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\(L^p\)-boundedness of the wave operator for the one dimensional Schrödinger operator.(English)Zbl 1127.35053

Summary: Given a one-dimensional perturbed Schrödinger operator \(H = - d^{2}/ dx ^{2} + V (x)\), we consider the associated wave operators \(W_{\pm}\), defined as the strong \(L^{2}\) limits \(\lim_{s\to\pm\infty}e^{isH}e^{-isH_{0}}\). We prove that \(W_{\pm}\) are bounded operators on \(L^{p}\) for all \(1 < p < \infty\), provided \((1+|x|)^{2}V(x)\in L^{1}\), or else \((1+|x|)V(x)\in L^{1}\) and 0 is not a resonance. For \(p = \infty\) we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.

MSC:

35Q40 PDEs in connection with quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47A40 Scattering theory of linear operators
47A50 Equations and inequalities involving linear operators, with vector unknowns
35B45 A priori estimates in context of PDEs

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References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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