24 votes

Applications of microlocal analysis?

Microlocal analysis is used in computed tomography and other tomographic imaging techniques e.g. in medicine . Specifically, it is used to describe which wavefront sets (here: boundaries of objects, e....
Dirk's user avatar
  • 12.3k
16 votes

Applications of microlocal analysis?

There are striking applications in dynamical systems, due to Dyatlov and Zworski, where dynamical zeta functions were analysed using techniques from microlocal analysis. These zeta functions have a ...
Anthony Quas's user avatar
  • 22.5k
15 votes

Applications of microlocal analysis?

Although microlocal analysis was developed originally exclusively for linear problems, it has played an increasingly important role in nonlinear PDE via what's known as paradifferential calculus. ...
Deane Yang's user avatar
  • 26.9k
12 votes

Applications of microlocal analysis?

(1) The older and more widely known applications are to regularity and solvability of PDEs of any order that are not necessarily elliptic. (2) The phenomenon of the propagation of singularities ...
T. Amdeberhan's user avatar
11 votes

Algebraic microlocal analysis and nonlinear PDE

Edit. I'have added a few other infos taken from the historical notes by Goro Kato and Daniele C. Struppa in [A1], to point out that the application of algebraic microlocal analytic (and thus ...
Daniele Tampieri's user avatar
8 votes

Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

Q1: No. You gave a proof above: $\overline{\langle f,\bar g\rangle} = \operatorname{Tr}(f\otimes g)$. Q2: No, if $\operatorname{Tr}$ is supposed to be continuous. Namely, $\mathcal S(\mathbb R^k)\...
Peter Michor's user avatar
8 votes

How to visualize the Microsupport of a Sheaf?

I'd been hoping for months that someone would come along and answer this question: every time I encounter the definition of microsupport, my brain responds with a flash of anger followed by a ...
Vidit Nanda's user avatar
  • 15.4k
6 votes
Accepted

Help understand a calculation involving RHom of sheaves on manifolds

In my experience finding an explicit resolution is rarely possible. Instead you want to learn how to use the six operations. A good reference is section 8.3 of Chriss and Ginzburg. First a few ...
Justin Hilburn's user avatar
5 votes

How to visualize the Microsupport of a Sheaf?

This might be too late, or out-of-date for you, but I've always understood the microsupport best in terms of Morse theory (or microlocal Morse theory, or the stratified Morse theory of Goresky-...
Brian Hepler's user avatar
5 votes

Applications of microlocal analysis?

A pretty recent and impressive application of microlocal analysis is to analytic number theory or automorphic forms, via the orbit method as developed by Nelson and Venkatesh. In loc. cit. they set ...
Desiderius Severus's user avatar
5 votes

Applications of microlocal analysis?

Broadly speaking, microlocal analysis helps one in 'geometrization' of certain results on the singularities of distributions. In this direction, one striking application of microlocal analysis is the ...
Uday's user avatar
  • 2,209
5 votes

Applications of microlocal analysis?

Search for papers by Maarten de Hoop (Colorado School of Mines, then Purdue, now a Simons chair at Rice University), who has used microlocal analysis very extensively to study problems in global and ...
Tom Dickens's user avatar
  • 1,077
5 votes
Accepted

do hyperfunction solutions always exist?

The answer to both questions is Yes. It has been known (Grainger, Kohn, Stein, Proc. Nat. Academy USA, Vol. 72, No. 9, 3287-3289 (1975)) that when $f$ is Baire category 1, then the Lewy operator is ...
Nagaraj Iyengar's user avatar
5 votes

Hörmander’s propagation of singularities in two variables

The Propagation-of-Singularities Theorem is telling you that for a real-principal type operator $P$, and a given equation $Pu=f$, $$ WF(u)\backslash WF(f)\quad \text{is invariant by the flow of $H_p$},...
Bazin's user avatar
  • 15.1k
4 votes

Intuition behind the Duistermaat-Guillemin version of Weyl's law

Small error: $N(\lambda)$ is a number of eigenvalues $< \lambda^m$ (where $m$ is an order of operator, obviously $m=2$ here). It is not a geometry but a Hamiltonian dynamics (which becomes a ...
Victor Ivrii's user avatar
4 votes
Accepted

Is every continuous microlocal operator a pseudo-differential operator?

This is more like a longish series of comments somewhat complementing Ilya Zakharevich's answers rather than an answer by itself. First of all, notice that since $\mathscr{S}(\mathbb{R}^n)$ embeds ...
Pedro Lauridsen Ribeiro's user avatar
4 votes

Is every continuous microlocal operator a pseudo-differential operator?

Preliminaries Averaging arguments show that the question boils down to taking convolutions. So one needs to find a generalized function F smooth outside of 0 such that the Fourier transform ℱF(ξ) ...
Ilya Zakharevich's user avatar
4 votes

Interesting (non) examples of singular support

The singular support of a sheaf is always coisotropic (Theorem 6.5.4 in Kashiwara and Schapira). In the example of the 2-dimensional conic subset of $T^\ast \mathbb R^2$ defined by a covector field ...
Sam Gunningham's user avatar
4 votes
Accepted

Is every endomorphism of the sheaf of holomorphic functions on a disk a differential operator?

Probably this paper answers your question negatively. In particular it shows that if $X$ is an open subset of $\mathbb{C}^n$ and the sheaf $\mathcal{O}$ of holomorphic functions on $X$ is given the ...
Simon Wadsley's user avatar
4 votes
Accepted

Characterisation of the wavefront set

Your intuitive characterization does not make sense: the function $u$ is defined on some neighborhood of $x_0$ and $(x_0,\xi_0)$ belongs to the sphere bundle. On the other hand, you may salvage part ...
Bazin's user avatar
  • 15.1k
4 votes

Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

In the "classical theory of topological vector spaces" the questions like this are intricated (in my opinion, this is an artifical complexity, the Nature can't be so complicated). But in the theory of ...
Sergei Akbarov's user avatar
4 votes

Weyl law for (non-semiclassical) Schrodinger operator

We are talking about a non-compact Riemannian manifold, right? Then Weyl's law may be incorrect, at least out of the box. Let us consider $H = -\Delta$ (so $V(x)=0$) in the domain $X \subset \mathbb{R}...
Victor Ivrii's user avatar
4 votes
Accepted

Equivalent Littlewood-Paley-type decompositions

The norms should be equivalent with a constant depending only on the ratio between the two bases (in this case, 2 and 3). I'll just consider the Besov case. I'll adopt the notation $\hat{f}_{n,b}(\xi)=...
Ben Johnsrude's user avatar
3 votes

Weyl law for (non-semiclassical) Schrodinger operator

I talked to several experts. Here is the punchline: apparently, only the case of $\mathbb{R}^n$ can be found in the literature (I would be glad to be wrong and would highly appreciate a reference). ...
Maxim Braverman's user avatar
3 votes
Accepted

Schwartz kernel theorem

By the Helffer-Sjöstrand formula, $$ f(-\Delta_D) = \frac1\pi \int_{\mathbb C} (\partial_{\bar z} \tilde f)(z) (-\Delta_D -z)^{-1}\,L(dz), $$ where $L(dz)$ is the Lebesgue measure and $\tilde f$ ...
ifw's user avatar
  • 1,191
3 votes

Is every continuous microlocal operator a pseudo-differential operator?

I do not remember how to prove the general statement I mentioned about gluing operators on ⟦-∞,0] and [0,∞⟧ which match on ∞-Jets at 0; neither can I prove micro-locality of the operator I described ...
Ilya Zakharevich's user avatar
3 votes

Is every continuous microlocal operator a pseudo-differential operator?

Not really an answer: but you do realize how many different notions of a ΨDO are there? So pick up one of them, and the example would not be a ΨDO in the sense of another definition. BTW: AFAIK, the ...
Ilya Zakharevich's user avatar
3 votes
Accepted

Wavefront set of characteristic function of rough set

Let me first begin with an elementary example, taking $X=[0,1]^2$ in $\mathbb R^2$. It is then easy to see directly that the wave-front-set of $\mathbb 1_X$ is everywhere the conormal bundle except at ...
Bazin's user avatar
  • 15.1k
2 votes

Microlocal proof of Wigner semicircle theorem?

You need the theory of $h$-pseudodifferential operators. Standard references are: "Spectral Asymptotics in the Semi-Classical Limit" by Dimassi and Sjöstrand. "An Introduction to Semiclassical ...
Abdelmalek Abdesselam's user avatar

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