Absolute continuity of the “even" periodic Schrödinger operator with nonsmooth coefficients
Absolute continuity of the “even" periodic Schrödinger operator with nonsmooth coefficients
复制标题
具有非光滑系数的“偶”周期薛定谔算子的绝对连续性
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
N. Filonov
中科院分区：
文献类型：
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作者：
M. Tikhomirov;N. Filonov
条件2.a)矩阵值函数g(度量)为正且有界，c01≤g(X)≤c11，0<；c0≤c1<；∞。B)磁势A和电势V属于下列类别：A∈Lq，loc，V∈Lq/2，loc，其中q=d，如果d≥3，则q>；2，当d=2时。在条件1和条件2下，hrd是半有界闭形式。对应于HRD的自伴算符H称为薛定谔算符。目前，在下列假设下，H的谱的绝对连续性是已知的。对于d=2，只要假设Det g∈W1Q，LOC，Q>；2就足够了(见[13])。对于d≥3，证明了当g(X)=a(X)1，其中a为标量函数，a∈C，A∈H loc，S>；(3D−2)/2，V∈Lp，loc，p=max{d/2，d−2}(见[9]).在[12]中，Friedlander考虑了在“均匀”的附加条件下的情况。
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”.