Absolute Continuity of the “even” Periodic Schrödinger Operator with Nonsmooth Coefficients

Condition 2. a) The matrix-valued function g (metric) is positive and bounded, c01 ≤ g(x) ≤ c11, 0 < c0 ≤ c1 < ∞. b) The magnetic potential A and the electric potential V belong to the following classes : A ∈ Lq,loc, V ∈ Lq/2,loc, where q = d if d ≥ 3, and q > 2 if d = 2. Under Conditions 1 and 2, hRd is a semibounded closed form. The selfadjoint operator H corresponding to hRd will be called the Schrödinger operator. At present, the absolute continuity of the spectrum of H is known under the following assumptions. For d = 2, it suffices to assume that det g ∈ W 1 q,loc, q > 2 (see [13]). For d ≥ 3, absolute continuity was proved if g(x) = a(x)1, where a is a scalar function, a ∈ C, A ∈ H loc, s > (3d− 2)/2, and V ∈ Lp,loc, p = max{d/2, d− 2} (see [9]). In [12], Friedlander considered the situation under an additional condition of “evenness”.