K. C. Shin 
On the reality of the eigenvalues for a class of PT-symmetric oscillators 
(76K, Latex2e)

ABSTRACT.  We study the eigenvalue problem -u"(z)-[(iz)^m+P(iz)]u(z)=\lambda u(z) 
with the boundary conditions that u(z) decays to zero as z tends to 
infinity along the rays \arg z=-\frac{\pi}{2}\pm \frac{2\pi}{m+2}, 
where P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z is a real polynomial 
and m\geq 2. We prove that if for some 1\leq j\leq\frac{m}{2}, 
we have (j-k)a_k\geq 0 for all 1\leq k\leq m-1, 
then the eigenvalues are all positive real. We then sharpen 
this to a slightly larger class of polynomial potentials. 
In particular, this implies that the eigenvalues are all positive real 
for the potentials \alpha iz^3+\beta z^2+\gamma iz when \alpha,\beta 
and \gamma are all real with \alpha\not=0 and \alpha \gamma \geq 0, 
and with the boundary conditions that u(z) decays to zero as z tends 
to infinity along the positive and negative real axes. 
This verifies a conjecture of Bessis and Zinn-Justin.