Study Of Discrete Spectrum Of Some Special Differential Operators

In this paper,2nth order differential operators with different coefficientsand n-item2nth order self-adjoint vector differential operator are studiedand some criterions for discreteness are obtained. These conclusionsenrich the results of spectrum theory of differential operator.Firstly, a class of2nth order differential operators with real valuedcoefficients which contain product of power function and exponentialfunction are considered. Some sufficient conditions for discreteness aregiven by means of the compactness of Sobolev spaces, the method ofdecomposition into direct sum of operator, comparison of quadratic formand oscillation theory of differential equations. Secondly,2nth ordersymmetric differential expressions with coefficients of logarithmicfunction and real valued coefficients are also considered. We know thatthe divergence speed of logarithmic function is slower than powerfunction. In order to solve this problem we use scaling of inequality andthe convergence of function to estimate the lower bound of differentialoperators considered, and get some sufficient and necessary conditions orsufficient conditions for the discreteness of the spectrum. Finally, an-item2nth order self-adjoint vector differential operator is studied. thedefinition of vector differential operator, boundary conditions and coefficient matrix form, and restrictive conditions are given in this part.At the same time, a sufficient condition is given by using minimaleigenvalue of matrix and scaling of inequality.