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
Tran Duong Anh-Tai,
Duc T. Hoang,
Thu D. H. Truong,
Chinh Dung Nguyen,
Le Ngoc Uyen,
Do Hung Dung,
Nguyen Duy Vy,
Vinh N. T. Pham
Affiliations
Tran Duong Anh-Tai
Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
Duc T. Hoang
Department of Physics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam
Thu D. H. Truong
Department of Physics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam
Chinh Dung Nguyen
Institute of Fundamental and Applied Sciences, Duy Tan University, 6 Tran Nhat Duat St., District 1, Ho Chi Minh City 700000, Vietnam
Le Ngoc Uyen
Department of Engineering Science, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
Do Hung Dung
Department of Natural Science, Dong Nai University, Dong Nai, Vietnam
Nguyen Duy Vy
Laboratory of Applied Physics, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Vinh N. T. Pham
Department of Physics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam
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=∫−∞+∞xαexp(−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.
