With Non-degenerate With Variable Coefficients Schr (?) Dinger Equation For The Study

In the present thesis, we consider the following Schodinger equation with non-degenerated coefficients,where q_ij(x) are asymptotic to constant , and the matrix (q_ij) is non-degenerated, and non-trapping condition is also satisfied. Two main theorems are obtained.Theorem A Let non-degenerated condition and non-trapping condition be satisfied . Then for any 0 < T <∞. above problem has the unique solution and the solution satisfieshere (q, r) are non-endpoint admissible pairs.Theorem B Let non-degenerated condition and non-trapping condition be satisfied . And q_ij are constant outside the bounded domain. Above problem has the unique solution and the solution satisfieshere (q, r) are admissible pairs.There are five chapters in this thesis. In chapter 1, we introduced the physical background of the Schodinger equation with non-degenerated coefficients. And non-trapping condition is introduced. The main result of our work is also presented .In chapter 2, we proved the global existence of Hamilton flow (x^t,ξ^t) under assumption of non-trapping condition. And the estimation of (x^t,ξ^t) is also derived. Through Kenig's method [35], we proved local smoothing property of Schrodinger equation with non-degenerated coefficients, where the regularity of coefficients isC^2+ε. In chapter 3, we derived the Strichartz estimate in the bounded domain. Through wave-packets transformation, we construct the exact parametrix of dispersive operator. Then we reduced the Strichartz estimate into the dyadic one,ie,(?),where S_λis the dyadic operator. Following Tataru's suggestion, we employed the exact parametrix in[58] and scaling method to prove above inequality. Compared with other work, our result is more general and the whole proof process is much simpler. It's just the local smoothing property that links the energy inequality and Strichartz inequality.In order to prove the Strichatz estimate in the unbounded domain, we need to simplify the problem. We first construct the elliptic operator of second order, such that [P,Q] = PQ - QP∈S⁰. Through a lemma of Chirst-Kiselev, we reduced the problem into proving inequality,whereχis a cut-off function. In the last chapter, above inequality is proved. Following Egorov's idea, we construct a phase function (?)(x,ξ), such that,Then we construct the whole space's Isozki-Kitada parametrix of twice order operator with non-degenerated coefficients. To complete the whole proof, we employed the TT^* argument and stationary phase method. Then we derived the Strichartz inequality,Our result is more general than works of Robbiano-Zuily and Bouclet-Tzvetkov.