The profile decomposition for the hyperbolic Schrödinger equation

Abstract

In this note, we prove the profile decomposition for hyperbolic Schrödinger (or mixed signature) equations on ${\mathbb{R}}^2$ in two cases, one mass-supercritical and one mass-critical. First, as a warm up, we show that the profile decomposition works for the ${\dot{H}}^{\frac{1}{2}}$ critical problem, which gives a simple generalization of for instance one of the results in Fanelli-Visciglia (2013). Then, we give the derivation of the profile decomposition in the mass-critical case by proving an improved Strichartz estimate. We will use a very similar approach to that laid out in the notes of Killip-Visan (2008), but we are forced to do a double Whitney decomposition to accommodate an extra scaling symmetry that arises in the problem with mixed signature.

Version 2 includes comments from an anonymous referee in particular with properly citing the proof of a similar estimate by Rogers and Vargas in Ref. 24. Ver. 3 contains a corrected version of the Appendix on Strichartz Extremizers thanks to Carneiro-Oliveira-Sousa in arXiv:1911.11796 (they are not Gaussians!). An Erratum is submitted to the journal version to note this as well