Some Schrödinger operators with dense point spectrum
Some Schrödinger operators with dense point spectrum
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一些具有稠密点谱的薛定谔算子
DOI：
10.1090/s0002-9939-97-03559-4
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发表时间：
1997
影响因子：
1
通讯作者：
B. Simon
中科院分区：
文献类型：
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作者：
B. Simon
给定任意正能量序列{en}n−1和(0，∞)上的任意单调函数g(R)且g(0)=1，→∞g(R)=∞，我们可以找到(−∞，∞)上的位势V(X)，使得{en}n=1是−d2dx2+V(X)和|V(X)|≤(|x|+1)−1g(|x|)的特征值.在文[7]中，Naboko证明了如下结果：定理1.设{κn}∞n=1是一个有理独立的正实数序列。设g(R)是[0，∞)上的单调函数，g(0)=1，→∞g(R)=∞。则在[0，∞)上存在位势V(X)，使得(1){κ2n}∞n=1是−d2dx2+V(X)在[0，∞)上的特征值，且u(0)=0。(2)|V(X)|≤g(X)(|x|+1)。这里我们的目标是构造允许证明下列定理的V‘S：定理2.设{κn}∞n=1是任意不同的正实数序列。设g(R)是[0，∞)上的单调函数，g(0)=1，→∞g(R)=∞。设{θn}∞n=1是[0，π)中的一个角序列。则在[0，∞)上存在位势V(X)，使得(1)对于每个n，(−D2dx2+V(X))u=κnu有一个解，它是无穷远的L且u‘(0)u(0)=cot(θn).(1)(2)|V(X)|≤g(X)|x|+1。∗本材料基于国家科学基金资助的工作。Dms-9401491。政府对这些材料有一定的权利。将提交给Proc。阿默。数学课。SoC。
Given any sequence {En}n−1 of positive energies and any monotone function g(r) on (0,∞) with g(0) = 1, lim r→∞ g(r) = ∞, we can find a potential V (x) on (−∞,∞) so that {En}n=1 are eigenvalues of − d 2 dx2 + V (x) and |V (x)| ≤ (|x| + 1)−1g(|x|). In [7], Naboko proved the following: Theorem 1. Let {κn}∞n=1 be a sequence of rationally independent positive reals. Let g(r) be a monotone function on [0,∞) with g(0) = 1, lim r→∞ g(r) = ∞. Then there exists a potential V (x) on [0,∞) so that (1) {κ2n}∞n=1 are eigenvalues of − d 2 dx2 + V (x) on [0,∞) with u(0) = 0 boundary conditions. (2) |V (x)| ≤ g(x) (|x|+1) . Our goal here is to construct V ’s that allow the proof of the following theorem: Theorem 2. Let {κn}∞n=1 be a sequence of arbitrary distinct positive reals. Let g(r) be a monotone function on [0,∞) with g(0) = 1 and lim r→∞ g(r) = ∞. Let {θn} ∞ n=1 be a sequence of angles in [0, π). Then there exists a potential V (x) on [0,∞) so that (1) For each n, (− d2 dx2 + V (x))u = κnu has a solution which is L at infinity and u′(0) u(0) = cot(θn). (1) (2) |V (x)| ≤ g(x) |x|+1 . ∗ This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The Government has certain rights in this material. To be submitted to Proc. Amer. Math. Soc.