Padé Approximants and the Anharmonic Oscillator
Yale University, 2020
Abstract
Several problems in physics, while in principle having been solved, do not admit closed form solutions. In particular, problems in perturbation theory, where in principle a problem may be solved to all orders in a perturbative parameter with sufficient effort. Padé approximation is a very simple and powerful generalisation of Taylor approximation. This method efficiently extracts quantitative and qualitative information about solutions from formal power series obtained perturbatively. One of the simplest non-trivial cases of this occurs while computing the energy eigenvalue expansion quartic quantum anharmonic oscillator, an oscillator system where the quadratic potential has a small quartic correction: \[ H = p^2 + x^2 + \beta x^4 \] where β is the small parameter in which perturbative expansions are generated.
By analytic continuation, a convergent power series can be used to determine a function everywhere up to a natural boundary (a dense set of singularities). However in practice, this convergence may be impractically slow, or may not converge at all to the point of interest. This can be overcome in certain cases through the theory of Padé approximants.
Recommended citation: Desai, Krish. (2020). "Padé Approximants and the Anharmonic Oscillator." Yale University. MS Mathematics Thesis.
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