Matrix representations of Atkinson-type Sturm-Liouville problems with coupled eigenparameter-dependent conditions

We investigate the Sturm-Liouville (S-L) operator with boundary and transfer conditions dependent on the eigen-parameter. By utilizing interval partitioning and factorization techniques of characteristic function, it is proven that this problem has a finite number of eigenvalues when the coefficients of the equation meet certain conditions, and some conditions for determining the number of eigenvalues are provided. The results indicate that the number of eigenvalues in this problem varies when the transfer conditions depend on the eigen-parameter. Furthermore, the equivalence between this problem and matrix eigenvalue problems is studied, and an equivalent matrix representation of the S-L problem is presented.