On the Eigenproblems of Pt -symmetric Oscillators

We consider the non-Hermitian Hamiltonian H = − d dx + P (x) − (ix) on the real line, where P (x) is a polynomial of degree at most n ≥ 1 with all nonnegative real coefficients (possibly P ≡ 0). It is proved that the eigenvalues λ must be in the sector | argλ| ≤ π 2n+3 . Also for the case H = − d dx − (ix) , we establish a zero-free region of the eigenfunction u and its derivative u and we find some other interesting properties of eigenfunctions. Preprint.