Christian Remling

Learn More
I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schrödinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a Denisov-Rakhmanov type theorem for the(More)
We present a direct and rather elementary method for defining and analyzing one-dimensional Schrödinger operators H = −d2/dx2 + μ with measures as potentials. The basic idea is to let the (suitably interpreted) equation −f ′′+μf = zf take center stage. We show that the basic results from direct and inverse spectral theory then carry over to Schrödinger(More)
We study the pointwise behavior of the Fourier transform of the spectral measure for discrete one-dimensional Schrödinger operators with sparse potentials. We find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model. We also present an improved version of known results on the spectrum of such operators.
If V is in L, virtually any perturbation technique allows one to control u (and, in fact, to show that all solutions of (1) are bounded as x → ∞). We are interested in cases where V is not L but is small at infinity in some sense. We want to generalize what has turned out to be a powerful set of tools in case V0 ≡ 0, namely, the use of modified Prüfer(More)
Consider the Schrödinger operator H = −d/dx + V (x) with powerdecaying potential V (x) = O(x−α). We prove that a previously obtained dimensional bound on exceptional sets of the WKB method is sharp in its whole range of validity. The construction relies on pointwise bounds on finite gap potentials. These bounds are obtained by an analysis of the Jacobi(More)