Let P (x) denote a 2 × 2 symmetric matrix-valued function defined on [0, 1]. We
prove that if there exists an infinite sequence {y(x; λnj )}∞
j=1 of Dirichlet eigenfunctions of the
operator − d2
dx2 + P (x) whose components all have zeros in common, then P (x) is simultaneously
diagonalizable on [0, 1]. This result can also be generalized to the general n-dimensional case.
