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Questions tagged [pseudo-differential-operators]

This tag is for questions regarding to pseudo-differential operators, which are generalizations of differential operators and Fourier multipliers. It satisfies estimates for the derivatives analogous to the estimates for derivatives of polynomials, which are symbols of differential operators.

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Limiting behavior of integral representation of $(\sqrt{\alpha^2-\partial_x^2}-\alpha)f(x)$

While studying pseudo-differential operators of type $\left(\sqrt{\alpha^{2} - \partial_{x}^{2}}-\alpha\right)\operatorname{f}\left(x\right)$, I came across the following integral representation of ...
Caesar.tcl's user avatar
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About evolution problem with variable coefficients.

I'm studying differential operators. For example, in the evolution equation \begin{align} u_t&=(1-\partial_x^2)u\\ u(0)&=u_0 \end{align} Question 1. Does this problem have any name in the ...
eraldcoil's user avatar
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Spectral theory for this kind of operators

Let $a(x,\xi)$ be the symbol for a differential operator $$|\partial^{\alpha}_{x}\, \partial^{\beta}_{\xi} a(x,\xi)| \leq C (1+|x|+|\xi|)^{m-|\alpha|-|\beta|}$$ $|x|+|\xi|\geq c$. Is there a spectral ...
zoran's user avatar
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Understanding the Relationship Between the Principal Symbol of $-\Delta$ and $\sqrt{-\Delta}$ and Geodesic Flow in Hamiltonian Systems

In the context of Hamiltonian systems in symplectic and Riemannian geometry, consider the following fact: Let $(M,g)$ be a Riemannian manifold and $(M,\omega,H)$ a Hamiltonian system with $$H(q,p)=\...
ayphyros's user avatar
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spectral theory of pseudo-differential operators of class $S^m$

I would be very grateful if you could give me the titles of books that deal with the spectral theory of pseudo-differential operators of class $S^m$ Thank you very much.
Fadil adil's user avatar
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Principal Symbol of the Fractional Laplacian on Manifolds

In the euclidean space $\mathbb{R}^n$, we can define the Fractional Laplacian as $$(-\Delta)^s f := \int |\xi|^{2s}\hat{f}(\xi)e^{ix\cdot\xi}d\xi.$$ The principal symbol is clearly $p(x,\xi)=|\xi|^{2s}...
ayphyros's user avatar
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Explicit form of Parametrix for 2nd Order Elliptic Linear PDE in Divergence Form

Suppose we are given the following elliptic operator: $$P(u) = -(a^{ij} (x) u(x)_j)_i $$ where $a^{ij}$ is positive, symmetric and bounded (uniformly elliptic) over a smooth bounded domain $\Omega \...
A. L.'s user avatar
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Example - Homogeneity

If $a(x,\xi)$ is of $C^{\infty}$ class and positively homogeneous of degree m for $|\xi|\geq1$, i.e., $$ a(x, t\xi) = t^{m} a(x, \xi), |\xi| \geq 1, t\geq1, $$ then $a \in S^{m}_{1,0} = S^{...
Uchiha Itachi's user avatar
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254 views

How to implement the Neumann boundary condition when solving the heat equation using Chebyshev's pseudo-spectral method

I am studying the Chebyshev pseudo-spectral method and having problems understanding how to implement the Neumann boundary condition when trying to solve a PDE. To understand better how to implement ...
FriendlyNeighborhoodEngineer's user avatar
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what does it mean for a sequence of function to converge pointwise but uniformly in $\epsilon$?

I am reading Ruzhansky's textbook on pseudodifferential operators and came across this passage: I have never seen a sentence before that says this sequence of functions converges pointwise, uniformly ...
Bill's user avatar
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Semigroup property between pseudodifferential operators and differential operators

Given a positive integer $n$ and a>0. Let consider the operators $\nabla^n (\cdot)= \sum_{i=1}^d \partial_i^{n}(\cdot) $, and $(1- \Delta)^{\frac a 2} $ defined at the Fourier level as (modulus ...
g.cooper's user avatar
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1 answer
35 views

Smoothness of heat kernel on Lipschitz and polygon (cornered) domain

I'm wondering about the spatial smoothness of the heat kernel $K(t,x,x_0)$ on Lipschitz and polygon domains (or cornered domains). It's well known that $K(t,x,x_0)$ is smooth in $t$ for very general ...
celebi's user avatar
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Why is the support of a linear differential operator the diagonal.

In Shubin's 'Differential operators and spectral theory' on page 16 he states that linear differential operators are properly supported as pseudo differential operators since the support of their ...
Pambra iskra's user avatar
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How does one show that the operator whose kernel is properly supported is a smoothing operator?

Proposition 1.7 (Properly supported smoothing operators). $L^{-\infty}=$ smoothing operators. Given $A \in L^{-\infty}$ with properly supported amplitude $a \in S^{-\infty}\left(\Omega \times \Omega \...
Pambra iskra's user avatar
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Estimate of Fourier transform of compactly supported function.

The following argument is quoted from a book about Pseudodifferential operator. I am confused about the estimate of Fourier transform of a compactly supported function. For a smooth function $p(x,\xi)$...
vent de la paix's user avatar
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Exercise of Pseudodifferential operators problem

Let $\varepsilon > 0$, $\Omega = \lbrace \zeta \in \mathbb{C}^n ; |Im\zeta| < \varepsilon |Re\zeta| \rbrace$ and $a(\zeta)$ a holomorphic function on $\Omega$ satisfying an estimate $|a(\zeta)| \...
Lilileaf's user avatar
2 votes
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108 views

The Hörmander symbol space $S^{-\infty}(\Omega \times \mathbb{R}^n) \subset S^{m}_{cl}(\Omega \times \mathbb{R}^n)$ is closed

This is Exercise 3.4) in Peter Hintz's Introduction to microlocal analysis Which I am using for exam preparation. Let $\Omega \subset \mathbb{R}^n$ open. Consider for $m \in \mathbb{Z}$ the space $S^m(...
Paul Joh's user avatar
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Reference request on basic properties of the principal symbols of pseudodifferential operators

I am looking for a reference which would treat some very elementary properties of the principal symbols of pseudodifferential operators, such as conjugation and products. In particular, I am ...
Cartesian Bear's user avatar
3 votes
1 answer
104 views

Reference Request for a theorem on translation invariant operators on $C_c^\infty(\mathbb{R}^n)$

I think a result with possible minor modifications in the hypothesis should be true. I am looking for a reference for such a result. Any leads are greatly appreciated. Let $\Lambda:C^\infty_c(\mathbb{...
Satwata Hans's user avatar
1 vote
0 answers
68 views

The support of the parametrix of laplacian

The Analysis of Linear Partial Differential Operator Vol.I 2nd Edition Page207 The proof of Lemma 7.6.3 says: By Theorem 7.1.22 the operator$(-\Delta)^s$ has a parametrix $E$ with support in the open ...
Vstal's user avatar
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Wavefront set of a distribution and elliptic points on a manifold

Let $M$ be a smooth closed manifold, $E$ a Hermitian vector bundle over $M$ and $P$ a pseudodifferential operator. Let $u\in D’(M,E)$ such that $Pu=0$. I want show that $$ WF(u) \subset T_0^{*}M \...
zarathustra's user avatar
1 vote
0 answers
95 views

Solving a fractional differential equation

I am working with a fractional Laplacian that is based on it's Fourier transform, namely $$(-\Box)^{\alpha}f(t) := \int_{-\infty}^\infty d\omega e^{i\omega t} |\omega|^\alpha \int_{-\infty}^\infty d\...
Audrique's user avatar
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3 votes
1 answer
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Proof of first step theorem 7.7.1. in Lars Hormander's "The Analysis of Linear Partial Differential Operators I"

I'm struggling to follow the "obvious" k=0 step in the proof of this theorem: THEOREM 7.7.1. Let $K\subset\mathbb{R}^n$ be a compact set, $X$ an open neighborhood containing $K$ and $j, k$ ...
Nathanael Schilling's user avatar
3 votes
0 answers
82 views

Intuition for the differences between two notions of quantum ergodicity: One given by weak-* convergence and one by pseudodifferential operators

Consider the two notions of quantum ergodicity of the Laplacian operator $\Delta$. (Phase space): $\Delta$ is said to be quantum ergodic (in the phase space) in a compact Riemannian manifold if there ...
Epsilon Away's user avatar
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1 vote
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30 views

Understanding an analogue between the classical ergodicty theorems and its QE version $\left<Au_j, u_j\right>\to \int\sigma(A)$

I am asking whether there is a formulation of quantum ergodicity property of pseudodifferential operators that has the following "form" of Birkhoff's/von Neumann's ergodicity theorems: A ...
Cartesian Bear's user avatar
5 votes
1 answer
112 views

Is there a smooth tempered distribution $u$ such that $\sum_{j=0}^\infty 2^{-j} u(x-j)$ is not smooth?

I found the following claim in page 3 of Weinstein, Alan, A symbol class for some Schrödinger equations on ${\mathbb{R}}^ n$, Am. J. Math. 107, 1-21 (1985). ZBL0574.35023: Operators whose symbols ...
Calvin Khor's user avatar
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1 answer
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Creating a bounded operator from an unbounded symbol ? - discussion around the Calderon-Vaillancourt theorem

Calderon-Vaillancourt theorem states that any quantization of a bounded symbol (in the sense of this article for example) is a bounded operator on the $L^2(\mathbb{R^n})$ space. I was wondering if ...
Vincent's user avatar
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2 votes
1 answer
38 views

Showing that $(|\left<a, b\right>| - \epsilon)^2 \leq |\left<a, Mb\right>|^2$ for self-adjoint operator $M$ such that $||(I - M)b|| < \epsilon$

Let $a, b$ be $L^2$ normalized functions and $M$ be a self-adjoint pseudodifferential operator such that $||(I - M)b|| < \epsilon$ and $\sigma(M) \leq 1$ (author of the paper I am reading has not ...
Cartesian Bear's user avatar
3 votes
0 answers
100 views

Reference needed: Symbol sequence for pseudodifferential operators

In Higsons's book Analytic K-Homology there is a section (subsection (b) in 2.8 "Geometric Examples of Extensions", starting from page 46) which discusses the following exact sequence called ...
ChenIteratedIntegral's user avatar
2 votes
2 answers
462 views

Good books and lecture notes to learn pseudo-differential operators and spectral theory

I am looking for a list of good books and lecture notes to learn pseudo-differential operators and spectral theory (for infinite dimensions.) I am familiar with introductory functional analysis, ...
Dimension Entangled's user avatar
1 vote
0 answers
109 views

Extending symbol of pseudodifferential operator defined outside of a ball

While reading Wong's book "An introduction to pseudo-differential operators" (3rd edition), I came across the following statement in the exercises : Let $\sigma \in C^\infty(R^n\times R^n) $...
Amd's user avatar
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1 answer
42 views

Simple expression for $(-\Delta)^{\frac{3}{2}}$ in the spatial domain?

I can write the harmonic and biharmonic operators as: $$-\Delta = -\sum_i \partial_{ii}, \quad (-\Delta)^2 = \sum_i\sum_j \partial_{ii}\partial_{jj}$$. Is there such a simple expression for $(-\Delta)^...
lightxbulb's user avatar
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2 votes
0 answers
40 views

Spectrum of the Weyl quantized operator $\mathrm{Op}\left(\sqrt{\frac{x^2+p^2}{2}}\right)$

Consider a 1D phase space whose generic points are denoted as $(x,p)$. We know that the Weyl quantization $\mathrm{Op}\left(\frac{x^2+p^2}{2}\right)$ is the harmonic oscillator Hamiltonian, whose ...
Laplacian's user avatar
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3 votes
1 answer
135 views

Defect measure associated to a sequence of exponentials

In my road to understand microlocal defect measures, at the beginning of Gerard's article Microlocal defect measures, there is an statement about (an example of) defect measures where I am struggling. ...
rebo79's user avatar
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1 vote
0 answers
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Composition of two differential operators . Laplacian operator.

I have a question with the composition of two operators. To contextualize. Let $p_1 = p_1 (x,D_x):= 1-\partial_ {x}^2 $ be the operator $ p_1: D(p_1)\subset L^2 \to L^2 $ where $$ D(p_1) = \left\{u \...
eraldcoil's user avatar
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2 votes
1 answer
86 views

Asymptotic expansions with compactly supported terms and smoothing operators

Suppose that we have a pseudodifferental operator $A\in \Psi^m(\mathbb{R}^n)$ with symbol $a\in S^m(\mathbb{R}^n\times\mathbb{R}^n)$, and $a$ has an asymptotic expansion $a\sim \sum\limits_{j=0}^\...
user900940's user avatar
1 vote
0 answers
70 views

Isomorphism of Differential operator involving the Laplacian

I have to show that $(1+\Delta)^s:H^k(U)\rightarrow H^{k-2s}(U)$ is an isomorphism where $\Delta$ is the Laplacian on $U\subset\mathbb{R}^n$ where $U$ has compact support and $H^k(U),H^{k-s}(U)$ are ...
Satwata Hans's user avatar
3 votes
1 answer
222 views

Asymptotic Expansions of Symbols Vs Asymptotic Expansion of Pseudodifferential Operators

Let $a_k(x,\xi)$ be a family of symbols on $\mathbf{R}^d \times \mathbf{R}^d$, where $a_k$ has order $\alpha_k$, and $\lim_{k \to \infty} \alpha_k = -\infty$. Then for another symbol $a(x,\xi)$, we ...
Jacob Denson's user avatar
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4 votes
0 answers
122 views

What is a Symbol (as in "Symbol Calculus")?

I have been reading through several papers on Deformation Quantization and the terms "symbol" and "symbol calculus" keep cropping up. I am somewhat well acquainted with most of the ...
J.V.Gaiter's user avatar
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2 votes
0 answers
60 views

Product with a smooth function, Sobolev regularity and pseudo-differential operator

I am working on microlocal Sobolev spaces using Hörmander's books, and while trying to write down the details of a proof, I have faced the following problem. It looks easy, but I can't find a correct ...
Thomas Guegamian's user avatar
1 vote
1 answer
41 views

Comparison of quantization procedures

Let $M$ be a compact manifold and let $U\subset M$ be an open set with a coordinate chart $\Phi:U\to V$ with $V\subset \mathbb{R}^d$. Suppose that $w\in C_c^\infty(U)$ is a smooth function supported ...
felipeh's user avatar
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2 votes
1 answer
74 views

Justifying Interchange of Integral

I am trying to show that if $P$ is a pseudo-differential operator with symbol given by $p(x,\xi)$ i.e. the operator $$P:\mathcal S\rightarrow \mathcal S$$ defined by $$Pf(x)=\int_{\mathbb R^n}e^{ix\...
user6's user avatar
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0 votes
0 answers
15 views

Laplacian raised to positive integer

In books about Pseudo-differential operators, they use many times $\triangle^k u$ but i have a question, what means this really option A $\displaystyle{\sum_{|\alpha|=k}}{\, \frac{k!}{\alpha!} \left({\...
user89940's user avatar
  • 388
2 votes
1 answer
277 views

How and when is the inverse Laplacian well-defined as a pseudo-differential operator?

I recently came across an interesting (mis-)use of formal equivalencies. First, the uncontroversial bits. By the Fourier derivative theorem, it is straightforward that the 2D Laplace operator can be ...
MrArsGravis's user avatar
1 vote
0 answers
215 views

What is the meaning of $\flat$ and $\sharp$ in this smoothing operator? $M^\sharp(x,\xi)=\sum_k \Psi_0(2^{-k\delta}D)m_k(x)\;\psi_{k+1}(\xi) $

I found this: $$M(x,\xi) = M^\sharp(x,\xi)+M^\flat(x,\xi),$$ in this paper, where $$M^\sharp(x,\xi)=\sum_k \Psi_0(2^{-k\delta}D)m_k(x)\;\psi_{k+1}(\xi) $$ and i'm not sure whether i understand it ...
Lord Commander's user avatar
2 votes
1 answer
175 views

Two Definition of Elliptic Symbols

There two definition of elliptic symbol. A smooth matrix function $p(x,\xi)$ is a elliptic symbol of order $m\in\mathbb{R}$ if exist a constant $c>0$ such that for all $|\xi|>c$ we have $p(x,\xi)...
user89940's user avatar
  • 388
2 votes
1 answer
285 views

Is it possible to construct a vector space on convolution

Let $S$ be a set of functions on $R$ such that for any two functions $f$ and $g$ in $S$, the convolution: $$ (f{\ast}g)(x)={\int}f(y)\ g(x-y)\ dy $$ exists. Since the Dirac delta is technically not a ...
user54738's user avatar
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3 votes
1 answer
264 views

Confusion about definition (in ΨDO theory) of Sobolev spaces on open sets in Euclidean space

I'm reading Pseudodifferential Operators by M. E. Taylor, where the author talks about $H^s(\Omega)$ for $s\in\mathbb{R}$ and $\Omega\subset\mathbb{R}^n$ an open set (for example, in the statement of ...
Yuxiao Xie's user avatar
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0 votes
1 answer
135 views

Fractional derivative via Continuous Functional Calculus

If we study C* algebras, at one point or another we are exposed to the idea of the Continuous functional Calculus, i.e. if $\mathcal{Y}$ is a unital C* algebra, $\xi \in \mathcal{Y}$ is normal, that ...
h3fr43nd's user avatar
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1 vote
0 answers
43 views

Why is the Order of a Classical Symbol Well-defined?

A function $a$ defined on $T^*\mathbb{R}^n$ is called a classical symbol of order $m$ provided it is smooth and there exists $\tilde{a}\in C^\infty(\mathbb{R}^n\times S^{n-1}\times\mathbb{R}_+)$ so ...
user269711's user avatar