Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner–von Neumann potential
Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner–von Neumann potential
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具有缓慢衰减的维格纳-冯·诺依曼势的薛定谔算子的谱密度零点
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发表时间：
2016
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通讯作者：
S. Simonov
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作者：
S. Simonov
我们考虑具有周期背景势和由两部分组成的扰动的半线上的Schrödinger算子$${mathcal {L}}_{alpha }$$ Lα：可和势和缓慢衰减的Wigner-von Neumann势$$frac{csin (2omega x+delta )}{x^{gamma }}$$ csin(2ωx+δ)xγ，其中$$gamma in (frac{1}{2},1)$$ γ∈(1,1)。该算子的连续谱与无扰动周期算子的连续谱具有相同的带隙结构。在每个波段都存在两个临界点，在这里特征函数方程有平方可求和的解。对于边界参数$$alpha =alpha _{cr}$$ α=αcr的某个值，每个临界点$$
u _{cr}$$ νcr是算子$${mathcal {L}}_{alpha }$$ Lα的特征值，特定于该特定点。证明了当$$alpha
e alpha _{cr}$$ α≠αcr时，算子$${mathcal {L}}_{alpha }$$ Lα的谱密度在$$
u _{cr}$$ νcr处为指数型零。
We consider the Schrödinger operator $${mathcal {L}}_{alpha }$$Lα on the half-line with a periodic background potential and a perturbation which consists of two parts: a summable potential and the slowly decaying Wigner–von Neumann potential $$frac{csin (2omega x+delta )}{x^{gamma }}$$csin(2ωx+δ)xγ, where $$gamma in (frac{1}{2},1)$$γ∈(12,1). The continuous spectrum of this operator has the same band-gap structure as the continuous spectrum of the unperturbed periodic operator. In every band there exist two points, called critical, where the eigenfunction equation has square summable solutions. Every critical point $$
u _{cr}$$νcr is an eigenvalue of the operator $${mathcal {L}}_{alpha }$$Lα for some value of the boundary parameter $$alpha =alpha _{cr}$$α=αcr, specific to that particular point. We prove that for $$alpha
e alpha _{cr}$$α≠αcr the spectral density of the operator $${mathcal {L}}_{alpha }$$Lα has a zero of the exponential type at $$
u _{cr}$$νcr.