Time Decay Estimates For Higher-order Schr(?)dinger Groups

The study for second-order Schrodinger operator-Δ+V originates from nonrelativistic quantum mechanics.The Schrodinger operator-Δ+V has become the core object of mathematical research after nearly a century of deep development.It not only has rich theoretical research contents,but also has extensive contacts and applications in many fields such as harmonic analysis,partial differential equation and differential geometry.Especially in the last 20 years,the deacy estimates of Schrodinger operator play an indispensable role in the study of the wellposedness and scattering theory of the nonlinear Schrodinger equation.The study for higher-order elliptic operator is a natural development of the theory for second-order Schrodinger operator,it is concerned and studied by a lot of mathematics.In this thesis,we are devoted to study the decay estimates for the following higher-order Schrodinger type operators:H=(-Δ)m+V(x),H0=(-Δ)m,where m≥2 and m∈N+,the potential V(x)is a real-valued measurable function in Rn which satisfies |V(x)|(?)<x>-β with β>0.The thesis consists of eight chapters,it mainly includes the following two parts work:In the first part(in chapters two to four),by the estimates of resolvent and the oscillatory integral method,we establish the following sharp L1-L∞ estimate for fourth-order Schrodinger groups e-it(Δ2+V)in dimension one:‖e-it(Δ2+V)Pac‖L1(R)→L∞(R)(?)|t|-1/4,t≠0,where Pac denotes the projection onto the absolutely continuous spectrum subspace of Δ2+V.Especially,we prove that zero resonances of Δ2+V do not affect the time decay rate of s-it(Δ2+V).The main results in this part have been published in Annales Henri Poincare.In the second part(in chapters five to seven),for higher-order Schrodinger type operators,we firstly establish the low energy asymptotic expansions for the resolvent Rv(z)=((-Δ)m+V-z)-1 of with the presence of zero resonances.Then,by the estimates of resolvent and the oscillatory integral method,we prove the following Kato-Jensen’s type estimates for higher-order Schr?dinger groups e-it(-Δ)m+V)with n>2m:‖<x>-σe-it((-Δ)m+V)Pac<x>-σ‖L2(Rn)→L2(Rn)(?)<t>-σ(m,n),where Pac denotes the projection onto the absolutely continuous spectrum subspace of(-Δ)m+V,σ(m,n)and s depend on the order m,the dimension n and the type of zero resonance for(-Δ)m+V.Finally,as an application of Kato-Jensen’s type estimates,we establish the time-space estimate—Strichartz’s typee estiamte for the solution of higher-order nonlinear Schrodinger equation when zero is a regular spectrum of(-Δ)m+V.The main results in this part have been published in Transactions of The American Mathematical Society.We point out that both of the above decay estimates depend on the asymptotic expansions near zero for the resolvent RV(z)=((-Δ)m+V-z)-1.As m≥2,the origin is the degenerate critical point for |ξ|2m which is the symbol of(-Δ)m,it makes the asymptotic expansions of RV(z)very complicated when zero resonances occur.The study for zero resonances classification and low energy asymptotic expansions of RV(z)is an important part of the thesis.