Decay Estimates For The Fourth-order Schr(?)dinger Operator And Its Applications In Nonlinear Dispersive Equations

The study of classical Schrodinger operator-? + V originates from nonrela-tivistic quantum mechanics.The Schrodinger operator-? + V has become the core object of mathematical research after more than half 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 dispersion estimates of Schrodinger operator play an indispensable role in the study of the wellposedness and scattering theory of the nonlinear Schrodinger equation.As a natural generalization of the second-order Schrodinger operator,we mainly discusses the fourth-order Schrodinger operator ?2 + V in this paper.It has im-portant applications in nonlinear fourth-order Schrodinger equation,beam equation and conformal geometry.In this paper,we systematically study various disper-sion estimators of fourth-order Schrodinger operator.At the same time,we discuss the embedded eigenvalue problem of the general higher order Schrodinger opera-tor.Finally,as an application,we study the scattering of the nonlinear fourth-order Schrodinger equation.The thesis consists of six chapters:In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some preliminary results and notations used in the whole thesis.In Chapter Two,we study the low energy asymptotic estimate and high energy-decay estimate of resolvent R(?2 + V;z).The low energy asymptotic estimate that is the asymptotic behaviour of resolvent as z ? 0 in the weighted Sobolev space H?s(Rd).The high energy decay estimate that is to obtain the decay rate of resolvent as z ? ? in the weighted Sobolev space H?s(Rd).Based on the resolvent estimates,by the limiting absorption principle,we derive the asymptotic behaviour of the spectral density dE(?)of operator ?2 + V as ? ? 0,and the decay estimate of dE(?)as ?? ?.In Chapter Three,we prove the local decay estimate and Kato-Jensen estimates for fourth-order Schrodinger operator ?2 + V based on the resolvent estimates.At the last section of this chapter,starting from the local decay estimate and using the abstract positive commutator method,we prove the Kato-Jensen type pointwise decay estimate for the fourth-order Schrodinger propagator eit(?2+V).In Chapter Four,using the local decay estimate and Kato-Jensen decay esti-mate,we prove Strichartz estimates for the fourth-order Schrodinger equation and the Lp-decay estimate,that is the L1 ? L2(Rd)?L? + L2(Rd)of propagator eit(?2+V).In the three dimensional case,we obtain the L1(R3)? L?(R3)decay estimate.In Chapter Five,we study the absence of positive embedded eigenvalues for higher order Schrodinger type operators P(D)+ V where P is a homogeneous elliptic polynomial of m degree.In the first section,for some higher order differential operators P(D),we construct potential V ? C0?(Rd)such that P(D)+ V still exist positive eigenvalue embedded in the continuous spectrum.In the second section,using the virial identity of operator P(D)+ V,we prove an absence of embedded eigenvalue criterion of operator P(D)+ V.In Chapter Six,we study scattering in energy space H2(Rd)of the following nonlinear fourth-order Schrodinger equation iut +(?2 + V)u + ?|u|p-1u = 0,(t,x)? R × Rd,u(0,x)= u0(x),Based on the Strichartz estimates which we have proved in Chapter Four,we de-rive Strichartz type estimates with gain of derivatives.Furthermore,we apply the Strichartz type estimates to obtain the global wellposedness of the above initial data problem.In the case d ? 7,using Morawetz estimate,we establish the scattering in the energy space for the above nonlinear fourth-order Schrodinger equation.