Boundary-Layer Theory, Strong-Coupling Series, and Large-Order Behavior

TL;DR

This paper investigates how lattice discretization affects boundary-layer problems in mathematical physics, using resummation techniques and large-order analysis to approximate continuum solutions.

Contribution

It compares Pade resummation and variational perturbation theory for two boundary-layer problems, analyzing their effectiveness and discrepancies through large-order behavior.

Findings

Both methods produce good approximations but deviate from exact solutions.

Large-order analysis reveals the nature of discrepancies in the approximations.

Resummation techniques are promising but have limitations in continuum limit recovery.

Abstract

The introduction of a lattice converts a singular boundary-layer problem in the continuum into a regular perturbation problem. However, the continuum limit of the discrete problem is extremely nontrivial and is not completely understood. This paper examines two singular boundary-layer problems taken from mathematical physics, the instanton problem and the Blasius equation, and in each case examines two strategies, Pade resummation and variational perturbation theory, to recover the solution to the continuum problem from the solution to the associated discrete problem. Both resummation procedures produce good and interesting results for the two cases, but the results still deviate from the exact solutions. To understand the discrepancy a comprehensive large-order behavior analysis of the strong-coupling lattice expansions for each of the two problems is done.