Aspects of perturbation theory in quantum mechanics: The BenderWu Mathematica ® package

Abstract

We discuss a general setup which allows the study of the perturbation theory of an arbitrary, locally harmonic 1D quantum mechanical potential as well as its multi-variable (many-body) generalization. The latter may form a prototype for regularized quantum field theory. We first generalize the method of Bender-Wu, and derive exact recursion relations which allow the determination of the perturbative wave-function and energy corrections to an arbitrary order, at least in principle. For 1D systems, we implement these equations in an easy to use MATHEMATICA ® package we call BenderWu. Our package enables quick home-computer computation of high orders of perturbation theory (about 100 orders in 10-30 seconds, and 250 orders in 1-2h) and enables practical study of a large class of problems in Quantum Mechanics. We have two hopes concerning the BenderWu package. One is that due to resurgence, large amount of non-perturbative information, such as nonperturbative energies and wave-functions (e.g. WKB wave functions), can in principle be extracted from the perturbative data. We also hope that the package may be used as a teaching tool, providing an effective bridge between perturbation theory and non-perturbative physics in textbooks. Finally, we show that for the multi-variable case, the recursion relation acquires a geometric character, and has a structure which allows parallelization to computer clusters.

Quasinormal modes of black holes and Borel summation

“…Instead, a much more economical way is known [16]. We do not repeat the argument in [16], but mention that there is a useful Mathematica package 4 [17] to compute the perturbative expansion of the spectrum for a given potential by this method. See these papers in detail.…”

Deconstructing zero: resurgence, supersymmetry and complex saddles

“…Hidden in (2.4) is the subscript zero, which labels the ground state level number N = 0. The recent Mathematica Package "BenderWu" [20] by Sulejmanpasic and one of us permits a simple computation, yielding in this model (see also [15]):…”

“…As for the ζ-deformed double-well potential of section 2.1, we can also use a parametric Bender-Wu analysis to find the large perturbative coefficients [20,21]. For the ground state:…”

“…The physical reason for this is explained below. A parametric Bender-Wu analysis (using [20]) permits extraction of the large-order behavior of the ζ-deformed perturbative coefficients a n (N ; ζ), including subleading corrections. For the ground state energy of the ζ-deformed theory one finds a n (N = 0; ζ) ∼ − 1 2π…”

“…The perturbative energy as a function of level number and coupling can be computed using a Bender-Wu analysis [20,21]:…”