On the eigenproblems of PT-symmetric oscillators

We consider the non-Hermitian Hamiltonian H=-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 greater than or equal to1 with all non-negative real coefficients (possibly P=0). It is proved that the eigenvalues lambda must be in the sector \arg lambda\less than or equal to pi/(2n+3). Also for the cubic case H=-d(2)/dx(2)-(ix)(3), we establish a zero-free region of the eigenfunction u and its derivative u' and we find some other interesting properties of eigenfunctions. (C) 2001 American Institute of Physics.