Bounds on embedded singular spectrum for one-dimensional Schrödinger operators

Remling, Christian

doi:10.1090/S0002-9939-99-05110-2

Bounds on embedded singular spectrum for one-dimensional Schrödinger operators
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Proc. Amer. Math. Soc. 128 (2000), 161-171 Request permission

Abstract:

We show that the solutions of the one-dimensional Schrödinger equation $-y''+Vy=Ey$ with potential $V(x)=O(x^{-\alpha })$ satisfy the WKB asymptotic formulae off a set of energies $E$ of Hausdorff dimension $\le 2(1-\alpha )$. 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.

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