Perturbation Methods

The Bender-Orszag canon: techniques for problems you can't solve in closed form. The trick is always the same — find a small parameter, expand around a solvable limit, and use the asymptotic structure of the expansion to extract more information than the leading order can give you on its own.

Most physically interesting problems are nonlinear, non-separable, or singular in a way that defeats exact solution. Asymptotic methods don't pretend to handle them globally; they handle the LIMITS — small coupling, large parameter, far field, late time — and stitch those limits together. The combined picture is often qualitatively complete, even when no closed-form expression for the answer exists.

Topics in this section

• Perturbation theory, divergent series, and resurgence — the perturbation series is almost always divergent. Why that's not a bug, why optimal truncation leaves an irreducible exponentially-small error, how Borel resummation extracts more, and how resurgence connects perturbative and non-perturbative physics into a single object. Worked example: the quartic anharmonic oscillator and the Stark effect.
• WKB and semiclassical methods — the $\hslash \to 0$ limit, the eikonal expansion, the $1/\sqrt{p(x)}$ amplitude, classical vs forbidden regions, Airy connection formulas at turning points, Bohr-Sommerfeld quantization with the Maslov index, tunneling rates and alpha decay, and the bridge to resurgence via the instanton action.
• Liouville-Lanczos linear-response TDDFT — first-order perturbation theory where the linear-response equation is solved by tridiagonalising the (non-Hermitian) Liouvillian superoperator with bi-orthogonal Lanczos and reading the dynamical polarizability off as a continued fraction. The Walker-Saitta-Gebauer-Baroni machinery behind Quantum ESPRESSO's turboTDDFT; the Timrov-Vast extension to electron energy loss spectroscopy in turboEELS.
• Padé approximants — replace a truncated power series by a rational function with the same Taylor expansion. Captures poles and singularities that polynomials cannot. Often extends the usable range of a divergent series by many orders of magnitude; central building block of every modern resummation.
• Borel-Padé resummation — the workhorse for factorially divergent asymptotic series in quantum mechanics and field theory. Tame the factorial growth via the Borel transform, Padé-approximate the much-better-behaved Borel transform, integrate back to recover the original function. Eight-digit accuracy on the AHO series at $g=0.1$ from coefficients whose partial sum has already diverged.
• Shanks transformation and Wynn's ε-algorithm — nonlinear acceleration of slowly convergent sequences. Each iteration of Shanks adds about two digits of accuracy to series like the Leibniz formula for $\pi /4$. The cheap recursive form (Wynn's epsilon) is ~20 lines of code.
• Variational perturbation theory — introduce a trial harmonic frequency $\Omega$ as a knob, demand the truncated series be stationary in it (Stevenson's Principle of Minimal Sensitivity), and the resulting sequence converges where Borel-Padé fails. The Kleinert technique that resums the AHO ground state to ~$10^{-30}$ at strong coupling.
• Hyperasymptotics — push past optimal truncation: re-expand the remainder around its saddle, get an asymptotic series for the remainder with its own optimal truncation at error $\sim e^{-2|x|/A}$, iterate. The Berry-Howls / Olde Daalhuis machinery underlying J. P. Boyd's "Devil's Invention" review; the resurgence-theoretic recipe for machine-precision evaluation of divergent series from a handful of coefficients.

Planned

• Method of steepest descent — asymptotic evaluation of oscillatory and exponentially-weighted integrals; the unifying perspective behind stationary phase, Laplace's method, and saddle points.
• Matched asymptotic expansions — boundary-layer problems where you solve the inner and outer limits separately and match in an overlap region. The canonical singular-perturbation tool.
• Multiple-scale analysis — when a problem has fast and slow timescales, treat them as independent variables and let the secular-condition equations pick the right slow evolution. Standard for nonlinear oscillators and weakly damped systems.
• Asymptotic expansion of integrals — Watson's lemma, Laplace's method, stationary phase, the method of steepest descent applied to special functions.

Each topic gets its own page when written. The unifying frame is that "asymptotic" doesn't mean "approximate" — it means the answer has a SPECIFIC analytic structure (formal power series, transseries, oscillatory expansion) that can be exploited to extract physics that closed-form methods can't access.