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

E. Caliceti, S. Graffi, J. Sjoestrand

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…