Spectral Analysis Of Fractional Schr(o|¨)dinger Operators

Schr(o|¨)dinger equation is a fundamental equation of quantum mechanics also is a basicassumption of quantum mechanics. Its theory is built on the basis of mathematical physics.There are very many research results, such as self-adjointness, semi-group property, scatter-ing theory, Strichartz estimation, smooth estimation, etc. Since in recent years, the fractionalSchr(o|¨)dinger operator has become more and more concerned by people. At present there arealready some conclusion, such as the nature of the direction of an ultra smooth estimates,nonlinear compressibility, situation of the existence and uniqueness of solutions, and spec-tral analysis, etc. At present the spectral analysis of Schr(o|¨)dinger operator is mainly directedagainst the integer times and in the case with peacekeeping potentials already has the relatedtheory.For the spectral analysis of Schr(o|¨)dinger operator, is mainly to the multi-particle Hamil-tonians operator. In the case with potentials, proof is to one fixed open interval, any pointspectrum of H has finite multiplicity and there is no accumulation point of point spectrum,and at the same time in the interval there is no singular continuous spectrum of H. Ourresearch mainly is based on the operator theory that three-body, N-body of the first-ordernegative Laplace operator with potentials, improved research is the fractional situation. Dif-ferent from the situation of integer times, we use the Fourier analysis method to calculatethe commutator of fractional operators. The potential putted forward properly will satisfythe compactness and boundedness in the space. Then we use Perry’s method of the integeroperator to analyze the properties of point spectrum and singular continuous spectrum.The structure of this paper are as follows. In the first chapter of the introduction, thispaper introduces some relevant theoretical basis and the existing main job. The secondchapter to formulate the main theorems, the conclusion of the first two parts, that is, toprove that any point spectrum of H in the interval has finite multiplicity and there is noaccumulation point of point spectrum. In the third chapter of this last will be the thirdconclusion, that in the interval there is no singular continuous spectrum of H.