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Singular integral operators and special functions in scattering theory

Research Project

Project/Area Number 21K03292
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12010:Basic analysis-related
Research InstitutionNagoya University

Principal Investigator

Richard Serge  名古屋大学, 教養教育院, 教授 (70725241)

Project Period (FY) 2021-04-01 – 2025-03-31
Project Status Granted (Fiscal Year 2023)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2024: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2023: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
KeywordsScattering theory / Singular integrals / Index theorems / Special functions / scattering theory / surface states / Wave operators / Surface states / Decay estimate / spectral theory / singular operators / special functions
Outline of Research at the Start

In this project, we want to study more systematically singular integral operators through their representations in terms of special functions, and derive new analytical estimates. The investigations are divided into 3 main tasks: the discovery part which corresponds to the systematic transcription of wave operators with special functions, the technical part in which refined estimates will be obtained through the study of regularity properties of C0-group in Banach spaces, the visionary part in which this program will be extended to representation theory, and ultimately to number theory.

Outline of Annual Research Achievements

The research activities can be summarized as follows:
1) The manuscript on scattering theory and an index theorem on the radial part of SL(2,R), jointly written with H. Inoue, has been revised and accepted in a fairly good mathematical journal.
2) The investigations on the 2D Schroedinger operator with threshold singularities have been successfully completed. This work provides a definitive answer to some questions and doubtful results raised about 40 years ago. The key of this paper is precisely the resolution of a singular integral operator in terms of simpler special functions. The resulting paper is accepted in an excellent mathematical journal.
3) The investigations on surface states led to new results for families of discrete magnetic operators. A long manuscript has been submitted, and contains results of different nature on scattering theory, K-theory, and on integrable models. A surface of resonances is also exhibited, probably for the first time. This project has been done with collaborators in Australia, Chili, and Japan.
4) New investigations on the scattering theory and on index theorems for quantum walks have also been initiated, and the completion of this work is expected in Fall 2024.

Current Status of Research Progress
Current Status of Research Progress

1: Research has progressed more than it was originally planned.

Reason

This research program has reached its maturity, and the relations between several topics have been established. These different topics and approaches complement each other and lead to a wide set of results. The participation of researchers of different origins and of students to this research project had also a positive impact.

Strategy for Future Research Activity

The research activity 4) will continue, and a collaboration project with N. Boussaid (started in 2022 but temporarily paused in 2023) will resume. A new project involving singular integral operators, special functions, and also some number theory is now under discussion with J. Faupin. This project would correspond to unexpected new developments of this research proposal and could open new directions of research in the future. Finally, the project of writing a book with my long term collaborator R. Tiedra de Aldecoa has started. This project will take a long time, but the anticipated book should become a reference in spectral and scattering theory.

Report

(3 results)
  • 2023 Research-status Report
  • 2022 Research-status Report
  • 2021 Research-status Report
  • Research Products

    (16 results)

All 2024 2023 2022 2021

All Journal Article (8 results) (of which Int'l Joint Research: 8 results,  Peer Reviewed: 8 results,  Open Access: 3 results) Presentation (8 results) (of which Int'l Joint Research: 7 results,  Invited: 8 results)

  • [Journal Article] Scattering theory and an index theorem on the radial part of SL(2,R)2024

    • Author(s)
      H. Inoue, S. Richard
    • Journal Title

      Journal of Topology and Analysis

      Volume: - Issue: 05 Pages: 1-35

    • DOI

      10.1142/s179352532450002x

    • Related Report
      2023 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Control simulation experiments of extreme events with the Lorenz-96 model2023

    • Author(s)
      Q. Sun, T. Miyoshi, S. Richard
    • Journal Title

      Nonlinear Processes in Geophysics

      Volume: 30 Issue: 2 Pages: 117-128

    • DOI

      10.5194/npg-30-117-2023

    • Related Report
      2023 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Analysis of COVID-19 in Japan with extended SEIR model and ensemble Kalman filter2023

    • Author(s)
      Q. Sun, T. Miyoshi, S. Richard
    • Journal Title

      Journal of Computational and Applied Mathematics

      Volume: 419 Pages: 114772-114772

    • DOI

      10.1016/j.cam.2022.114772

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Analysis of COVID-19 Spread in Tokyo through an Agent-Based Model with Data Assimilation2022

    • Author(s)
      C. Sun, S. Richard, T. Miyoshi, N. Tsuzu
    • Journal Title

      Journal of Clinical Medicine

      Volume: 11 Issue: 9 Pages: 2401-2401

    • DOI

      10.3390/jcm11092401

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Decay Estimates for Unitary Representations with Applications to Continuous- and Discrete-Time Models2022

    • Author(s)
      S. Richard, R. Tiedra de Aldecoa
    • Journal Title

      Annales Henri Poincare

      Volume: 24 Issue: 1 Pages: 1-36

    • DOI

      10.1007/s00023-022-01199-5

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Spectral and scattering theory for topological crystals perturbed by infinitely many new edges2022

    • Author(s)
      S. Richard, N. Tsuzu
    • Journal Title

      Reviews in Mathematical Physics

      Volume: 33 Issue: 04

    • DOI

      10.1142/s0129055x22500106

    • NAID

      120007190679

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Discrete Laplacian in a half-space with a periodic surface potential I: Resolvent expansions, scattering matrix, and wave operators2022

    • Author(s)
      H. S. Nguyen, S. Richard, R. Tiedra de Aldecoa
    • Journal Title

      Mathematische Nachrichten

      Volume: in press Issue: 5 Pages: 912-949

    • DOI

      10.1002/mana.201900430

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Scattering Operator and Wave Operators for 2D Schroedinger Operators with Threshold Obstructions2021

    • Author(s)
      S. Richard, R. Tiedra de Aldecoa, L. Zhang
    • Journal Title

      Complex Analysis and Operator Theory

      Volume: 15 Issue: 6

    • DOI

      10.1007/s11785-021-01153-z

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Levinson's theorem for two-dimensional scattering systems: it was a surprise, it is now topological!2023

    • Author(s)
      Serge Richard
    • Organizer
      2nd Chile-Japan Workshop on Mathematical Physics and PDE, Santiago (Chile)
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Scattering theory and an index theorem on the radial part of SL(2,R)2023

    • Author(s)
      Serge Richard
    • Organizer
      Spectral and Scattering Theory and Related Topics, Rims Kyoto (JP)
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological Levinson's theorem applied to group theory: a starter2023

    • Author(s)
      Serge Richard
    • Organizer
      Seminar of operator algebras and noncommutative geometry, Wollongong, Australia
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Scattering theory and an index theorem on the radial part of SL(2,R)2022

    • Author(s)
      Serge Richard
    • Organizer
      Seminar of mathematical physics, Kyushu University
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Scattering theory and an index theory theorem on the radial part of SL(2,R)2022

    • Author(s)
      Serge Richard
    • Organizer
      Conference Asymptotic Analysis and Spectral Theory, Oldenburg, Germany
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Theorie de la diffusion et un theoreme d'indice sur la partie radiale de SL(2,R)2022

    • Author(s)
      Serge Richard
    • Organizer
      Seminar of EDP, University of Besancon, France
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Theorie de la diffusion et un theoreme d'indice sur la partie radiale de SL(2,R)2022

    • Author(s)
      Serge Richard
    • Organizer
      Seminar of analysis and EDP, University of Cergy-Pontoise, France
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Scattering theory and non-commutative geometry: A walk from the parentheses of Levinson to the hexagon of Cordes2021

    • Author(s)
      Serge Richard
    • Organizer
      Global Noncommutative Geometry Seminar
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2021-04-28   Modified: 2024-12-25  

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