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.
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Acknowledgements
We would like to thank Professor Kaori Nagatou for the helpful suggestions. Special thanks go to Professor Yasumasa Nishiura and Takashi Teramoto for many valuable comments, and we are also grateful to the referees for suggesting several improvements.
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Sekisaka, A., Nii, S. (2016). Computer Assisted Verification of the Eigenvalue Problem for One-Dimensional Schrödinger Operator. In: Nishiura, Y., Kotani, M. (eds) Mathematical Challenges in a New Phase of Materials Science. Springer Proceedings in Mathematics & Statistics, vol 166. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56104-0_8
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DOI: https://doi.org/10.1007/978-4-431-56104-0_8
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