一类离散Schr dinger算子本征函数的次指数衰减
Sub-exponential decay of eigenfunctions for some discrete Schrödinger operators
摘要：
按照Froese和Herbst的方法，我们证明了对于一类势V，当λ的估计在E处成立时，在\mathbb{Z}^d上的多维离散薛定谔算子H = $Δ$ + V的特征值为E的特征函数$ψ$是次指数衰减的.在一维情况下，我们进一步证明了这个本征函数至少以cosh^{--1}（（E --2）/（$θ$ \_E --2））的速率指数衰减，其中$θ$ \_E是H位于E和2之间的最近阈值。后一个结果的后果是2和高于和低于该值的最近阈值之间不存在本征值。对离散薛定谔算子的Combes-Thomas方法也作了评述。
英文摘要：
Following the method of Froese and Herbst, we show for a class of potentials V that an eigenfunction $ψ$ with eigenvalue E of the multi-dimensional discrete Schrödinger operator H = $Δ$ + V on \mathbb{Z}^d decays sub-exponentially whenever the Mourre estimate holds at E. In the one-dimensional case we further show that this eigenfunction decays exponentially with a rate at least of cosh^{--1}((E -- 2)/($θ$\_E -- 2)), where $θ$\_E is the nearest threshold of H located between E and 2. A consequence of the latter result is the absence of eigenvalues between 2 and the nearest thresholds above and below this value. The method of Combes-Thomas is also reviewed for the discrete Schrödinger operators.
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