Publication Date:
2012-05-01
Description:
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 ). Content Type Journal Article Pages 1-26 DOI 10.1007/s00023-012-0180-1 Authors Masahiro Kaminaga, Department of Electrical Engineering and Information Technology, Tohoku Gakuin University, Tagajo, 985-8537 Japan Takuya Mine, Graduate School of Science and Technology, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, 606-8585 Japan Journal Annales Henri Poincare Online ISSN 1424-0661 Print ISSN 1424-0637
Print ISSN:
1424-0637
Electronic ISSN:
1424-0661
Topics:
Mathematics
,
Physics
