Abstract
We consider the Schrödinger operator on \({\mathbb{R}^2}\) with a locally square-integrable periodic potential V and give an upper bound for the Bethe–Sommerfeld threshold (the minimal energy above which no spectral gaps occur) with respect to the square-integrable norm of V on a fundamental domain, provided that V is small. As an application, we prove the spectrum of the two-dimensional Schrödinger operator with the Poisson type random potential almost surely equals the positive real axis or the whole real axis, according as the negative part of the single-site potential equals zero or not. The latter result completes the missing part of the result by Ando et al. (Ann Henri Poincaré 7:145–160, 2006).
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Communicated by Anton Bovier.
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Kaminaga, M., Mine, T. Upper Bound for the Bethe–Sommerfeld Threshold and the Spectrum of the Poisson Random Hamiltonian in Two Dimensions. Ann. Henri Poincaré 14, 37–62 (2013). https://doi.org/10.1007/s00023-012-0180-1
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DOI: https://doi.org/10.1007/s00023-012-0180-1