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
Source abstract
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=∫−∞+∞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.
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Research context
OpenAlex topics: Physics of Superconductivity and Magnetism · Quantum, superfluid, helium dynamics · Cold Atom Physics and Bose-Einstein Condensates
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Affiliations
Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University 1 , Onna, Okinawa 904-0495, Japan
Department of Physics, Ho Chi Minh City University of Education 2 , Ho Chi Minh City, Vietnam
Institute of Fundamental and Applied Sciences, Duy Tan University 3 , 6 Tran Nhat Duat St., District 1, Ho Chi Minh City 700000, Vietnam
Faculty of Natural Sciences, Duy Tan University 4 , 03 Quang Trung St., Hai Chau District, Danang 550000, Vietnam
Department of Engineering Science, The University of Electro-Communications 5 , Chofu, Tokyo 182-8585, Japan
Department of Natural Science, Dong Nai University 6 , Dong Nai, Vietnam
Laboratory of Applied Physics, Advanced Institute of Materials Science, Ton Duc Thang University 7 , Ho Chi Minh City, Vietnam
Faculty of Applied Sciences, Ton Duc Thang University 8 , Ho Chi Minh City, Vietnam
Funding
Ministry of Education and Training of Vietnam
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