An invariant method of stationary phase - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:44:55Z http://mathoverflow.net/feeds/question/114697 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114697/an-invariant-method-of-stationary-phase An invariant method of stationary phase Kofi 2012-11-27T19:55:33Z 2012-11-28T08:41:05Z <p>The method of stationary phase is very well-known and employed in many areas of physics and mathematics, and, of course, included in various versions as theorem in textbooks, especially on pseudors and microlocal analysis.</p> <p>However, it always is somewhat dependent on local coordinates and the Fourier transform, despite being a quite invariant problem. To be precise, the question would be the following.</p> <blockquote>Let $M$ be a manifold and $\phi: M \longrightarrow \mathbb{C}$ be a smooth function with values in the closed right half plane. Let $u$ be an volume density on $M$ with compact support in $M$. Determine an asymptotic expansion as $t \rightarrow \infty$ of the integral $$I(\phi, u, t) = \int_M e^{-\phi t} u$$ under some nondegeneracy conditions on $\phi$. </blockquote> <p>(For example, one could require $\phi$ to be Morse or, more general, require that the set where it vanishes is a submanifold $C$ of $M$ and that at a point $p \in C$, the Hessian of $\phi$ is non-degenerate on the space $T_pM/T_pC$.)</p> <p>It is well-known that in these cases $I(\phi, u, t)$ has an asymptotic expansion of the form $$I(\phi, u, t) = (t/\pi)^{-(n-k)/2}\sum_{j=0}^\infty t^{-j} \int_C s_j,$$ where $k$ is the dimension of $C$ and the $s_j$ are certain volume densities on $C$. <strong>In fact, they have to be certain universal terms, depending only on the $2j$-th jets of $\phi$ and $u$ at $C$.</strong> This is not stated in most textbooks. </p> <p>I wonder if it is possible to find these terms $s_j$ using Invariance theory alone. I would like if someone ever thought about this and knows a reference to this more invariant, geometric approach.</p> <hr> <p>/Edit: To clarify my question: I was wondering if it is possible to determine the constants by <strong>invariance theory</strong>, i.e. some argument like "there is only one polynomial on the $2j$-jets of $u$ and $\phi$ that is invariant under coordinate transformation" or so. For the first term, this goes like this, supposed that $\phi$ is purely real:</p> <p>Define the $n-k$-density $\mathrm{H}\phi$ on $C$ by setting $$\mathrm{H}\phi[X_1, \dots, X_{n-k}] := \sqrt{\left|\det \bigl( D^2\phi[X_i, X_j] \bigr)_{ij}\right|},$$ where $D^2\phi$ is the (on $C$ well-defined) Hessian of $\phi$. Now $u/\mathrm{H}\phi$ is a $k$-density on $C$ -- this is $s_0$.</p> <p>Now there should be similar characterizations of the higher $s_j$ (which obviously can get arbitrarily complicated).</p> http://mathoverflow.net/questions/114697/an-invariant-method-of-stationary-phase/114700#114700 Answer by Liviu Nicolaescu for An invariant method of stationary phase Liviu Nicolaescu 2012-11-27T20:28:05Z 2012-11-27T20:28:05Z <p>Check Proposition 1.2.4 from the book <em>Fourier Integral Operators</em> by the late great Duistermaat. This result applies in the case when the phase $\phi$ is Morse. If the phase is not Morse, but the critical points are finitely determined (finite Milnor number) then things are a bit more complicated. The vol 1 book of Arnold-Gusein-Zade Varchenko <em>Singularities of differentiable maps</em> is a good source. You can also have a look at the <a href="http://www.nd.edu/~lnicolae/Zach-thesis.pdf" rel="nofollow">senior thesis of a Zach Lamberty</a>, a former student of mine. There he deals with the $2$-dimensional case ($\dim M =2$) and he essentially works out the toric resolution trick of Arnold and comp. for a special and quite degenerate two variable phase.</p> http://mathoverflow.net/questions/114697/an-invariant-method-of-stationary-phase/114702#114702 Answer by Bazin for An invariant method of stationary phase Bazin 2012-11-27T21:05:10Z 2012-11-27T21:05:10Z <p>$\phi=\Re \phi+i\Im \phi$. You have assumed $\Re \phi\ge 0$ and you deal with a complex phase function. Note that the standard notation is not yours, since what is usually called the real stationary phase method coincides here with the case $\phi$ purely imaginary. </p> <p>Never mind, let's follow your notations and note that $t\rightarrow+\infty$ (the $+$ is missing in your formulation and is quite important since $\Re \phi\ge 0$).</p> <p>(1) Let us assume that $\Im \phi$ does not have a stationary point on the support of the amplitude $u$($d\Im \phi\not=0$ on $supp u\cap${$\Re \phi= 0$}: then $I(\phi, u,t)=O(t^{-N})$ for any $N>0$.</p> <p>(2) Let us assume that $\Im \phi$ is such that $$d\Im \phi=0\text{ at}\quad supp u\cap{\text{{\Re \phi= 0}}\text{}}\Longrightarrow \text{Hessian}(\Im \phi) \text{ non-singular}$$ then $I(\phi, u,t)\sim ct^{-n/2}$, where $n=dim M$. The constant $c$ can be computed explicitly in terms of the indices of the Hessian at the stationary points, the value of the amplitude there and appears as a finite sum corresponding to the finite number of stationary points of $\Im \phi$ on the compact $supp u\cap${$\Re \phi= 0$}. A complete expansion is available in Hormander ALPDO first volume, Chapter 7, section devoted to the complex stationary phase method. To sum-up the simple case exposed here: the integral is largest when $\Re \phi$ vanishes at a critical point of $\Im \phi$, and if that critical point is non-degenerate, you find a behavior in $t^{-n/2}$. No coordinate choice is involved here.</p> <p>I should say that the real stationary phase method is easier to understand: it corresponds here to your case with $\phi$ purely imaginary (!). You may for instance assume that $i\phi$ is a real-valued Morse function and the Morse lemma is providing a normal form (No such thing exists for a complex valued function). You find a finite number of stationary points on the support of $u$, and you can take advantage of the normal form on a neighborhood of each critical point. Anyhow the contribution elsewhere is $O(t^{-\infty}).$ Morse lemma reduces the problem to an integral with an exactly quadratic phase for which you have a full expansion since you know explicitly the Fourier transform of a Gaussian function.</p>