Probing crystal defects with scattering theory and non-commutative topology

• Project/Area Number: 26707005
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Partial Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Nagoya University
• Principal Investigator: Richard Serge 名古屋大学, 多元数理科学研究科(国際), G30特任教授 (70725241)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥6,500,000 (Direct Cost: ¥5,000,000、Indirect Cost: ¥1,500,000); Fiscal Year 2016: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2015: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000); Fiscal Year 2014: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
• Keywords: functional analysis / spectral theory / scattering theory / index theorem / wave operators / quantum walk / periodic systems / index theorems

Outline of Final Research Achievements

Spectral and scattering results for Schroedinger operators with various type of electric or magnetic potentials have been obtained. Our investigations naturally lead to index theorems in scattering theory. Such results correspond to the exhibition of quantities which are stable under perturbations. We have also fully developed the spectral analysis of 1 dimensional quantum walk with anisotropic behavior, and provided a construction for embedding an arbitrary number of eigenvalues in the continuous spectrum of a Schroedinger operator.

Research Products

• [Journal Article] On Schroedinger operators with inverse square potentials on the half-line (2017)
  • Author(s): J. Derezinski, S. Richard
  • Journal Title: Ann. Henri Poincare Volume: 18 Issue: 3 Pages: 869-928
  • DOI: 10.1007/s00023-016-0520-7
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Spectral and scattering properties at thresholds for the Laplacian in a half-space with a periodic boundary condition (2017)
  • Author(s): S. Richard, R. Tiedra de Aldecoa
  • Journal Title: J. Math. Anal. Appl. Volume: 446 Issue: 2 Pages: 1695-1722
  • DOI: 10.1016/j.jmaa.2016.09.045
  • NAID: 120005971271
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Low energy spectral and scattering theory for relativistic Schroedinger operators (2016)
  • Author(s): S. Richard, T. Umeda
  • Journal Title: Hokkaido Mathematical Journal Volume: 45 Pages: 141-179
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Resolvent expansions and continuity of the scattering matrix at embedded thresholds: the case of quantum waveguides (2016)
  • Author(s): S. Richard, R. Tiedra de Aldecoa
  • Journal Title: Bull. Soc. math. France Volume: 144 Pages: 251-277
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Continuity of the spectra for families of magnetic operators on Z^d, (2016)
  • Author(s): D. Parra, S. Richard
  • Journal Title: Anal. Math. Phys. Volume: 6 Issue: 4 Pages: 327-343
  • DOI: 10.1007/s13324-015-0121-5
  • NAID: 120005946585
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Schroedinger operators with n positive eigenvalues: an explicit construction involving complex valued potentials (2016)
  • Author(s): Serge Richard, Jun Uchiyama, Tomio Umeda
  • Journal Title: Proceedings of Japan Academy Volume: 92, Ser. A Issue: 1 Pages: 7-12
  • DOI: 10.3792/pjaa.92.7
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] One-dimensional Dirac operators with zero-range interactions: Spectral, scattering, and topological results (2014)
  • Author(s): K. Pankrashkin, S. Richard
  • Journal Title: Journal of mathematical physics Volume: 55 Issue: 6 Pages: 0623051-17
  • DOI: 10.1063/1.4884417
  • Peer Reviewed
• [Presentation] Spectral and scattering theory on perturbed crystal lattices (2017)
  • Author(s): S. Richard
  • Organizer: Seminar talk
  • Place of Presentation: WPI AIMR Sendai
  • Year and Date: 2017-02-23
  • Invited
• [Presentation] Operateurs de Schroedinger presque homogenes (2016)
  • Author(s): S. Richard
  • Organizer: Seminar of mathematical physics
  • Place of Presentation: University of Orsay
  • Year and Date: 2016-09-21
  • Invited
• [Presentation] Operateurs de Schroedinger presque homogenes (2016)
  • Author(s): S. Richard
  • Organizer: Seminar of mathematical physics
  • Place of Presentation: University of Grenoble, France
  • Year and Date: 2016-09-19
  • Invited
• [Presentation] Operateurs de Schroedinger presque homogenes (2016)
  • Author(s): S. Richard
  • Organizer: Seminar of EDP, analysis and applications
  • Place of Presentation: University of Lorraine, France
  • Year and Date: 2016-09-16
  • Invited
• [Presentation] Continuite du spectre pour des familles d'operateurs magnetiques sur Z^d (2016)
  • Author(s): S. Richard
  • Organizer: Seminar of analysis, geometry and algebra
  • Place of Presentation: University of Lorraine, France
  • Year and Date: 2016-09-15
  • Invited
• [Presentation] Theorie spectrale et de la diffusion sur des cristaux topologiques perturbes (2016)
  • Author(s): S. Richard
  • Organizer: Seminar of analysis
  • Place of Presentation: University of Bordeaux, France
  • Year and Date: 2016-09-12
  • Invited
• [Presentation] Continuity of the spectra for families of magnetic operators on Z^d (2016)
  • Author(s): S. Richard
  • Organizer: Spectral and Scattering Theory and Related Topics
  • Place of Presentation: Rims, Kyoto
  • Year and Date: 2016-01-20
  • Int'l Joint Research / Invited
• [Presentation] Continuity of the spectra for families of magnetic operators on Z^d (2015)
  • Author(s): S. Richard
  • Organizer: Seminar of analysis
  • Place of Presentation: University of Hyogo, Himeji
  • Year and Date: 2015-11-28
  • Invited
• [Presentation] Towards scattering theory and index theorems in periodic systems (2015)
  • Author(s): S. Richard
  • Organizer: Workshop: quantum walks, random walks and scattering theory on periodic systems
  • Place of Presentation: Nagoya University
  • Year and Date: 2015-01-21 – 2015-01-22
  • Invited
• [Presentation] Asymptotic expansion at embedded thresholds (2014)
  • Author(s): S. Richard
  • Organizer: Seminar of analysis
  • Place of Presentation: University of Tsukuba
  • Year and Date: 2014-12-10
  • Invited
• [Presentation] Levinson's theorem: an index theorem in scattering theory (2014)
  • Author(s): S. Richard
  • Organizer: International Conference on Spectral Theory and Mathematical Physics
  • Place of Presentation: Santiago, Chili
  • Year and Date: 2014-11-24 – 2014-11-28
  • Invited
• [Presentation] Back-and-forth between scattering theory and index theorems (2014)
  • Author(s): S. Richard
  • Organizer: Seminar of operator algebras
  • Place of Presentation: University of Tokyo
  • Year and Date: 2014-10-01
  • Invited
• [Book] Spectral Theory and Mathematical Physics (2016)
  • Author(s): S. Richard
  • Total Pages: 54
  • Publisher: Springer