Zeros of eigenfunctions of some anharmonic oscillators

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.