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.