On Uniqueness Properties Of Solutions For Higher-order Schr(o|¨)dinger Equations

Since the twenty century, theory of Schr(o|¨)dinger equations has constantly been an im-portant subject of Mathematical Physics,which has covered scattering estimates, spectralanalysis, maximal operator’s estimates,local smoothing estimates, self-adjiont propertiesand so on,the study on this theory is of profound theoretical in mathematical and quan-tum mechanics. this thesis will discuss uniqueness continuation Properties of Schr(o|¨)dingerequations with time. At present, the case with generalized Schr(o|¨)dinger equation is relativelymature,scholars extend the methods of Hardy’s uncertainty principles to Schr(o|¨)dinger equa-tions with non-constant coefficients, use the behavior of the solution at two different times,t0, t1,to control the behavior of the solution in [tβx0, t1],so that∥eu(x, t)∥L2has lowerbound, then we choose reasonable potential V (x, t) and the solution,which guarantee thatthe solution is identically equal zero. Compared with generalized Schr(o|¨)dinger equation, theunique properties on the integral order of higher order Schr(o|¨)dinger equation is not perfect,but we have operator’s heat kernel estimates,△~m(m≥2is integer),which show that thedecay index can be increase to e|x|2m2m1.In order to get lower bound estimate, the key ofthis thesis is to choose reasonable weighted index and deal with lower derivative term.Based on the theories of generalized Schr(o|¨)dinger equation and KDV equation, thewhole thesis will continue to study the unique properties on the integral order of higherorder Schr(o|¨)dinger equation, the main arrangements are as follows: the fist of all, we intro-duce some basic concepts, major marks, and inequality; secondly, we prove the solutionof Schr(o|¨)dinger equation is finite, make a reasonable weighted index to get logarithmicconvexity, which can obtain lower bound of the solution, as well as Carleman inequality;finally, using these lemma we prove higher order Schr(o|¨)dinger equation also has uniquesolution, whose method is similar to generalized Schr(o|¨)dinger equation’s. Then, we pointsome areas that can be improved.