Spectral Studies On Several Types Of The Left-Definite Differential Operator

In this paper, we have given in the Hilbert space self-adjoint operator A is bounded, if there exists a number of meet (Ax,x)≥k(x,x),(x∈D(A)) Left Hilbert space and satisfy a given set of five left-related condition, called continuous Hilbert space on(H,A), self-adjoint operator is called left-definite operator on[H,A). The set of differential operators in this paper, we study several specific differential equations, and obtained its associated set of left-definite space, left-definite operator and its spectrum. Including the classic fourth-order Laguerre Differential expression; non-classical Laguerre Differential expression while a=-2, the fourth-order Legendre Differential expression and1[y]=-y"+ky differential expression related to a given left-definite space and left-definite operator and its spectrum.First of all, study the history of differential operators and left-definite differential operators theory, the significance and relevance of the conclusions and the status of in-depth investigation, and summarizes the relevant differential operator and left-definite spectral theory of differential operators the methods and results, and other relevant analyzes the significance and value of research for future study provides a strong basis.In order to better understand the contents of this article, in this chapter we will introduce in this paper relate to some of the basic concepts, lemma:such as left-definite space, left-definite operator, self-adjoint operators and spectrum and so on.Based on the theory given in the second chapter, the first in the space onH=L2((0,∞), e-t) to discuss fourth-order Laguerre type differential equation The formula has the characteristic function (Laguerre polynomial) According to this operator are bounded, we give the left-definite space and first left-definite operator to the problem.Secondly, using a similar approach was discussed specific type of non-classical Laguerre differential expression while a=-2, given the formula left-definite space and first left-definite operator. Then, we used a method different from the literature [21] discussed the correlation given left-definite space fourth-order Legendre differential equations, left-definite operator and spectrum. Finally, the boundary conditions in certain differential operator1[y]=-y"+ky, obtained under bounded operator, while the first and second left-definite space and left-definite operator was given.Finally summarizes Hilbert space left in a series of important results given spectral theory, given space as left and left to establish a given operator, point spectrum calculation, etc. Second, fourth-order and higher-order differential equations for N-th left given space and related theoretical equation needs further study.