考虑一类耦合边界条件和转移条件均含谱参数的二阶复系数微分算子的J-自伴性和格林函数。在适当的Hilbert空间上定义一个与问题相关的线性算子,将所研究的问题转化为对此空间中算子的研究,并证明该算子是J-自伴的。另外,通过构造微分方程的基本解得到问题的格林函数。In this paper, we consider the J-self-adjointness and Green’s function of a class of discontinuous second-order complex coefficient differential operator with eigenparameters in boundary and transmission conditions. By introducing a linear operator related to the problem in a suitable Hilbert space, the considered problem can be interpreted as the study of the operator in this space, and this operator is proved to be J-self-adjoint. In addition, the Green’s function of the problem is obtained by constructing the fundamental solutions of the differential equation.
The boundness and compactness of products of multiplication,composition and differentiation on weighted Bergman spaces in the unit ball are studied.We define the differentiation operator on the space of holomorphic functions in the unit ball by radial derivative.Then we extend the Sharma's results.
This paper focuses on the direct and inverse problems for a third-order self-adjoint differential operator with non-local potential and anti-periodic boundary conditions.Firstly,we obtain the expressions for the characteristic function and resolvent of this third-order differential operator.Secondly,by using the expression for the resolvent of the operator,we prove that the spectrum for this operator consists of simple eigenvalues and a finite number of eigenvalues with multiplicity 2.Finally,we solve the inverse problem for this operator,which states that the non-local potential function can be reconstructed from four spectra.Specially,we prove the Ambarzumyan theorem and indicate that odd or even potential functions can be reconstructed by three spectra.