Spectra Of Differential Operators Cf2n-tn Order With Periodic Coefficients

The spectra theory of differential operators is an important part in differentialoperator theory, also is the main research part of functional analysis. Manymathematical problems in quantum mechanics and physics have been due to theoperator problems on the linear space. The operator theory not only is a powerfultool for solving the problems of modern mathematics, but also is an important pillarof modern physics and quantum mechanics. Until now, many Chinese and foreignexcellent mathematical workers in differential operator areas continue to expand thisareas, the differential operator theory has risen rapidly in last fifty or sixty years andhas formed a set of theoretical. At the same time, many new problems have beenraised with the development of the differential operator theory.M.S.P. Eastham, a Scottish mathematics expert, has a great contribution to thespectrum of the second-order Sturm-Liouville differential operators with realperiodic coefficients (i.e. the Hill’s operators), draw the conclusion that the Hill’soperators are the self-adjoint operators, their spectra are purely continuous, and thespectra consist of infinite closed intervals in real axis. The closed intervals aregenerally separated by the gaps, these gaps are equivalent to the stable interval of thecorresponding S-L equation. To M.S.P. Eastham’s research as the foundation, thespectra of the second-order S-L differential operators with complex periodiccoefficients have studied in the same way in domestic, obtains these differentialoperators are J-self-adjoint operators, their spectra are purely continuous, and thevast majority spectra composed of analytic segment on the complex plane. Here, wecannot but suspect that, promotes to the higher order situation whether still had thesimilar conclusion? The answer is affirmative. Present thesis will demonstrate it indetail.This thesis mainly divides into four parts: the first part is the introduction, wemainly introduce the development of the differential operator areas, as well as thepurpose, the anticipated result, the practical significance; The second chapter is thepreparation knowledge, state some useful concepts, the theorem and the conclusion,including the Folquet theory, the solution of differential equation with complex periodic coefficients and so on; The third chapter and the fourth chapter are core, thethird chapter splited up into three parts:(1) The2nth-order symmetric differentialoperator with real periodic coefficients is the essence self-adjoint operator;(2) Theirspectra are purely continuous;(3) The distributed situation of their spectra; Thefourth chapter has the similar structure with the third chapter:(1) The2nth-orderJ-symmetric differential operator with complex periodic coefficients is the essenceJ-self-adjoint operator;(2) Their spectra are also purely continuous;(3) Thedistributed situation of the spectra.