Bounds on embedded singular spectrum for one-dimensional Schrödinger operators

Source abstract

We show that the solutions of the one-dimensional Schrödinger equation − y + V y = E y -y+Vy=Ey with potential V ( x ) = O ( x − α ) V(x)=O(x^{-\alpha }) satisfy the WKB asymptotic formulae off a set of energies E E of Hausdorff dimension ≤ 2 ( 1 − α ) \le 2(1-\alpha ) . This result gives restrictions on the structure of possible embedded singular spectrum. The proof relies on new norm estimates for the integral transform associated with the WKB method.