Explicit bounds for the asymptotics of oscillatory integrals

Question

Recall the following theorem (c.f. LC Evans, M Zworski, "Lectures on semiclassical analysis", Theorem 3.15, depending on the version):

Theorem: Let $\varphi: \mathbb R^n \to \mathbb R$ be smooth and $a: \mathbb R^n \to \mathbb R$ smooth with compact support $K$. Suppose that there exists $x_0 \in K$ with $\partial \varphi(x_0) = 0$ and $\det \partial^2 \varphi(x_0) \neq 0$, and suppose that $\partial \varphi \neq 0$ on $K\smallsetminus \{ x_0\}$. For positive $\hbar$, define: $$ I_\hbar = \int_{\mathbb R^n} e^{i\varphi(x)/\hbar} \, a(x)\,dx $$ Then for $k=0,1,\dots$, there exists differential operators $A_{2k}(x,\partial)$ of order $\leq 2k$, and constants $C_N$, all depending on $\varphi$, such that for each $N$ we have: $$ \left| I_\hbar - \hbar^{n/2} \,e^{i\varphi(x_0)/\hbar} \sum_{k=0}^{N-1} A_{2k}(x,\partial) \, a(x_0)\,\hbar^k\right| \leq C_N\, \hbar^{N + \frac n 2} \sum_{|\alpha| \leq 2N + n+1} \sup_K | \partial^\alpha a|$$ where $\partial^\alpha$ is shorthand for some product of $\frac{\partial}{\partial x^i}$s.

My question: I know how to give the operators $A_{2k}$ explicitly; they depend only on the Taylor expansion of $\varphi$ at $x_0$, and are succinctly described combinatorially by ``Feynman diagrams''. What I would like to know is how explicitly the $C_N$ can be given? For example, can $C_N$ be taken to depend on the maximum values of some finite list (depending on $N$, of course) of derivatives of $\varphi$?

The reason I'm asking is that the above theorem gives $I_\hbar$ to an accuracy of $O(\hbar^\infty)$, but I would like to vary $\varphi$ and study $I_\hbar$ in some limit, and to know that my $O(\hbar^\infty)$ estimates still hold, I need to swap some limits, which requires more explicit description of the estimates.

As with any post, feel free to re-tag as appropriate.

Answer 1

I didn't realize this had such connections with physics... Anyhow, I didn't work out every detail but I think this works: First note that one can write $a(x) = a(x)\psi_1(x) + a(x)\psi_2(x)$ where $\psi_1$ is 1 on a neighborhood of x_0 to be described in a moment, and where the second term does not affect the asymptotics due to the nonvanishing gradient condition.

Next, note by Morse's lemma if $\psi_1$ was chosen correctly, one can change coordinates on the support of $a(x)\psi_1(x)$ to make the phase just $\sum_{i = 1}^n x_i^2$. The key is that the coordinate change can be given by the composition of a finite set of quite explicit coordinate changes (see p.347 of Stein's Harmonic Analysis for example). You'll get a Jacobian factor and $a(x)\psi_1(x)$ gets replaced by its composition with the coordinate change. Since the phase function is now fixed, the standard asymptotic methods will give $C_N$ in terms of finitely many derivatives of the components of the coordinate change as well as the cutoff $\psi_1$

So the question becomes how to describe the derivatives of the components of the coordinate change. This is where you should check the details carefully, but what I believe is that if you use the formulas on p.347 or something similar, using minors to explicitly write the inverse function, in addition to suprema of finitely many derivatives of $\phi(x)$ you will get an additional factor of a negative power of the smallest eigenvalue of the Hessian, if the support of $\psi_1$ is small enough. So ultimately, your constants should be a constant times a product of a) the suprema of finitely many derivatives of $\phi(x)$, b) a negative power of the smallest eigenvalue of the Hessian, and c) a function of the size of the support of $\psi_1$.

You can't get rid of b)... consider the case where your phase is just a quadratic polynomial. I can't see offhand if there's a better way to deal with the cutoff (or if you care about it anyhow).

Answer 2

Back in the mid 80's, Jonathan Goodman and I studied a similar question. We needed smooth tame estimates for a Fourier integral operator, so we could solve nonlinear PDE's of nonlinear principal type. Specifically, we wanted to prove a local isometric embedding of a 4-dimensional Riemannian manifold.

To do this, we needed a finite version, including an explicit error term, of the asymptotic expansion found by Hormander of the parametrix (which is a Fourier integral operator) of a differential operator of real principal type.

Unfortunately, we never published the paper and by now neither Jonathan nor I have a complete copy of the paper. It was written using a now-obselete Mac word processor (the irony is that I was one of the first people to type up a Ph.D. thesis in TeX but I then abandoned it for Mac GUI word processors before returning to LaTeX). If anybody has a copy lying around somewhere, I'd be very grateful if you could send me a copy.

Anyway, I can still offer some rather vague advice on this. I suggest trying to work through the proof of the asymptotic expansion but using a finite Taylor expansion and the explicit integral error formula for the Taylor polynomial. It is especially instructive to work out first the 1-term Taylor expansion with explicit error term, i.e. the fundamental theorem of calculus. I don't know if this is at all useful (since I do not know the theorem you refer to), but it worked beautifully for Jonathan and me.

Answer 3

After a bit of digging, I traced the exposition of your Theorem 3.14/15 first to Fedoryuk, then to Erdelyi. However Erdelyi's treatment of the problem is different than what you need:

The main purpose of the present paper is to supply explicit expressions for the error terms associated with the expansions of [4] from which realistic bounds are readily computable. The derivations of Erdelyi do not lend themselves readily to the construction of error bounds owing to the somewhat artificial nature of the neutralizer functions employed in the analysis. Our approach is based instead on Hardy’s theory of generalized integrals [7], [8].

This is from a paper of Olver available here. While it deals only with the one-dimensional case, I would expect that to give you a decent starting point.

Answer 4

The following article by Jorge Rezende, treats a general case of the method of stationary phase on a Hilbert space. An estimate of the "remainder" term of the asymptotic series is given in which the dependence on N is explicit. The expression assumes absolute convergence of the moments of the Fourier transform of a(x) and the nonquadratic part of phi(x).