Narrow Results

We review the issue of Borel summability in the framework of multiscale analysis and renormalization group, by discussing a proof of Borel summability of the massive Euclidean planar theory; this result is not new, since it was obtained by Rivasseau and 't Hooft. However, the techniques that we use have already been proved effective in the analysis of various models of consended matter and field theory; therefore, we take the planar theory as a toy model for future applications.

We present a new formulation of perturbation theory for quantum systems, designated here as: “mean field perturbation theory” (MFPT), which is free from power-series-expansion in any physical parameter, including the coupling strength. Its application is thereby extended to deal with interactions of arbitrary strength and to compute system-properties having non-analytic dependence on the coupling, thus overcoming the primary limitations of the “standard formulation of perturbation theory” (SFPT). MFPT is defined by developing perturbation about a chosen input Hamiltonian, which is exactly solvable but which acquires the nonlinearity and the analytic structure (in the coupling strength) of the original interaction through a self-consistent, feedback mechanism. We demonstrate Borel-summability of MFPT for the case of the quartic- and sextic-anharmonic oscillators and the quartic double-well oscillator (QDWO) by obtaining uniformly accurate results for the ground state of the above systems for arbitrary physical values of the coupling strength. The results obtained for the QDWO may be of particular significance since “renormalon”-free, unambiguous results are achieved for its spectrum in contrast to the well-known failure of SFPT in this case.

We extend Borel summability methods to the analysis of the 3D Navier–Stokes initial value problem,

This implies in particular local existence of a classical solution to (*) for t ∈ (0, T), where T depends on and . Global existence of the solution to NS follows if ‖Û(⋅, q)‖_l¹ has subexponential bounds as q → ∞.

If f = 0, then the converse is also true: if NS has global solution, then there exists n ≥ 1 for which ‖Û(⋅, q)‖ necessarily decays. More generally, if the exponential growth rate in q of Û is α, then a classical solution to NS exists for t ∈ (0, α^-1/n).

We show that α can be better estimated based on the values of Û on a finite interval [0, q₀]. We also show how the integral equation can be solved numerically with controlled errors.

Preliminary numerical calculations of the integral equation over a modest [0, 10], q-interval for n = 2 corresponding to Kida ([21]) initial conditions, though far from being optimized or rigorously controlled, suggest that this approach gives an existence time for 3D Navier–Stokes that substantially exceeds classical estimate.