On Unique Continuation For The Higher Order Schr(o|¨)dinger Equations In N-dimension

Schr(o|¨)dinger equations are the fundamental equations of Quantum Mechanics, whichhave the important research value in Quantum Mechanics and their application field. Theyhave been aslo well paid widespread attention to in mathematics and physics. Many domes-tic and international scholars have made a lot of research about Schr(o|¨)dinger equations andhave made brilliant achievements, especially the unique continuation of the solutions whichis a hot problem now. At present, people mostly further study the unique continuation of thesolutions against the second order. However, for the higher Schr(o|¨)dinger equations, peopleonly discuss the situation in one dimension.Based on the existing knowledge and the theory of Schr(o|¨)dinger equations, thepaper talks about the unique continuation for the higher order Schr(o|¨)dinger equations inn-dimension. The goal is to obtain abundant conditions on the behavior of the solution uat two different times t0=0and t1=1, which ensure that u≡0is the unique solution.In chapter I, we introduce the theory about Schr(o|¨)dinger equations, the main results ofthe article as well as some marks and inequalities. In chapter II, we make a suitableCarleman estimates to the higher Schr(o|¨)dinger operator firstly. Then, we obtain the uniquecontinuation of solutions of the higher Schr(o|¨)dinger equations in a neighborhood of zero byabove Carleman estimates. In chapter III, we mainly study the global unique continuationof the higher Schr(o|¨)dinger equations. By proving the logarithmic convexity of certainquantities which can measure the appropriate exponential decay at infinity, we get theexponential decay weighted estimates. At last, we can obtain the unique continuation forthe higher order Schr(o|¨)dinger equations in n-dimension in Rn×[0,1] by the exponentialdecay weighted estimates and the local unique continuation of chapter II.