FINITE SPECTRUM OF STURM-LIOUVILLE PROBLEMS WITH $ N $ TRANSMISSION CONDITIONS AND SPECTRAL PARAMETERS IN THE BOUNDARY CONDITIONS

In this paper, we mainly study the finite spectrum of Sturm-Liouville problems with $ n $ transmission conditions and spectral parameters in the boundary conditions. For any positive integer $ n $ and a set of positive integers $ m_{i}, i=0, 1, \cdots, n $, it has at most $ m_{0}+m_{1}+\cdots+m_{n}+2n+1 $ eigenvalues. And further we show that these $ m_{0}+m_{1}+\cdots+m_{n}+2n+1 $ eigenvalues can be distributed arbitrarily throughout the complex plane in the non-self-adjoint case and anywhere along the real line in the self-adjoint case. The key to this analysis is an iterative construction of the characteristic function, the main tool used in this paper is Rouche's theorem and iterative construction of initial value.