Computer assisted verification of the eigenvalue problem for one-dimensional Schrödinger operator

Ayuki Sekisaka, Shunsaku Nii

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We propose a rigorous computational method for verifying the isolated eigenvalues of one-dimensional Schrödinger operator containing a periodic potential and a perturbation which decays exponentially at ±∞. We show how the original eigenvalue problem can be reformulated as the problem of finding a connecting orbit in a Lagrangian-Grassmanian. Based on the idea of the Maslov theory for Hamiltonian systems, we set up an integer-valued topological measurement, the rotation number of the orbit in the resulting one-dimensional projective space. Combining the interval arithmetic method for dynamical systems, we demonstrate a computer-assisted proof for the existence of isolated eigenvalues within the first spectral gap.

Original languageEnglish
Title of host publicationMathematical Challenges in a New Phase of Materials Science
EditorsYasumasa Nishiura, Motoko Kotani
PublisherSpringer New York LLC
Pages145-157
Number of pages13
ISBN (Print)9784431561026
DOIs
Publication statusPublished - Jan 1 2016
EventInternational Conference on Mathematical Challenges in a New Phase of Materials Science, 2014 - Kyoto, Japan
Duration: Aug 4 2014Aug 8 2014

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume166
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

OtherInternational Conference on Mathematical Challenges in a New Phase of Materials Science, 2014
CountryJapan
CityKyoto
Period8/4/148/8/14

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Sekisaka, A., & Nii, S. (2016). Computer assisted verification of the eigenvalue problem for one-dimensional Schrödinger operator. In Y. Nishiura, & M. Kotani (Eds.), Mathematical Challenges in a New Phase of Materials Science (pp. 145-157). (Springer Proceedings in Mathematics and Statistics; Vol. 166). Springer New York LLC. https://doi.org/10.1007/978-4-431-56104-0_8