Some Schrödinger Operators with Dense Point Spectrum

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.