Research On Spectrum Of Some Differential Operators

In this paper, we study an important problem in the field of differential operators: spectral analysis. First we analyze the spectrum of a class of high-order singular left-definite Sturm-Liouville operators with an indefinit weight function. With operator theory method, we obtain that its spectrum is consisting of real eigenvalues which are unbounded from below and from above,and can be indexed to satisfy the inequalities≤λ_-2≤λ_-1≤λ₀<0 <λ₀≤λ₁≤λ₂≤.In the second part of this paper, we study regular approximations of its spectrum on the interval ( a ,b) by using interval truncation ( a r ,br) andThen we consider essential spectrum of a class of left-definite Sturm- Liouville operators with periodic coefficients and an indefinit weight function. We obtain that its essential spectrum has a band structure and the endpoints of these bands are eigenvalues of periodic and semi-periodic boundary conditi- ons over an interval those length is the fungamental period with Floquet theory.This paper contains four parts. The first part: an introduction of the background of the problems we investigate and main results we obtain in this paper. The second part: the spectrum of a class of high-order singular left- definite Sturm-Liouville operators. The third part: regular approximations of the spectrum of a class of high-order singular left-definite Sturm- Liouville operators. The fourth part: essential spectrum of a class of left- definite Sturm-Liouville operators with periodic coefficients and an indefinit weight function.