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We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity.
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The effect of metric behavior at spatial infinity on pointwise wave decay in the asymptotically flat stationary setting
abstract: The current work considers solutions to the wave equation on asymptotically flat, stationary, Lorentzian spacetimes in $(1+3)$ dimensions. We investigate the relationship between the rate at which the geometry tends to flat and the pointwise decay rate of solutions. The case where the spacetime tends toward flat at a rate of $$|x|^{-1}$$ was studied by Tataru (2013), where a $$t^{-3}$$ pointwise decay rate was established. Here we extend the result to geometries tending toward flat at a rate of $$|x|^{-\kappa}$$ and establish a pointwise decay rate of $$t^{-\kappa-2}$$ for $$\kappa\in\Bbb{N}$$ with $$\kappa\ge 2$$. We assume a weak local energy decay estimate holds, which restricts the geodesic trapping allowed on the underlying geometry. We use the resolvent to connect the time Fourier Transform of a solution to the Cauchy data. Ultimately the rate of pointwise wave decay depends on the low frequency behavior of the resolvent, which is sensitive to the rate at which the background geometry tends to flat.
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- Award ID(s):
- 2002132
- PAR ID:
- 10568456
- Publisher / Repository:
- American Journal of Mathematics
- Date Published:
- Journal Name:
- American Journal of Mathematics
- Volume:
- 146
- Issue:
- 1
- ISSN:
- 1080-6377
- Page Range / eLocation ID:
- 47 to 105
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1353/ajm.2024.a917539
