We consider the $n$-dimensional vectorial Sturm-Liouville problem under some coupled boundary conditions. In some special cases the potential can be reconstructed from one spectrum. We prove that if the spectrum is the same as the spectrum belonging to the zero potential, under periodic boundary conditions or semi-periodic boundary conditions in which case an additional condition on the potential is imposed, then the potential is actually zero.
"AMBARZUMYAN’S THEOREMS FOR VECTORIAL STURM-LIOUVILLE SYSTEMS WITH COUPLED BOUNDARY CONDITIONS." Taiwanese J. Math. 14 (4) 1429 - 1437, 2010. https://doi.org/10.11650/twjm/1500405958