Timeline for An invariant method of stationary phase
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 2, 2012 at 19:23 | comment | added | Bazin | You are certainly right that my notations are poorly chosen: it is indeed true that $c$ can depend on $t$, via the value of $\psi$ at the critical points. Again think about multiplying everything by $e^{it}$. | |
| Dec 1, 2012 at 19:22 | comment | added | Matthias Ludewig | So the constant depends on $t$? | |
| Nov 28, 2012 at 19:44 | comment | added | Bazin | The $e^{it\psi}$ must be there: the value of the phase at a critical point has to be taken into account. Imagine that you multiply everything by $i$. The signature is invariantly defined as well as the square root of the determinant, which is a half-density et the critical set. | |
| Nov 28, 2012 at 15:06 | comment | added | Matthias Ludewig | Yes, this is basically what I wrote in my edit to the above post, except that I considered the real case (I think the $e^{it\psi(x)}$ is wrong in your definition of $c$?). But the higher coefficients always make use of a chart. | |
| Nov 28, 2012 at 13:05 | comment | added | Bazin | The notation $\vert \psi''(x)\vert^{1/2}$ means $$ \vert\det \psi''(x)\vert^{1/2}. $$ | |
| Nov 28, 2012 at 13:02 | comment | added | Bazin | The $c$ in my answer is, assuming $\phi=-i\psi$, $\psi$ real-valued Morse function $$ c=\sum_{x\in supp u,d\psi(x)=0}e^{it\psi(x)}\frac{e^{i\frac{\pi}{4}sign \psi''(x)}}{\vert \psi''(x)\vert^{1/2}} u(x), $$ a coordinate-free expression. Note that $sign \psi''(x)$ stands for the signature of the quadratic form $\psi''(x)$, that is the number of positive eigenvalues minus the number of negative eigenvalues. The sum above is finite by compactness of $supp u$. | |
| Nov 27, 2012 at 21:39 | comment | added | Matthias Ludewig | Yes, this is the standard way. It works just as well in the case that $\phi$ is purely real (and gets even simpler). However, it describes the asymptotic expansion via differential operators given in Morse coordinates. And there is a vast amount of Morse coordinates for a given function. Of course, the term cannot depend on this choice, however when looking at its definition, this is not obvious at all. I am looking for a description of these terms such that one can see their invariance of the choice of Morse chart by its definition. | |
| Nov 27, 2012 at 21:05 | history | answered | Bazin | CC BY-SA 3.0 |