On the Rayleigh–Schrödinger coefficients for the eigenvalues of regular perturbations of an anharmonic oscillator

We identify a class of perturbations of a complex anharmonic oscillator \(H\) for which the known formulas for the Rayleigh–Schrödinger coefficients can be significantly simplified. We investigate the effect of the spectral instability of the operator \(H\) on the behavior of the sequence of first perturbative corrections. We show that if \(H\) is not self-adjoint and the perturbation is finite and has finite smoothness at the right end of its support, then this sequence exponentially increases at infinity.