Upper Bound for the Bethe-Sommerfeld Threshold and the Spectrum of the Poisson Random Hamiltonian in Two Dimensions

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).