  We study eigenfunctions of Schrodinger operators -y"+Py on the real line with
zero boundary conditions, whose potentials P are real even polynomials with
positive leading coefficients. For quartic potentials we prove that all zeros
of all eigenfunctions belong to the union of the real and imaginary axes.
Similar result holds for sextic potentials and their eigenfunctions with
finitely many complex zeros. As a byproduct we obtain a complete classification
of such eigenfunctions of sextic potentials.
