The Classical Harmonic Oscillator on ℝ Perturbed by a Certain Scalar Potential

We investigate the perturbation A = H + V , where H = 1 2 -d 2 dx 2 + x 2 represents the harmonic oscillator in R, and V is a specific scalar potential.Let k denote the k th eigenvalue of the operator H.The eigenvalues of the perturbed operator L are given by k + k where k accounts for the perturbative effects of the potential V .The primary result of this study is to provide an asymptotic expansion of k and to establish a connection between the coefficients of this expansion and a particular transform of the potential V .