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We establish dispersive estimates and local decay estimates for the time evolution of non-self-adjoint matrix Schrödinger operators with threshold resonances in one space dimension. In particular, we show that the decay rates in the weighted setting are the same as in the regular case after subtracting a finite rank operator corresponding to the threshold resonances. Such matrix Schrödinger operators naturally arise from linearizing a focusing nonlinear Schrödinger equation around a solitary wave. It is known that the linearized operator for the 1D focusing cubic NLS equation exhibits a threshold resonance. We also include an observation of a favorable structure in the quadratic nonlinearity of the evolution equation for perturbations of solitary waves of the 1D focusing cubic NLS equation.
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On pointwise decay of waves
This paper introduces some of the basic mechanisms relating the behavior of the spectral measure of Schrödinger operators near zero energy to the long-term decay and dispersion of the associated Schrödinger and wave evolutions. These principles are illustrated by means of the author’s work on decay of Schrödinger and wave equations under various types of perturbations, including those of the underlying metric. In particular, we consider local decay of solutions to the linear Schrödinger and wave equations on curved backgrounds that exhibit trapping. A particular application is waves on a Schwarzschild black hole spacetime. We elaborate on Price’s law of local decay that accelerates with the angular momentum, which has recently been settled by Hintz, also in the much more difficult Kerr black hole setting. While the author’s work on the same topic was conducted ten years ago, the global semiclassical representation techniques developed there have recently been applied by Krieger, Miao, and the author to the nonlinear problem of stability of blowup solutions to critical wave maps under non-equivariant perturbations.
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- Award ID(s):
- 1902691
- PAR ID:
- 10331677
- Date Published:
- Journal Name:
- Journal of mathematical physics
- Volume:
- 62
- Issue:
- 6
- ISSN:
- 0022-2488
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1063/5.0042767
