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.
CITATION STYLE
Sekisaka, A., & Nii, S. (2016). Computer assisted verification of the eigenvalue problem for one-dimensional Schrödinger operator. In Springer Proceedings in Mathematics and Statistics (Vol. 166, pp. 145–157). Springer New York LLC. https://doi.org/10.1007/978-4-431-56104-0_8
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