Stability Of Essential Spectra Of Singular Hamiltonian Differential Operators Under Perturbations Small At Infinity

The spectral theory for singular differential operators,especially the linear singular Hamiltonian operators,have been investigated extensively by using many methods and many good results have been established cf.,e.g.[10,18,19].Furthermore,the pertur-bation theory is an important part of the spectral theory.The theory of perturbation was created by Rayleigh and Schrodinger.Rayleigh gave the formula for calculating the natural frequency when he researched a small perturbation of the oscillate system.This method can be understood as using the solution of the simple eigenvalue problem to obtain an approximate solution of the complex eigenvalue problem.Schrodinger use a similar method to research the eigenvalue problems in physics.Many scholars develope-d the perturbation theory for linear operators as times goes on.In 1973,Jorgens and Weidmann proposed the concept of perturbation small at infinity and gave the sufficient conditions for the stability of the essential spectrum of the Schrodinger operator under perturbation small at infinity[8].Sun Huaqing and Qi Jiangang have given a sufficient condition for the stability of essential spectra of the singular Sturm-Liouville differen-tial operators under perturbations small at infmity[20].In terms of the conditions that Jorgens and Weidmann put forward in his artical the requirements in[20]for operators are relatively weak and more convenient to apply.This paper use the method in[20]to obtain the related results to the Hamiltonian system and given some simple applications.This paper give the stability of the essential spectrun of the Hamiltonian operator un-der a small perturbations at infinity based on the research of[20].We given a,sufficient conditions to guarantee that the multiplication operator is small at,infinity perturbation relative to the singular linear Hamilton operator.We give the distribution of the essential spectrum of the minimum operator defined by some two dimensional Dirac system under perturbations small at infinity.What's more,we give the relationship of essential spectra under different weight functions.In this paper,we consider the stability of essential spectra of singular Hamiltonian systems which may be formally non-symmetric.The main results are as follows:In the second chapter,we give the preparatory knowledge of the singular linear Hamil-ton system and its adjoint system.In the third chapter,it is proved that the essential spectrum of the corresponding minimal operator is preserved by perturbation small at infinity with respect to the unper-turbed operator.It is interesting to find that there are no restrictions on operators except the operator after the perturbation is closable.In particular,the pre-minimal operator may be non-symmetric.For the multiplication operator,a sufficient condition is given to ensure that it is small at infinity perturbation relative to the singular linear Hamiltonian operator.We restrict the coefficient function of the multiplication operator and the po-tential function of the Hamiltonian operator.In this way,the multiplication operator is the perturbation small at infinity relative to the singular linear Hamilton operator.In the forth chapter,we give the distribution of the essential spectrum of the minimum operator defined by some two dimensional Dirac system under perturbations small at infinity.What's more,we give the relationship of essential spectra under different weight functions.