Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials

Krieger

2016

Memoirs of the AMS

Abstract. We study the Cauchy problem for the one-dimensional wave equationThe potential V is assumed to be smooth with asymptotic behaviorWe derive dispersive estimates, energy estimates, and estimates involving the scaling vector field t∂t + x∂x, where the latter are obtained by employing a vector field method on the "distorted" Fourier side. In addition, we prove local energy decay estimates. Our results have immediate applications in the context of geometric evolution problems. The theory developed in this paper is fundamental for the proof of the co-dimension 1 stability of the catenoid under the vanishing mean curvature flow in Minkowski space, see [23].

mentioning

confidence: 99%

A vector field method on the distorted Fourier side and decay for wave equations with potentials

Krieger

2016

Memoirs of the AMS

“…• Theorem 1.2 yields control over all derivatives with respect to E of the involved quantities. This is the most salient feature of the result and it unveils the main difference to Schrödinger operators with potentials that exhibit power law decay where one loses powers of E upon differentiating with respect to E, see e.g., [4]. Such behavior is in stark contrast to the situation here where one only loses powers of and such a loss is in fact negligible compared to the size of e − 1 .…”

Section: Introductionmentioning

confidence: 67%

“…As a consequence, the Regge-Wheeler potential is a prominent example from mathematical physics where our results apply, at least for x ≤ 0. In the case x ≥ 0, where the potential decays like an inverse square, one has to rely on [4]. In fact, in [7] it is shown how to synthesize [4] and the present results to obtain the sharp t −3 decay for linear perturbations of Schwarzschild without symmetry assumptions on the data.…”

Section: Introductionmentioning

confidence: 81%

“…However, we find that the available results are not suitable for our purposes. The only paper we are aware of that studies the double asymptotics E, → 0+ in a way similar to us is [4] which deals with inverse square potentials. Since the Regge-Wheeler potential exhibits inverse square decay towards spatial infinity, [4] is used to deal with the far field region in our main application which we now describe in more detail.…”

Section: Introductionmentioning

confidence: 99%

“…In the latter case one speaks of the existence of a zero energy resonance. However, we are not faced with this complication since it is not hard to see that the positivity assumption on the potential already excludes the existence of a zero energy resonance, see [4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Semiclassical Low Energy Scattering for One-Dimensional Schrödinger Operators with Exponentially Decaying Potentials

Costin

Schlag

et al. 2011

Ann. Henri Poincaré

Self Cite

Abstract. We consider semiclassical Schrödinger operators on the real line of the form) with > 0 small. The potential V is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions f±(·, E; ) with error terms that are uniformly controlled for small E and , and construct the scattering matrix as well as the semiclassical spectral measure associated to H( ). This is crucial in order to obtain decay bounds for the corresponding wave and Schrödinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta ℓ where the role of the small parameter is played by ℓ −1 . It follows from the results in this paper and [7], that the decay bounds obtained in [8], [6] for individual angular momenta ℓ can be summed to yield the sharp t −3 decay for data without symmetry assumptions.

On Pointwise Decay of Linear Waves on a Schwarzschild Black Hole Background

Schlag

Soffer

2011

Commun. Math. Phys.

Self Cite

117

121

We prove sharp pointwise t −3 decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates t −4 , and t −6 , respectively. We proceed by decomposition into angular momentum ℓ and summation of the decay estimates on the Regge-Wheeler equation for fixed ℓ. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in ℓ is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.1 1 The notation a± stands for a± ε where ε > 0 is arbitrary (the choice determines the constants involved). Also, instead of (/ ∇ 10 , / ∇ 9 ) in (1.3) one needs less, namely (/ ∇ σ+1 , / ∇ σ ) where σ > 8 is arbitrary, see the proof in Section 5 for details.2 From the point of view of the decay estimates in [17], these values need to be excluded as they are precisely the ones that give rise to a zero energy resonance.