Stationary Phase and Semiclassical Rigor

A stationary-phase approximation becomes a theorem when its remainder is controlled under stated hypotheses. For path integrals, a further issue appears: the number of integration variables grows when the time partition is refined. An estimate proved at each fixed dimension need not survive that limit. Stationary Phase develops the general leading formula and Hessian phase; here we examine remainder bounds and their uniformity.

Required background. Oscillatory Integrals fixes the Fresnel prescription. Stationary Phase in Quantum Mechanics explains the physical interpretation of saddles and their widths.

A quadratic phase gives an explicit remainder estimate

Take $a\in\mathcal S(\mathbb R)$ and $h>0$. Define

$$I(h)=\int_{\mathbb R}e^{ix^2/(2h)}a(x)\,dx, \qquad \widehat a(\xi)=(2\pi)^{-1/2}\int e^{-ix\xi}a(x)\,dx.$$

The amplitude makes the integral absolutely convergent. Fourier inversion and the regulated Gaussian formula yield

$$I(h)=\sqrt{2\pi h}\,e^{i\pi/4} (2\pi)^{-1/2}\int e^{-ih\xi^2/2}\widehat a(\xi)\,d\xi.$$

The exchange of integrals can be justified first with Gaussian damping in $x$, then by dominated convergence using rapid decay of $\widehat a$.

For real $s$, the exponential Taylor remainder obeys

$$\left|e^{-is}-\sum_{j=0}^{N-1}\frac{(-is)^j}{j!}\right| \leq\frac{|s|^N}{N!}.$$

Consequently, for every integer $N\geq1$,

$$I(h)=\sqrt{2\pi h}\,e^{i\pi/4} \sum_{j=0}^{N-1}\frac{(ih/2)^j}{j!}a^{(2j)}(0)+R_N(h),$$

with the concrete bound

$$|R_N(h)|\leq \frac{h^{N+1/2}}{2^NN!} \int_{\mathbb R}|\xi|^{2N}|\widehat a(\xi)|\,d\xi.$$

The derivatives of $a$ enter through Fourier moments, rather than through an informal assertion that the amplitude is slowly varying. For a family of amplitudes, a common error estimate requires a common bound on those moments.

General stationary phase needs uniform geometric hypotheses

For a fixed smooth real phase and a compactly supported smooth amplitude in $d$ dimensions, assume the support contains one critical point, its Hessian is invertible, and there are no other critical points on the support. The standard theorem gives the leading term of order $h^{d/2}$ and an error of order $h^{d/2+1}$.

The proof separates a neighbourhood of the critical point from the nonstationary region. A local Morse-coordinate change reduces the phase to a nondegenerate quadratic form; repeated integration by parts controls the complement. This decomposition identifies what its error constant depends on: derivative bounds for phase and amplitude, a lower bound on Hessian invertibility, and a lower bound on the phase gradient away from the critical neighbourhood.

When these bounds deteriorate with a parameter, the usual fixed-parameter remainder cannot be called uniform. For example, the saddles of $x^3/3-\mu x$ coalesce as $\mu\downarrow0$. Their Hessians vanish, and the local Gaussian description must be replaced by a uniform Airy-type construction.

A finite time partition is another parameter

With $n$ interior coordinates, a time-sliced propagator can involve an integral in $\mathbb R^{dn}$. At each fixed $n$, a stationary-phase theorem may apply. Nevertheless the dimension, derivative seminorms, determinant factors, and inverse Hessian bounds all depend on $n$.

An assertion $R_n(h)=O_n(h)$ gives no uniform estimate as $n\to\infty$. Even an error model $|R_n(h)|\leq nh$ fails to control sequences with $n$ comparable to $1/h$. This example is a logical warning about two limits, not an estimate claimed for every propagator.

A rigorous continuum semiclassical construction therefore needs estimates uniform in the partition, a separately constructed Fourier-integral propagator with controlled remainder, or another theorem relating the two constructions. Merely applying the finite-dimensional theorem repeatedly supplies none of these automatically.

Asymptotic expansions need not converge

The remainder statement is for each fixed truncation order as $h\to0$. It does not assert convergence of the infinite series in powers of $h$. The constants and high derivatives can grow with the order. It also does not imply relative accuracy at points where the leading coefficient vanishes or where several saddle contributions nearly cancel.

These distinctions preserve the usefulness of semiclassical expansions: a controlled finite truncation can be highly accurate without defining a convergent series or resolving exponentially small contributions.

Exercise: an amplitude varying on the saddle scale

Let $\chi\in C_c^\infty(\mathbb R)$ equal one near zero and put $a_h(x)=\chi(x)e^{ix/\sqrt h}$. Show why retaining only $a_h(0)=1$ in the quadratic-phase leading term gives the wrong answer.

Solution

Complete the square:

$$\frac{x^2}{2h}+\frac{x}{\sqrt h} =\frac{(x+\sqrt h)^2}{2h}-\frac12.$$

The stationary point moves to $-\sqrt h$, where $\chi=1$ for small $h$. Stationary phase therefore gives $I(h)=\sqrt{2\pi h}\,e^{i\pi/4}e^{-i/2}+o(\sqrt h)$. The missing order-one phase is not a small remainder. Derivatives of $a_h$ grow with negative powers of $h$, so the fixed-amplitude remainder bound was never uniform for this family.

References

• Hörmander, Lars, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 1990, Chapter 7 — stationary-phase expansions with derivative-dependent remainders.
• Zworski, Maciej, Semiclassical Analysis, Graduate Studies in Mathematics 138, American Mathematical Society, 2012, Chapter 3; see also Lawrence C. Evans and Maciej Zworski, Lectures on Semiclassical Analysis — quantitative stationary phase and semiclassical operator constructions.