Perturbation theory develops an expression for the desired solution in terms of a formal power series known as a perturbation series in some "small" parameter, that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, while further terms describe the deviation in the solution, due to the deviation from the initial problem. Formally, we have for the approximation to the full solution A, a series in the small parameter (here called ε), like the following:

A = A0 + ε^1 A1 + ε^2 A2 + ⋯


In this example, A0 would be the known solution to the exactly solvable initial problem and A1, A2, ... represent the first-order, second-order and higher-order terms, which may be found iteratively by a mechanistic procedure. For small ε these higher-order terms in the series generally (but not always) become successively smaller. An approximate "perturbative solution" is obtained by truncating the series, often by keeping only the first two terms, expressing the final solution as a sum of the initial (exact) solution and the "first-order" perturbative correction

A ≈ A0 + εA1 ( ε → 0 )


Some authors use big O notation to indicate the order of the error in the approximate solution:

A = A0 + εA1 + O ( ε^2 )


If the power series in ε converges with a nonzero radius of convergence, the perturbation problem is called a regular perturbation problem. In regular perturbation problems, the asymptotic solution smoothly approaches the exact solution. However, the perturbation series can also diverge, and the truncated series can still be a good approximation to the true solution if it is truncated at a point at which its elements are minimum. This is called an asymptotic series. If the perturbation series is divergent or not a power series (e.g., the asymptotic expansion has non-integer powers ε^1/2 or negative powers ε^−2 then the perturbation problem is called a singular perturbation problem. Many special techniques in perturbation theory have been developed to analyze singular perturbation problems. If you finished reading this, please stop stalking me.

https://www2.ph.ed.ac.uk/~ldeldebb/docs/QM/lect17.pdf