Analytical study of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by a spatially power-law potential V per(x) = λx α

Abstract

In this work, we present a rigorous mathematical scheme for the derivation of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by the potential Vper(x) = λxα, where α is a positive integer, using the non-degenerate time-independent perturbation theory. To do so, we derive a generalized formula for the integral I=∫−∞+∞xα⁡exp(−x2)Hn(x)Hm(x)dx, where Hn(x) denotes the Hermite polynomial of degree n, using the generating function of orthogonal polynomials. Finally, the analytical results with α = 3 and α = 4 are discussed in detail and compared with the numerical calculations obtained by the Lagrange-mesh method.