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I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional SchrÃ¶dinger 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 theâ€¦ (More)

This paper deals with general structural properties of one-dimensional SchrÃ¶dinger operators with some absolutely continuous spectrum. The basic result says that the Ï‰ limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for suchâ€¦ (More)

We present an approach to de Brangesâ€™s theory of Hilbert spaces of entire functions that emphasizes the connections to the spectral theory of differential operators. The theory is used to discuss the spectral representation of one-dimensional SchrÃ¶dinger operators and to solve the inverse spectral problem.

We present a direct and rather elementary method for defining and analyzing one-dimensional SchrÃ¶dinger operators H = âˆ’d2/dx2 + Î¼ with measures as potentials. The basic idea is to let the (suitably interpreted) equation âˆ’f â€²â€²+Î¼f = zf take center stage. We show that the basic results from direct and inverse spectral theory then carry over to SchrÃ¶dingerâ€¦ (More)

- Denis Krutikov, Christian Remling
- 2001

We study the pointwise behavior of the Fourier transform of the spectral measure for discrete one-dimensional SchrÃ¶dinger 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.

We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a special type? For example, can we approximate by periodic operators?

More specifically, we study the spectral properties of the associated selfadjoint operators HÎ² = âˆ’d2/dx2 + V (x), acting on the Hilbert space L2(0,âˆž). The index Î² âˆˆ [0,Ï€) refers to the boundary condition y(0)cosÎ² + yâ€²(0)sin Î² = 0. For the general theory, see, for example, [29]. These questions are of relevance in quantum mechanics; for more backgroundâ€¦ (More)

We link recently developed approaches to the inverse spectral problem (due to Simon and myself, respectively). We obtain a description of the set of Simonâ€™s A functions in terms of a positivity condition. This condition also characterizes the solubility of Simonâ€™s fundamental equation.

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 â†’ âˆž). 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 â‰¡ 0, namely, the use of modified PrÃ¼ferâ€¦ (More)

Consider the SchrÃ¶dinger operator H = âˆ’d/dx + V (x) with powerdecaying potential V (x) = O(xâˆ’Î±). 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 Jacobiâ€¦ (More)