Decay estimates for biSchrödinger operators in dimension one
Abstract
This paper is devoted to study the time decay estimates for biSchrödinger operators $H=\Delta^{2}+V(x)$ in dimension one with decaying potentials $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ at zero energy threshold without/with the presence of resonances, and then characterize these resonance spaces corresponding to different types of zero resonance in suitable weighted spaces $L_s^2({\mathbf{R}})$. Next we use them to establish the sharp $L^1L^\infty$ decay estimates of Schrödinger groups $e^{itH}$ generated by biSchrödinger operators also with zero resonances. As a consequence, Strichartz estimates are obtained for the solution of fourthorder Schrödinger equations with potentials for initial data in $L^2({\mathbf{R}})$. In particular, it should be emphasized that the presence of zero resonances does not change the optimal time decay rate of $e^{itH}$ in dimension one, except at requiring faster decay rate of the potential.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.15966
 Bibcode:
 2021arXiv210615966S
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics
 EPrint:
 49 pages. This is a final version which will be published in Annales Henri Poincare