Lower Bound Estimates For A Class Of Higher-order Schr(o|")dinger Equation

Since the1920s, Schrodinger operator theory has been one of the central topics of mod-ern mathematic and physics. With further research on Harmonic Analysis, people gradually come to realize the importance of Schrodinger operator theory, which has been widely ap-plied to the linear and nonlinear problems. Higher order Schrodinger operator P(D)+V, as a generalization of Schrodinger operator—Δ+V, not only retains parts of profound results of lower order case, but also has its own characteristics which are more abundant and difficult. Although research methods and conclusions of the second-order provide ideas and approaches for higher order, plenty of results in higher order case are not just simple extension of the second order, more often,we are compelled to seek for more effective mea-sures to deal with higher order Schrodinger operator. Therefore, the study of higher order Schrodinger operator would in turn show its value by injecting new vitality into the second-order Schrodinger operator theory. It helps to deepen our understanding of Schrodinger operator and would make Schrodinger operator theory more dazzling and prolific.The upper bound of solution operator of a class of higher order Schrodinger equation can be estimated in the method developed by Miyachi, meanwhile the lower bound of solution operator, estimated by Philip Brenner’s approach, can be extended to high-er order situation. It is consoling that the index of the upper and lower bound are all optimal.