$PT$ symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum

Consider in $L^2(R^d)$, $d\ge 1$, the operator family $H(g):=H_0+igW$. $\ds H_0= a^\ast_1a_1+… +a^\ast_da_d+d/2$ is the quantum harmonic oscillator with rational frequencies, $W$ a $P$ symmetric bounded potential, and $g$ a real coupling constant. We show that if $\mathrm{\mid }g\mathrm{\mid }<\rho$, $\rho$ being an explicitly determined constant, the spectrum of $H(g)$ is real and discrete. Moreover we show that the operator $\text{ds}H(g)=a_1^{\ast }{a}_1+a_2^{\ast }{a}_2+ig{a}_2^{\ast }{a}_1$ has real discrete spectrum but is not diagonalizable.