Self-adjointness And Essential Spectra Of Fractional Power Of Schr(o|¨)dinger Operator

The theory of Schr（o|¨）dinger equation has been the central topic of analysis mathemat-ics because of its intuitive physical background and its application value. Therefore theself-adjointness and spectral analysis, as the basic problems of Schr（o|¨）dinger equation, arealso concerned by many mathematicians. The study on the second Schr（o|¨）dinger operatorhas made great achievements, and the method of study has become perfect, so the studyof self-adjointness and spectra also has become perfect. For example, for the problems ofself-adjointness, they can be solved by perturbation theorem or Kato’s inequality or decom-position of tensor product. Fractional power of Schr（o|¨）dinger operator, as the nature extensionof the second classical case, has attracted the interest of mathematicians recently. However,the study on it is more diffcult, because of lack of necessary technical means, as well asa less intuitive physical background. But the study has also gotten some results becauseof the efforts of mathematicians, such as the smooth estimation of Schr（o|¨）dinger equation.Certainly there exist many problems which is to be solved.In the paper, we mainly studyself-adjointness and spectral analysis of fractional power of Schr（o|¨）dinger equation.For the fractional power of Schr（o|¨）dinger equation, we study its adjointness and spectraaccording to different potentials. The paper is divided into four parts. First, the frst chapteris the introduction, the background and development of fractional power of Schr（o|¨）dingerequation are introduced briefy; in the second chapter, some classical conclusions and meth-ods of second-order Schr（o|¨）dinger operator are quoted; in the third chapter, self-adjonintnessof fractional power of Schr（o|¨）dinger operator under two different potentials are studyed; inthe last part, a property of essential spectrum of fractional power of Schr（o|¨）dinger operator isdiscussed. On the proof of main conclusions, the methods in classical case are promotedand applicated to the case of fractional power of Schr（o|¨）dinger operator. After main content,some shortcomings of conclusions and the content which can be continued to improved arepointed out.