The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions

For any positive integer n and a set of positive integers mi{m}_{i}, i=1,2,…,n+1i=1,2,\ldots ,n+1, we construct a class of quadratic eigenparameter-dependent boundary Sturm-Liouville problems with n transmission conditions, which have at most ∑i=1n+1mi+n+5{\sum }_{i=1}^{n+1}{m}_{i}+n+5 eigenvalues. The key to this analysis is still the division of intervals and an iterative construction of the characteristic function. Further, some examples are given for a simple explanation.