Some Studies On The Decay Estimates Of A Class Of Sixth-order Schr(?)dinger Operators Under The Resonance Conditions

In this paper,we devote to study the decay estimates of the sixth-order Schr???dinger operator H=?-??³+ V?x?with a decaying potential V?x?in the Eyropean space R⁶.Specifically,we mainly prove the Kato-Jensen decay estimates of the Schr???dinger group e^-itH about the time t under various resonance assumptions at the origin.The related estimates play an important role in the study about the well-posedness of the solution of higher-order nonlinear Schr???dinger equations.The main organizations of this paper are as follows:Firstly,we obtain the lower energy asymptotic expansions of R₀?z?according tothe lower energy asymptotic expansions of the free Schr???dinger resolvent.Then we calculate the expansions of M???^-1 under the various resonance assumptions at the origin,the lower energy asymptotic expansions of R_v?z?is obtained by the symmetric resolvent formula.Secondly,we give a description of the resonance space S_iL²,1?i?i?5,that is itsrelationship with the distribution solution of the equation H?=0.Then we obtain the higher energy decay estimates of R₀?z?according to the higher energy decay estimates of the free Schr???dinger resolvent,so as to obtain the relevant estimates of R_V?z?;and continuously extend R_v?z?to the positive semi-axis by the limiting absorption principal and study its boundary value behavior.In the end,we obtain the lower energy asymptotic expansions and the higher energy decay estimates of the spectral density from the Stone's formula;and then using the oscillation integration technique,we prove the Kato-Jensen decay estimates of the Schr???dinger group e^-it??-??3+v?.