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.
125 questions
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Is there a definition of the half Laplacian for Manifold-valued functions?
I want to know if there is a generalization for the fractional Laplacian for manifold valued functions. I am trying to define a fractional, parabolic-like PDE intrinsically where solutions map from $\...
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The adjoint of $\partial^{a + ib}$ on $H^k([0, \infty))$ for functions that do not vanish at $x = 0$.
Let's say I have a pseudodifferential operator $\partial^{a + ib}$ for $a + ib \in \mathbb C$ that is defined on the Sobolev space $H^k([0, \infty))$ for a sufficiently large $k$. (In fact, I really ...
- 1,348
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Locality and pseudolocality of a shifted Laplacian
Denote the differential operator $A=\Delta+1$, $\Delta$ is Laplacian. We know that $A$ is invertible, its inverse can be written using Fourier transform: $A^{-1}u=\mathcal{F}^{-1}\circ(1+|\cdot|^2)^{-...
- 41
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Meromorphic symbol pseudodifferential operator
Context: For some function $f\in\mathcal{M}(\mathbb C)$ (meromorphic function on $\mathbb C$), I am interested in linear operators $T_f$ that act on functions of the form $g_a:x\mapsto \exp(ax)$ in ...
- 2,017
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98
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Commutator and pseudo differential calculus
Let $j$ be a smooth bounded function yielding a pseudo differential operator $j(p)$ of order $0$ (where $p=-i\nabla$). I stumbled upon an article saying that the commutator $[j(p), |x|]$ is a bounded ...
- 107
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Question on pseudodifferential operators
I have conceptual question on pseudodifferential operators. Let $U\subset\mathbb{R}^{d}$ be open. A pseudodifferential operator $A:C^{\infty}_c(U)\to C^{\infty}(U)$ is an operator of the form
$$A\psi(...
user674359
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Composition of symbols of pseudo-differential operators
For $m \in \mathbb{R}$, denote the set $S^m$ to be the set of all $\sigma \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ such that for any $\alpha, \beta \in \mathbb{N}^n$, there exists $C_{\alpha, ...
user1428375
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pseudodifferential operator, singular integral operator, and oscillatory integral operator
I am confused by the three integral operators in the title.
I am currently reading a note found online, and from its development, my impression is that the kernels of pseudodifferential operators have ...
- 431
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Limiting behavior of integral representation of $(\sqrt{\alpha^2-\partial_x^2}-\alpha)f(x)$
While studying pseudo-differential operators of type $\left(\sqrt{\alpha^{2} - \partial_{x}^{2}}-\alpha\right)\operatorname{f}\left(x\right)$, I came across the following integral representation of ...
- 905
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Understanding the Relationship Between the Principal Symbol of $-\Delta$ and $\sqrt{-\Delta}$ and Geodesic Flow in Hamiltonian Systems
In the context of Hamiltonian systems in symplectic and Riemannian geometry, consider the following fact: Let $(M,g)$ be a Riemannian manifold and $(M,\omega,H)$ a Hamiltonian system with $$H(q,p)=\...
- 463
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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.
- 11
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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}...
- 463
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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 \...
- 235
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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^{...
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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 ...
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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 ...
- 4,751
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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 ...
- 75
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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 ...
- 81
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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 ...
- 141
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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 \...
- 141
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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 $p(x,\xi)$...
- 475
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156
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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(...
- 599
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68
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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 ...
- 1,673
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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 $\Lambda:C^\infty_c(\mathbb{...
- 631
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121
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The support of the parametrix of laplacian
The Analysis of Linear Partial Differential Operator Vol.I 2nd Edition Page207 The proof of Lemma 7.6.3 says:
By Theorem 7.1.22 the operator$(-\Delta)^s$ has a parametrix $E$ with support in the open ...
- 384
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84
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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 $$ WF(u) \subset T_0^{*}M \...
- 181
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128
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Solving a fractional differential equation
I am working with a fractional Laplacian that is based on it's Fourier transform, namely
$$(-\Box)^{\alpha}f(t) := \int_{-\infty}^\infty d\omega e^{i\omega t} |\omega|^\alpha \int_{-\infty}^\infty d\...
- 121
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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 $j, k$ ...
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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 ...
- 1,080
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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 ...
- 1,673
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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 ...
- 36.7k
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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 ...
- 13
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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 ...
- 1,673
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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 ...
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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, ...
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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) $...
- 130
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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)^...
- 2,438
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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 ...
- 2,418
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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.
...
- 509
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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 \...
- 3,936
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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}^\...
- 105
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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 ...
- 631
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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 ...
- 2,291
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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 ...
- 2,724
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71
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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 ...
- 399
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60
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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 ...
- 4,034
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1
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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\...
- 4,329
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22
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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({\...
- 388
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390
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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 ...
- 345
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289
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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 ...
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