Dispersive Estimates for Schrödinger Operators in Dimension Two

Schlag, Wilhelm

doi:10.1007/s00220-004-1262-9

Cited by 152 publications

(241 citation statements)

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“…Statements of this type appear in the work of Murata [9] and Buslaev-Perelman [2], with weighted L 2 (R) as the underlying space. A weighted L 1 → L ∞ bound was proven recently by Schlag [11]. Our first theorem is a refinement of Schlag's result.…”

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confidence: 55%

Transport in the one-dimensional Schrödinger equation

Goldberg

2007

Proc. Amer. Math. Soc.

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Abstract. We prove a dispersive estimate for the Schrödinger equation on the real line, mapping between weighted L p spaces with stronger time-decay (|t|2 ) than is possible on unweighted spaces. To satisfy this bound, the long-term behavior of solutions must include transport away from the origin. Our primary requirements are that x 3 V be integrable and −∆+V not have a resonance at zero energy. If a resonance is present (for example, in the free case), similar estimates are valid after projecting away from a rank-one subspace corresponding to the resonance.In one dimension, the linear propagator of the free Schrödinger equation is given by the explicit convolutionThis immediately gives rise to the dispersive estimateSuch an estimate cannot be true in general for the perturbed operator H = −∆ + V (x). Even small perturbations of the Laplacian may lead to the formation of bound states, i.e. functions f j ∈ L 2 satisfying Hf j = −E j f j . Bound states with strictly negative energy are known to possess exponential decay, hence they belong to the entire range of L p (R), 1 ≤ p ≤ ∞. For each of these bound states f j , the associated evolution e itH f j = e −itE j f j clearly violates (1). It is well known [3,10] that if V ∈ L 1 (R), then the pure-point spectrum of H consists of at most countably many eigenvalues −E j < 0. The absolutely continuous spectrum of H is the entire positive half-line, and there is no singular continuous spectrum. Bound states can therefore be removed easily via a spectral projection, suggesting that one should look instead for dispersive estimates of the formThe condition V ∈ L 1 does not always guarantee regularity at the endpoint of the continuous spectrum. We say that zero is a resonance of H if there exists a bounded solution to the equation Hf = 0. Since resonances are not removed by the

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confidence: 55%

Transport in the one-dimensional Schrödinger equation

Goldberg

2007

Proc. Amer. Math. Soc.

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“…In fact, in that case the harder L 1 (R d ) → L ∞ (R d ) estimate is now known in all dimensions d ≥ 1 under the condition that zero energy is neither an eigenvalue nor a resonance (and there are now also results in the case when the latter assumption does not hold). The seminal paper for this class of estimates is [14] and we refer the reader to [25] for a survey of more recent work.…”

Section: Introductionmentioning

confidence: 99%

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $\mathbb{R}^3$

2008

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Abstract. We present a novel approach for bounding the resolvent offor large energies. It is shown here that there exist a large integer m and a large number λ 0 so that relative to the usual weighted L 2 -norm,for all λ > λ 0 . This requires suitable decay and smoothness conditions on A, V . The estimate (2) is trivial when A = 0, but difficult for large A since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and then sum over all possible combinations of cones. Chains of cones that all point in the same direction lead to a Volterra-type gain of the form (m!) −ε with ε > 0 fixed. On the other hand, cones that are not aligned contribute little due to the assumed decay of A. We make no use of micro-local analysis, but instead rely on classical phase space techniques. As a corollary of (2), we show that the time evolution of the operator in R 3 satisfies global Strichartz and smoothing estimates without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance.

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“…Several results are available for the equations i∂ t u − ∆u + V (x)u = 0, u + V (x)u = 0. We cite among the others [8], [15], [16], [19], [32] and the recent survey [33] for Schrö-dinger; and [5], [6], [10], [12], [13] for the wave equation. We must also mention the wave operator approach of Yajima (see [2], [39], [40], [41]) which permits to deal with the above equations in a unified way, although under nonoptimal assumptions on the potential in dimensions 1 and 3.…”

Section: Introductionmentioning

confidence: 99%