2016年

# Computer Assisted Verification of the Eigenvalue Problem for One-Dimensional Schrodinger Operator

MATHEMATICAL CHALLENGES IN A NEW PHASE OF MATERIALS SCIENCE
• Ayuki Sekisaka
• ,
• Shunsaku Nii

166

145

157

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

SPRINGER JAPAN

We propose a rigorous computational method for verifying the isolated eigenvalues of one-dimensional Schrodinger operator containing a periodic potential and a perturbation which decays exponentially at +/-infinity. 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.

リンク情報
DOI
https://doi.org/10.1007/978-4-431-56104-0_8
Web of Science