This paper describes the $N$-dimensional vectorial Sturm-Liouville problem with coupled boundary conditions. We first derive the asymptotic expressions of large eigenvalues for the vectorial Sturm-Liouville operator with smooth coefficients. In addition, the regularized trace formula for the operator is calculated with residue techniques in complex analysis. These formulae are then used to obtain some results of inverse eigenvalue problems in the spirit of Ambarzumyan.
"Estimates of large eigenvalues and trace formula for the vectorial Sturm-Liouville equations." Rocky Mountain J. Math. 43 (6) 2049 - 2078, 2013. https://doi.org/10.1216/RMJ-2013-43-6-2049