An approximate representation of a solution to a problem as a series in a small dimensionless parameter ε, with successive terms computed by solving linearized or order-by-order equations.

Principle

Assume the desired quantity is analytic or asymptotic in ε; substitute a power series ansatz into the governing equations and solve at each order to obtain corrections.

Demonstration

Compute the energy shift in a system with Hamiltonian H = H0 + εV by expanding eigenvalues and eigenvectors in powers of ε (perturbation theory) to obtain first- and second-order corrections.

Misapplication

Extending a formal perturbation series beyond its radius of validity or treating an asymptotic divergent series as convergent without applying resummation or nonperturbative analysis.

Consequence

Yields systematic corrections and error estimates when ε is small; clarifies dependence on a small parameter and often reveals secular terms that signal breakdown and the need for resummation or multiple-scale methods.

Reversal

Nonperturbative effects (e.g., exponentially small contributions or bifurcations) that vanish at all orders in ε and require methods beyond series expansion.

Boundary

Requires an identifiable small parameter and either convergence or asymptotic control; fails when no small parameter exists or when the solution has essential singularities in ε.

Semantic Tension

Contrast with purely numerical approximations: perturbation expansions give analytic dependence on a parameter and hierarchical corrections, but may be less reliable asymptotically than numerics when the series diverges.

Synthesis

A perturbation expansion decomposes a problem around a solvable limit into an ordered series in a small parameter, producing successive analytic corrections while signalling when nonperturbative physics must be incorporated.