Decay Estimates for Bi-Schrodinger Operators in Dimension One

Authors：Soffer, Avy^*; Wu, Zhao; Yao, Xiaohua

Source：Annales Henri Poincare, 2022, 23(8): 2683-2744.

DOI：10.1007/s00023-021-01147-9

• ZERO-ENERGY
• INVERSE SCATTERING
• WAVE-OPERATORS
• ELLIPTIC PDES
• TIME-DECAY
• EQUATION
• EIGENFUNCTIONS
• EXPANSIONS

This paper is devoted to study the time decay estimates for bi-Schrodinger 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)(R). Next, we use them to establish the sharp L-1 - L-infinity decay estimates of Schrodinger groups e(-itH) generated by bi-Schrodinger operators also with zero resonances. As a consequence, Strichartz estimates are obtained for the solution of fourthorder Schrodinger equations with potentials for initial data in L-2(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.