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

• Published in: AIP advances Vol. 11; no. 8
• Main Authors: Anh-Tai, Tran Duong, Hoang, Duc T., Truong, Thu D. H., Nguyen, Chinh Dung, Uyen, Le Ngoc, Dung, Do Hung, Vy, Nguyen Duy, Pham, Vinh N. T.
• Format: Journal Article
• Language: English
• Published: 01.08.2021
• ISSN: 2158-3226; 2158-3226
• DOI: 10.1063/5.0059800

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.