Convergent quantum normal forms, $\mathcal PT$-symmetry and reality of the spectrum

Abstract

A class of non-selfadjoint, $\mathcal PT$-symmetric operators is identified similar to a selfadjoint one, thus entailing the reality of the spectrum. The similarity transformation is explicitly constructed through the method of the quantum normal form, whose convergence (uniform with respect to the Planck constant) is proved. Further consequences of the uniform convergence of the quantum normal form are the establishment of an exact quantization formula for the eigenvalues and the integrability of the classical hamiltonian corresponding to the given $\mathcal PT$-symmetric operator.