Analytical study of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by a spatially power-law potential Vper(x) = λxα
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 V-per(x) = lambda x(alpha), where alpha is a positive integer, using the non-degenerate time-independent perturbation theory. To do so, we derive a generalized formula for the integral I = integral(+infinity)(-infinity)x(alpha)exp(-x(2))H-n(x)H-m(x)d(x), where H-n(x) denotes the Hermite polynomial of degree n, using the generating function of orthogonal polynomials. Finally, the analytical results with alpha = 3 and alpha = 4 are discussed in detail and compared with the numerical calculations obtained by the Lagrange-mesh method
