Questions tagged [micro-local-analysis]
The micro-local-analysis tag has no usage guidance.
100
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Applications of microlocal analysis?
What examples are there of striking applications of the ideas of microlocal analysis?
Ideally I'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/...
20
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1
answer
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Algebraic microlocal analysis and nonlinear PDE
Though originating in the study of linear partial differential equations, microlocal analysis has become an invaluable tool in the study of nonlinear pde. Of particular importance has been the ...
13
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4
answers
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Is every continuous microlocal operator a pseudo-differential operator?
Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space.
Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal.
By being microlocal I mean that the wave ...
12
votes
2
answers
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How to visualize the Microsupport of a Sheaf?
I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...
11
votes
2
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427
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Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$
I asked this question on Math StackExchange, but it did not receive an answer, despite my offering a bounty to attract attention. I am unsure whether it is appropriate for this venue, but I thought ...
11
votes
3
answers
3k
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Characterization of inverse differential operators
If I have a partial differential operator $p(D)$, where $p$ is a polynomial with constant coefficients and $D$ is the derivative in Euclidean space. Its inverse is easily described in Fourier space: $\...
10
votes
2
answers
795
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Weyl law for (non-semiclassical) Schrodinger operator
The Weyl law for a semiclassical Schrodinger operator
$$ A_h\ := \ -h^2\Delta+V(x) $$
on an $d$-dimensional complete Riemannian manifold $M$
says that the number $N(A_h,1)$ of eigenvalues of $A_h$ ...
10
votes
1
answer
375
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Is every endomorphism of the sheaf of holomorphic functions on a disk a differential operator?
Let $D= \{z\in \mathbb{C}:|z| < 1\}$ be the unit disk. And consider the sheaf of holomorphic functions $\mathcal{O}_{D}$.
Question (?) : Is there a sheaf endomorphisms $\phi : \mathcal{O}_D \to \...
9
votes
2
answers
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Fourier transform of compactly supported distribution is smooth
My advisor made the comment that if $u\in \mathcal{E}'$ is a compactly supported distribution, then $\hat{u}(\xi)\in C^{\infty}(\mathbb{R}^n)$ is actually a smooth function (not merely a distribution ...
9
votes
0
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358
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Bernoulli-like polynomials
Let $\psi_0 (x,t)=\frac{te^{xt}}{1-e^{-t}}$. Then
$$\psi_0(0,t)=\frac{t}{1-e^{-t}};$$
$$\psi_0(x,t)=1+\sum_{n=1}^\infty \frac{t^n}{n!} B_n(x)$$
where $B_n$ is a monic polynomial of degree $n.$
Now ...
8
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3
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Characteristic Variety of the Principal Symbol solves PDE system?
In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see for example, http://www.sciencedirect.com/science/...
7
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1
answer
859
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Pseudo-differential operators which are independent of lower order perturbations
In the area of pseudo-differential operators, we know that for elliptic type or real principal type operators reductions are independent of lower order terms. For example, if $P$ is zeroth order real ...
7
votes
0
answers
618
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What's the definition of a microlocal sheaf?
I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general.
In this paper of ...
7
votes
0
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233
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Smoothness of solution map for PDE
I am wondering what sort of results are available for the following sort of problem, or where to look in the literature for work dealing with such problems, especially in the degenerate elliptic ...
6
votes
1
answer
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propagation of singularities & the Schrodinger equation
I've been thinking about the following propagation of singularities result:
Let $X$ be a compact manifold, and let $P$ be a differential operator (of, say, order $m$) on $X$ whose principal symbol $\...
6
votes
1
answer
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How to understand the Fourier-Sato transform and microlocalization functors?
Given a smooth real vector bundle $\pi: E \to M$ I can look at the (bounded from below) derived category of sheaves on $E$. Since $E$ admits a very natural action of $\mathbb{R}^{\geq 0}$ by scaling, ...
6
votes
1
answer
800
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Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold
I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
6
votes
0
answers
193
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Product of distributions under wavefront set condition is zero
Assume $u, v \in \mathcal{D}'(\mathbb{R}^n)$ are distributions with compact support. Denote by $\operatorname{WF}(\bullet) \subset T^*\mathbb{R}^n \setminus 0$ the wavefront set of a distribution $\...
6
votes
0
answers
374
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Non-characteristic maps (ala D-modules)
I am trying to understand a `well known' fact (see Kashiwara'sIntroduction to microlocal analysis page 63, remark 4.8) about non-characteristic morphisms. Here is ...
5
votes
1
answer
359
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Wavefront set of characteristic function of rough set
It is a standard exercise to show that if $X\subseteq\mathbb{R}^n$ has smooth boundary, then the characteristic function $1_X$ has wavefront set $$\{(x,\xi)\in\partial X\times\mathbb{R}^n\setminus\{0\}...
5
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1
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635
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Kashiwara's watermelon theorem and Microlocal version of Helgason's (support) and Holmgren's theorems
I would like to find good references for the theorems mentioned above in the title. I am reading chapter VIII of Hörmander's classic, but I wonder whether there is something more up-to-date.
My ...
5
votes
1
answer
208
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Why do people study Weyl asymptotics and partial-spectral-projections?
The major focus of the research that my advisor has me doing centers around the idea of asymptotic behavior of partial-spectral-projections on compact manifolds. In a few sentences, here is the ...
5
votes
1
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392
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C^\infty versus semiclassical wavefront sets
Zworski states that if $u$ is a compactly supported distribution, independent of the semiclassical parameter $h$, then the relationship between the $C^\infty$ and semiclassical wavefront sets of $u$ ...
5
votes
1
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Does the 'reproducing kernel formula' for a bounded open set $U$ define an equivalent norm on the Sobolev space $H^1_0(U)$
We refer to the 'reproducing convolution formula with a kernel' for an open bounded domain $U$ of $R^n$, $n \geq 2$ discussed in the paper of G. Talenti (Annali de Matematica, Dec 1976) on Best ...
4
votes
1
answer
249
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Interesting (non) examples of singular support
I'm trying to better understand singular support of sheaves on smooth manifolds---to this end: What are examples of conical subsets of $T^*X$ that cannot arise as the singular support of a sheaf on $...
4
votes
1
answer
222
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Real-analytic variant of theorem 4.2.5 of Duistermaat's "FIO", 1996
Theorem 4.2.5 of Duistermaat's "Fourier Integral Operators", 1996, states:
Let $A \in I^m(X,Y,C)$ be an elliptic Fourier Integral Operator of order $m$, associated to a bijective canonical ...
4
votes
1
answer
252
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Interpretation of the integral "with respect to a plane wave" in terms of Radon transform
This question might have a formulation in higher dimensions, but for now let's deal with the 2 dimensional Radon transform:
$\newcommand{\R}{\mathbb{R}}$
$$
Rf(\varphi,s)=\int_{-\infty}^\infty f(s\...
4
votes
1
answer
286
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Sharpest version of semiclassical Calderon-Vaillancourt theorem
Let $S$ be the space of symbols defined by $$S:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}\},$$ ...
4
votes
1
answer
193
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Distribution boundary value of analytic function and wave front sets
Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
4
votes
0
answers
68
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Local energy estimate in a semiclassical regime
Let us consider $h_n=(2n+1)^{-1/2}\to 0$ as $n\to \infty$ be a small parameter, which we just write as $h$ for convenience, and $u_h : \mathbb{R} \to \mathbb{R}$ be functions satisfying $Pu_h=0$ (I ...
4
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0
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What role do semiclassical methods play in the study of Ginzburg--Landau-type equations?
As far as I understand, semiclassical limits are used in quantum mechanics to analyse equations that depend on a small parameter $\hbar$. Apparently studying properties of the PDE as $\hbar \to 0$ ...
4
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0
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If theorem valid for compactly supported distribution then is it also valid for tempered distribution?
I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution.
For instance,
Theorem: Any $A \in \Psi^{m}$ ...
4
votes
0
answers
192
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Can the existence of geodesics be deduced from properties of the Laplacian?
As I understand it, the semiclassical trace formula in particular relates lengths of geodesics to eigenvalues of the Laplacian.
Is it possible to prove that every compact Riemannian manifold has a ...
4
votes
0
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Properties of microlocalization
Let $i: M\hookrightarrow X$ be the inclusion of a closed submanifold in a smooth manifold $X$. I denote by $T_MX$ the normal bundle to $M$ in $X$, by $T^{\ast}_MX$ its dual bundle, and by $D^b(X)$ the ...
4
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0
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One-parameter unitary group preserving invariant domain of infinitesimal generator
Let $\mathcal{H}$ be a separable Hilbert space (e.g. $L^{2}(\mathbb{R}^{d}))$, and let $\mathcal{D}_{1}\subset\mathcal{H}$ be a dense subspace (e.g. $\mathcal{S}(\mathbb{R}^{d})$). Suppose that an ...
4
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0
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242
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What's a good resource for Hormander symbols of type (1/2, 1/2)?
I'm currently working with some pseudodifferential operators of Hormander class $L^{m}_{\frac{1}{2},\frac{1}{2}}$ and unfortunately many of the usual tools break down, due to difficulties with their ...
3
votes
1
answer
402
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Help understand a calculation involving RHom of sheaves on manifolds
I am reading a paper and there is some computation of RHom of sheaves that I don't understand. I hope this is the right place to ask.
It is this paper, example 3.10 , page 25
arxiv.org/pdf/1005.1517v4....
3
votes
1
answer
234
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do hyperfunction solutions always exist?
I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an ...
3
votes
2
answers
261
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Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1
In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then $\...
3
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1
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Equivalent Littlewood-Paley-type decompositions
The theory of Besov and Triebel-Lizorkin spaces usually proceeds by taking a dyadic decomposition of unity, i.e. some non-negative functions $\psi_0,\psi \in C_c^\infty(\mathbb{R})$ such that
\begin{...
3
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1
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546
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Characterisation of the wavefront set
I am totally new to microlocal analysis, and have been studying Jared Wunsch's notes. I have been puzzling over the properties of the wavefront set.
Let $X$ be a compact Riemannian manifold, and $\...
3
votes
1
answer
591
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Intuition behind the Duistermaat-Guillemin version of Weyl's law
The theorem in question (see this paper), after a modification by Ivrii (see this paper) states the following:
Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 2$. Assume that the ...
3
votes
1
answer
492
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Wave front set from the FBI or Segal-Bergman transform (and a motivation)
In the André Martinez's notes "Introduction to microlocal and semiclassical analysis" the Wave Front Set is defined as the complement of the set of points having neighborhoods where the FBI transform ...
3
votes
2
answers
312
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Reference for commutator estimate
I'm interested in Sobolev space estimates for commutators involving a pseudodifferential operator and a Fourier multiplier. More specifically, suppose $p = p(x,\xi) \in S_{1,0}^{m_1}$ and let $q = q(\...
3
votes
1
answer
767
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Application of Egorov's Theorem for Pseudodifferential Operators
Let $\;P_{0} \in OPS^{m}_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic ...
3
votes
1
answer
377
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Elliptic Fourier integral operators
I know what it means for a pseudodifferential operator $A\in\Psi(\mathbb{R}^n)$ to be elliptic at a point $(x,\xi)\in T^*\mathbb{R}^n$: the principal symbol of $A$ is non-vanishing at the point.
But ...
3
votes
1
answer
315
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Wavefront set and Duhamel's principle
Consider the Cauchy problem:
$$
\frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0,
$$
where $A$ has real principal ...
3
votes
0
answers
46
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microlocalisation of a Lagrangian state
Let
$f : \mathbb{R} \to \mathbb{R}$ a smooth function of exponential decay at infinity (I think of something like a Gaussian) ;
$g$ a polynomial of $\mathbb{R}$.
I would like to have a quantitative ...
3
votes
0
answers
43
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Conditions ensuring that the paraproduct remainder is well-defined
In short, my question is: are there conditions that one can impose on two tempered distributions $u$ and $v$ that will guarantee that the paraproduct remainder $R(u,v)$ is well-defined and is "...
3
votes
0
answers
97
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Definition clarification: "regular directed distributions"
(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of ...