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Journals and Conferences
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)
This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω 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 such… (More)
We present an approach to de Branges’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 Schrödinger operators and to solve the inverse spectral problem.
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.
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 Hβ = −d2/dx2 + V (x), acting on the Hilbert space L2(0,∞). The index β ∈ [0,π) refers to the boundary condition y(0)cosβ + y′(0)sin β = 0. For the general theory, see, for example, . These questions are of relevance in quantum mechanics; for more background… (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 Simon’s A functions in terms of a positivity condition. This condition also characterizes the solubility of Simon’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 → ∞). 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)