Consider the vectorial Sturm-Liouville problem:
−y''(x) + P(x)y(x) = λIdy(x)
Ay(0) + Idy(0) = 0
By(1) + Idy(1) = 0
where P(x)=[pij(x)]d i,j=1 is a continuous symmetric matrix-valued function
defined on [0, 1], and A and B are d×d real symmetric matrices. An eigenfunction
y(x) of the above problem is said to be of type (CZ) if any isolated zero
of its component is a nodal point of y(x). We show that when d = 2, there
are infinitely many eigenfunctions of type (CZ) if and only if (P(x), A, B) are
simultaneously diagonalizable. This indicates that (P(x), A, B) can be reconstructed when all except a finite number of eigenfunctions are of type (CZ).
The results supplement a theorem proved by Shen-Shieh (the second author)
for Dirichlet boundary conditions. The proof depends on an eigenvalue estimate,
which seems to be of independent interest.
Proceedings of the American Mathematical Society 133(5), pp.1475-1484