Abstract:

This paper mainly discusses the self-adjointness and eigenvalue dependence of a class of second-order differential operator with internal point conditions containing an eigenparameter. First, a problem-related linear operator $T$ is defined in an appropriate Hilbert space, and the study of the problem to be transformed into the research of the operator $T$ in this space, and the operator $T$ is proved to be self-adjoint according to the definition of self-adjoint operator. In addition, on the basis of self-adjoint, it is proved that the eigenvalues are not only continuously dependent but also differentiable on each parameter of the problem, and the corresponding differential expressions are given. Meanwhile, the monotonicity of the eigenvalues with respect to the part parameters of the problem is also discussed.

Key words: Internal point conditions, Eigenparameters, Self-adjointness, Dependence of eigenvalue, Monotonicity of eigenvalues