Perturbative, Variational, and Asymptotic Thinking

Perturbative, variational, and asymptotic calculations make different promises. A perturbative calculation finds coefficients near a chosen limit. A variational calculation searches a family of states and may give a bound. An asymptotic calculation describes a limit through finite remainders, even when its infinite series diverges. Understanding the promise helps you decide what an answer does—and does not—establish.

Helpful background. Small parameters and error estimates distinguishes a dimensionless diagnostic, a remainder estimate, and a rigorous bound.

Perturbation theory describes a neighborhood

Perturbative reasoning starts from a family such as

$$H(\lambda)=H_0+\lambda V,$$

where $H_0$ is sufficiently understood and $\lambda=0$ is the reference point. For an observable or spectral quantity $Q$, one writes

$$Q(\lambda) = Q^{(0)} +\lambda Q^{(1)} +\lambda^2Q^{(2)} +\cdots.$$

The coefficients answer local questions: how does $Q$ respond at the reference point, and which structures appear order by order? They do not automatically answer whether the series converges at the physical value of $\lambda$.

What controls the expansion?

The bookkeeping parameter must be converted into dimensionless physical ratios. For nondegenerate eigenstate mixing, a relevant diagnostic is

$$\epsilon_{mn} = \frac{\lvert\lambda V_{mn}\rvert} {\lvert E_n^{(0)}-E_m^{(0)}\rvert}.$$

The numerator alone does not determine smallness. A weak operator can cause strong mixing across a small gap, while a large dimensional matrix element can be harmless compared with a much larger separation.

Perturbation theory is therefore local in two senses:

• local in parameter space, because it expands around a chosen value of $\lambda$;
• local in spectral structure, because isolated levels, degenerate multiplets, continua, and thresholds require different starting organizations.

Regular and singular changes

In a regular perturbation, the limiting solution retains the qualitative structure needed for nearby parameter values. In a singular perturbation, the limit changes the order of an equation, its boundary conditions, its spectrum, or the relevant scales. Boundary layers, turning points, continuum thresholds, and secular time dependence are warning signs that a naive power series is not uniform.

The appearance of a large coefficient is often a message from the physics. A denominator approaching zero may announce a level crossing or resonance. A term proportional to time may announce that fixed-order perturbation theory has outlived its time window. The right response is reorganization: degenerate perturbation theory, multiple-scale analysis, resummation, or an effective reduced model.

Include the remainder

A useful perturbative answer has the form

$$Q(\lambda) = \sum_{n=0}^{N} \lambda^n Q^{(n)} +R_N(\lambda),$$

with the regime and status of $R_N$ stated. Depending on the problem, the remainder may have a rigorous bound, an asymptotic order estimate, or only a diagnostic based on the next term and numerical comparison.

Start the technical development at Nondegenerate Perturbation Theory and Degenerate Perturbation Theory.

Variational calculations restrict the trial states

Variational reasoning does not require a nearby solvable Hamiltonian. It chooses a family $\mathcal M$ of admissible states and optimizes an exact functional over that family.

For a self-adjoint Hamiltonian bounded below, the Rayleigh quotient is

$$\mathcal R[\psi] = \frac{\langle\psi\rvert H\lvert\psi\rangle} {\langle\psi\vert\psi\rangle}.$$

For nonzero trial states in the appropriate operator or quadratic-form domain (interpreting the numerator as the quadratic form when needed),

$$E_0 \le \inf_{\psi\in\mathcal M} \mathcal R[\psi].$$

The inequality is the source of trust. The optimized energy can be useful even when there is no small coupling.

The trial family is the approximation

The Hamiltonian has not been expanded. Instead, the full admissible state space has been replaced by a smaller manifold or subspace. Its design encodes physical judgment:

• exact symmetries and boundary conditions;
• short- and long-distance behavior;
• expected nodes and angular structure;
• correlation patterns;
• relevant length and energy scales.

Enlarging nested trial spaces cannot worsen the ground-state upper bound. In a finite basis, the Rayleigh–Ritz method converts the optimization into a generalized eigenvalue problem. Excited-state bounds require the min–max structure or correctly imposed orthogonality; simply minimizing again does not produce the first excited state.

Why energy can look better than the state

Let the exact nondegenerate ground state be $\lvert0\rangle$, and write a normalized trial state as

$$\lvert\psi\rangle = \sqrt{1-\delta^2}\,\lvert0\rangle + \delta\lvert\chi\rangle,$$

where

$$\langle0\vert\chi\rangle=0, \qquad \langle\chi\vert\chi\rangle=1.$$

Then

$$\mathcal R[\psi]-E_0 = \delta^2 \left( \langle\chi\rvert H\lvert\chi\rangle-E_0 \right).$$

The energy error is second order in the amplitude of the orthogonal contamination, while a general observable can have a first-order error through interference terms. This is why an excellent variational energy need not imply an equally excellent density, tail, or correlation function.

Bounds versus stationary principles

Not every method called variational supplies an upper bound. Time-dependent variational principles and action-based stationary principles can yield useful equations on a trial manifold without ordering the result above or below the exact answer. The guarantee must be identified from the particular functional, not inferred from the word variational.

See Variational Principle, Rayleigh–Ritz Method, and Trial Wavefunctions.

Asymptotic expansions compare fixed truncations

Asymptotic analysis asks how a quantity behaves as a dimensionless parameter approaches a limit. For $\epsilon\to0$, a Poincaré expansion may satisfy, for every fixed $N$,

$$f(\epsilon) - \sum_{n=0}^{N} a_n\epsilon^n = O\!\left(\epsilon^{N+1}\right).$$

This statement concerns each finite partial sum as $\epsilon$ tends to zero. It does not say that the infinite series converges for a fixed nonzero $\epsilon$.

Every convergent Taylor expansion is asymptotic as the argument approaches its expansion point, but an asymptotic expansion need not converge. The distinction matters because the correct numerical procedure for a divergent asymptotic series is usually finite, often near the least term.

A factorially divergent example

For $\epsilon\gt0$, consider

$$I(\epsilon) = \int_0^\infty \frac{e^{-t}}{1+\epsilon t}\,dt.$$

The finite geometric identity

$$\frac{1}{1+\epsilon t} = \sum_{n=0}^{N} (-\epsilon t)^n + \frac{(-\epsilon t)^{N+1}} {1+\epsilon t}$$

gives

$$I(\epsilon) = \sum_{n=0}^{N} (-1)^n n!\,\epsilon^n +R_N(\epsilon),$$

where

$$R_N(\epsilon) = (-\epsilon)^{N+1} \int_0^\infty \frac{e^{-t}t^{N+1}} {1+\epsilon t}\,dt.$$

Because $1+\epsilon t\ge1$,

$$\lvert R_N(\epsilon)\rvert \le (N+1)!\,\epsilon^{N+1}.$$

The factorial coefficients make the infinite power series divergent for every nonzero $\epsilon$, yet every fixed truncation is an asymptotic approximation as $\epsilon\to0$. The ratio of successive term magnitudes is $(n+1)\epsilon$, so terms decrease only until $n$ is of order $1/\epsilon$.

Beyond all algebraic orders

A contribution such as

$$e^{-A/\epsilon}, \qquad A\gt0,$$

is smaller than every power of $\epsilon$ as $\epsilon\to0$. Its ordinary power-series asymptotic coefficients all vanish. Tunneling splittings and instanton weights often have this structure. Calling such a term nonperturbative means that no finite algebraic order in the chosen parameter can produce it; it does not mean the term is unknowable or uncontrolled.

Uniformity matters

An approximation can be asymptotically correct at every fixed point and still fail in a parameter-dependent region. WKB fails near a turning point because the local momentum vanishes. A large-time expansion may fail near a coalescing saddle. A low-energy scattering expansion may fail at a threshold pole.

Identify the region where the approximation fails and find a local description that can be matched to the outer solution. For the mathematical definitions and integral methods, see Asymptotic Analysis; their semiclassical realization begins at WKB Approximation.

Effective descriptions can be exact before expansion

An effective description retains selected degrees of freedom and reproduces specified predictions within a scale window. Eliminating variables algebraically can be exact; expanding the resulting energy dependence or omitting operators is a further approximation. Feshbach projection and the Schrieffer–Wolff transformation illustrate these two stages.

Scattering supplies another useful distinction. An asymptotic state describes preparation or detection far from an interaction. It is not the same concept as an asymptotic expansion. Scattering amplitudes can be exact properties of a model; Born or semiclassical methods approximate them. See scattering amplitudes for that definition.

One Two-Level Model, Three Viewpoints

Consider

$$H = \begin{pmatrix} 0 & g\\ g & \Delta \end{pmatrix}, \qquad \Delta\gt0.$$

The lower eigenvalue, also used in the error-bound example, is

$$E_- = \frac{\Delta-\sqrt{\Delta^2+4g^2}}{2}.$$

Take $g$ real. This elementary model makes the different claims easy to compare.

Perturbative viewpoint

For $x=g/\Delta$, expansion around $x=0$ gives

$$E_- = -\frac{g^2}{\Delta} +\frac{g^4}{\Delta^3} -\frac{2g^6}{\Delta^5} +O\!\left(\frac{g^8}{\Delta^7}\right).$$

The complex branch points of the square root set the radius of convergence to $1/2$. The power series converges inside that disk; its boundary behavior requires a separate check. This example is a useful antidote to the claim that every perturbative series diverges.

Variational viewpoint

For the normalized real trial family

$$\lvert\psi(\theta)\rangle = \cos\theta\,\lvert a\rangle - \sin\theta\,\lvert b\rangle,$$

the Rayleigh quotient is

$$E(\theta) = \Delta\sin^2\theta -2g\sin\theta\cos\theta.$$

Its stationary condition is

$$\tan(2\theta) = \frac{2g}{\Delta}.$$

Choose the minimum with $\cos(2\theta)>0$ and the sign of $\sin(2\theta)$ matching $g$. The tangent equation alone also allows maxima. Because a real symmetric matrix has real eigenvectors and this family includes every normalized real direction, minimization gives the exact $E_-$, even though arbitrary complex states are not parameterized. If the trial space were restricted to $\lvert a\rangle$ alone, it would give the valid but crude upper bound $E_0\le0$.

Effective viewpoint

Retain $\lvert a\rangle$ and eliminate $\lvert b\rangle$. The exact projected equation is

$$E = \frac{g^2}{E-\Delta}.$$

For $\lvert E/\Delta\rvert\ll1$,

$$\frac{g^2}{E-\Delta} = -\frac{g^2}{\Delta} \left( 1+\frac{E}{\Delta}+\cdots \right).$$

Solving iteratively reproduces the perturbative low-energy expansion. The effective viewpoint emphasizes why the correction appears: the retained state makes virtual excursions into the eliminated state. The perturbative viewpoint emphasizes its order in $g/\Delta$. The variational viewpoint emphasizes the energy bound and quality of the trial space.

No viewpoint changes the exact answer. Each makes a different structure visible.

Combining methods

The labels can overlap. A perturbative series may also be asymptotic. A variational ansatz can be motivated by weak-coupling physics, yet supply an energy bound at stronger coupling. The Schrieffer–Wolff method is both perturbative and effective. Variational perturbation theory reorganizes a perturbative calculation using an adjustable reference problem; its higher orders do not automatically retain a variational bound.

State the parameter and the object being approximated. “Perturbative in the interaction” and “variational within a Gaussian family” are more informative than either adjective alone.

Compare the same prediction in the same regime

Align the Hamiltonian, boundary conditions, observable, and parameter values before comparing methods. An upper bound is not a symmetric error bar. Agreement of two calculations is most informative when they have different possible errors: for example, perturbation theory compared with direct diagonalization, or WKB compared with exact levels as the excitation increases.

The two-level error guide supplies a quantitative example. The failure-mode guide explains what to try when the comparisons disagree.

Exercises

Energy error and state error

Using the decomposition

$$\lvert\psi\rangle = \sqrt{1-\delta^2}\,\lvert0\rangle + \delta\lvert\chi\rangle,$$

derive the energy error formula and explain why a general observable can have an error of order $\delta$.

Solution

Because $H\lvert0\rangle=E_0\lvert0\rangle$ and $\langle0\vert\chi\rangle=0$, the cross terms in the energy vanish. Therefore

$$\langle\psi\rvert H\lvert\psi\rangle-E_0 = \delta^2 \left( \langle\chi\rvert H\lvert\chi\rangle-E_0 \right).$$

For a general observable $A$, the expectation contains interference terms

$$2\delta\sqrt{1-\delta^2}\, \operatorname{Re}\langle0\rvert A\lvert\chi\rangle,$$

which are generically first order in $\delta$. Energy stationarity suppresses the leading energy error; it does not suppress every observable error.

A divergent but useful expansion

For the integral $I(\epsilon)$ above, show that the magnitude of successive formal terms stops decreasing when $n$ is of order $1/\epsilon$.

Solution

The magnitude of the $n$th term is

$$T_n=n!\,\epsilon^n.$$

Thus

$$\frac{T_{n+1}}{T_n} =(n+1)\epsilon.$$

Terms decrease while $(n+1)\epsilon\lt1$ and increase after that ratio exceeds one. The least term therefore occurs near $n\sim1/\epsilon$. For fixed small $\epsilon$, summing far beyond this order worsens the asymptotic approximation.

Reproduce the two-level expansion

Starting from the effective equation

$$E=\frac{g^2}{E-\Delta},$$

assume $E=-g^2/\Delta+c g^4/\Delta^3+O(g^6/\Delta^5)$ and determine $c$.

Solution

Expand the right-hand side:

$$\begin{aligned} \frac{g^2}{E-\Delta} &= -\frac{g^2}{\Delta} \frac{1}{1-E/\Delta} \\ &= -\frac{g^2}{\Delta} \left( 1+\frac{E}{\Delta}+\cdots \right) \\ &= -\frac{g^2}{\Delta} +\frac{g^4}{\Delta^3} +O\!\left(\frac{g^6}{\Delta^5}\right). \end{aligned}$$

Hence $c=1$, in agreement with the exact eigenvalue expansion.

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, 1999. Asymptotic expansions and singular perturbations.
• F. W. J. Olver and R. Wong, “Definitions and Elementary Properties,” §2.1, in NIST Digital Library of Mathematical Functions, Chapter 2: Asymptotic Approximations. Precise remainder and uniformity definitions.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978. Variational and min–max principles for spectral bounds.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020. Perturbative and variational methods.