On the eigenproblems of PT-symmetric oscillators

We consider the non-Hermitian Hamiltonian H= -\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a polynomial of degree at most n \geq 1 with all nonnegative real coefficients (possibly P\equiv 0). It is proved that the eigenvalues \lambda must be in the sector | arg \lambda | \leq \frac{\pi}{2n+3}. Also for the case H=-\frac{d^2}{dx^2}-(ix)^3, we establish a zero-free region of the eigenfunction u and its derivative u^\prime and we find some other interesting properties of eigenfunctions.