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

Ayuki Sekisaka, Shunsaku Nii

研究成果: 著書/レポートタイプへの貢献会議での発言

抄録

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.

元の言語英語
ホスト出版物のタイトルMathematical Challenges in a New Phase of Materials Science
編集者Yasumasa Nishiura, Motoko Kotani
出版者Springer New York LLC
ページ145-157
ページ数13
ISBN(印刷物)9784431561026
DOI
出版物ステータス出版済み - 1 1 2016
イベントInternational Conference on Mathematical Challenges in a New Phase of Materials Science, 2014 - Kyoto, 日本
継続期間: 8 4 20148 8 2014

出版物シリーズ

名前Springer Proceedings in Mathematics and Statistics
166
ISSN(印刷物)2194-1009
ISSN(電子版)2194-1017

その他

その他International Conference on Mathematical Challenges in a New Phase of Materials Science, 2014
日本
Kyoto
期間8/4/148/8/14

Fingerprint

Eigenvalue Problem
Connecting Orbits
Computer-assisted Proof
Eigenvalue
Interval Arithmetic
Rotation number
Spectral Gap
Periodic Potential
Operator
Projective Space
Computational Methods
Hamiltonian Systems
Dynamical system
Orbit
Decay
Perturbation
Integer
Demonstrate

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

これを引用

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

Computer assisted verification of the eigenvalue problem for one-dimensional Schrödinger operator. / Sekisaka, Ayuki; Nii, Shunsaku.

Mathematical Challenges in a New Phase of Materials Science. 版 / Yasumasa Nishiura; Motoko Kotani. Springer New York LLC, 2016. p. 145-157 (Springer Proceedings in Mathematics and Statistics; 巻 166).

研究成果: 著書/レポートタイプへの貢献会議での発言

Sekisaka, A & Nii, S 2016, Computer assisted verification of the eigenvalue problem for one-dimensional Schrödinger operator. : Y Nishiura & M Kotani (版), Mathematical Challenges in a New Phase of Materials Science. Springer Proceedings in Mathematics and Statistics, 巻. 166, Springer New York LLC, pp. 145-157, International Conference on Mathematical Challenges in a New Phase of Materials Science, 2014, Kyoto, 日本, 8/4/14. https://doi.org/10.1007/978-4-431-56104-0_8
Sekisaka A, Nii S. Computer assisted verification of the eigenvalue problem for one-dimensional Schrödinger operator. : Nishiura Y, Kotani M, 編集者, Mathematical Challenges in a New Phase of Materials Science. Springer New York LLC. 2016. p. 145-157. (Springer Proceedings in Mathematics and Statistics). https://doi.org/10.1007/978-4-431-56104-0_8
Sekisaka, Ayuki ; Nii, Shunsaku. / Computer assisted verification of the eigenvalue problem for one-dimensional Schrödinger operator. Mathematical Challenges in a New Phase of Materials Science. 編集者 / Yasumasa Nishiura ; Motoko Kotani. Springer New York LLC, 2016. pp. 145-157 (Springer Proceedings in Mathematics and Statistics).
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