Study Of Essential Spectrum Of Some Differential Operators

Study of differential operator theory originated in the early19thcentury, with all kinds of math problems and physical problems, spectraltheory of differential operators are important in the theory of differentialoperators. Qualitative analysis of the spectrum in the spectral theorywhich studied more active issues, it focuses on the distribution of theessential spectrum, the discrete spectrum and so on. But now the study ofthe essential spectrum are rare, many problems remain to be solved.In this article, the essential spectrum is studied, and we focuse on twoaspects of the work:One side, coefficients of differential operators are discussed on theessential spectrum of issues by many mathematical workers, but theymainly focused on working with constant coefficients, power factor andcoefficient differential operator. Through a unitary transformation ondifferential equations, this article is based on Schultze Study on theessential spectrum theorem, studying essential spectrum of a class ofsymmetric differential operators with exponential functions and logarithmic functions of multiplicative coefficients is discussed, and atthe time, nuclear space dimensions of he biggest operator is got.The other side, Professor Wang Zhong began an in-depth study of J self-adjoint differential operators to discuss the essential spectrum of aclass of second order symmetry differential operators with integralcoefficients. On this basis, the study is promoted in this article, with thehelp of operator of direct sum decomposition method and comparativemethods of quadratic. The article discusses four-order differentialexpression with integral coefficients, the scope of essential spectrumof self-adjoint Sturm-Liouville differential operators is generatedbyT0of the differential expression. And we cite an example. Afterfurthermore, the above problem is extended to higher order, and we getthe scope of essential spectrum.