TY - GEN

T1 - Computer assisted verification of the eigenvalue problem for one-dimensional Schrödinger operator

AU - Sekisaka, Ayuki

AU - Nii, Shunsaku

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84978805913&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84978805913&partnerID=8YFLogxK

U2 - 10.1007/978-4-431-56104-0_8

DO - 10.1007/978-4-431-56104-0_8

M3 - Conference contribution

AN - SCOPUS:84978805913

SN - 9784431561026

T3 - Springer Proceedings in Mathematics and Statistics

SP - 145

EP - 157

BT - Mathematical Challenges in a New Phase of Materials Science

A2 - Nishiura, Yasumasa

A2 - Kotani, Motoko

PB - Springer New York LLC

T2 - International Conference on Mathematical Challenges in a New Phase of Materials Science, 2014

Y2 - 4 August 2014 through 8 August 2014

ER -