on the following webpage:

http://www.maebashi-it.ac.jp/˜niikuni/slide/20230905.pdf

・MORIOKA, Hisashi（Ehime University） 

     Title : A shape resonance model for quantum walks

    Abstract : Some properties of resonances for multi-dimensional quantum walks are studied. Resonances for quantum walks are defined as eigenvalues of complex translated time evolution operators in the pseudo momentum space. A quantum walk with a non-penetrable barrier has some eigenvalues. We can see an instability of eigenvalues under a perturbation of the non-penetrable barrier. However, we also see the stability as resonance poles. This is an analogue of shape resonance model for Schrödinger operators. This is a joint work with Kenta Higuchi (Ehime University / JSPS research fellow PD).

 ・PARRA, Daniel（Universidad de Santiago de Chile）

     Title : Spectral and Scattering properties of discrete Dirac operators

     Abstract :  In recent times, there has been a growing interest in the study of Dirac analogs in the discrete setting and Dirac-Type operator on graphs and networks. In this talk, we will start by presenting different definitions of a discrete Dirac operator and show some examples on periodic graphs. We will discuss briefly how our proposed Dirac-Hodge operator can recover the continuous one when the mesh size goes to 0. For the scattering part of the talk, we will present results on the asymptotic completeness of the wave operator for periodic graphs under metric perturbations and asymptotics for the spectral shift function in the particular case Z^2.

Organizer : TAIRA, Kouichi (Ritsumeikan University)  ktaira [at] fc.ritsumei.ac.jp