Questions tagged [pseudo-differential-operators]
The pseudo-differential-operators tag has no usage guidance.
0
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10 views
Boundary value problem for Hemlholtz equation with pseudospectral method.
I have an equation of the form $(1-0.1 \Delta)f=1$ with boundary condition $f=0$. I need to solve it for $f$ by pseudospectral method in python. Apparently, it should be $f=\frac{1}{1-0.1\Delta}$. I ...
2
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1answer
38 views
What does constant along characteristic mean
When we have linear or quasilinear first order pde
$$ a(x,y,u) u_x + b(x,y,u) u_y = c(x,y,u) $$
And suppose we have found characteristics with prescribed Cauchy data $u |_{\text{curve}} = \phi$
I ...
2
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1answer
53 views
About pseudo-differential operators
Let $\Omega$ be an open and connect subset of $\mathbb{R}^2$,we denote by $\partial \Omega$ its boundary the latter is supposed to be smooth ($\mathcal{C}^\infty)$, its outword normal vector is ...
1
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1answer
30 views
The symbol of a pseudifferential operator: how to reconstruct the symbol from an operator?
To simplify let us assume that $U \subset \mathbb{R}^n$ is an open set. Let $P$ be a (scalar) psuedodifferential operator of order $d$ and $p$ be its symbol. Therefore it may be expressed as follows:
...
2
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0answers
23 views
Understanding Y U Egorov's theorem
I was reading the paper
"Hörmander, L., Fourier integral operators. I, Acta Math. 127, 79-183 (1971). ZBL0212.46601." and on page 3, there is a statement saying the following thing:
Can someone ...
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55 views
Integration by parts for an oscillatory integral
I want to understand a certain integration by parts argument in Treves' book Introduction to Pseudodifferential and Fourier Integral Operators (It appears in the proof of Theorem 4.1). Here is a self-...
0
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1answer
17 views
Differentiation of an operator raised to a variable power
I am studying first order systems of the form
\begin{equation}
L=\partial_t+K(t,x,D_x)\text{ where }D_x=-i\partial_x
\end{equation}
There is a change of variable and operator of our concern becomes ...
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1answer
35 views
Pseudo Differential Operator example
I was reading a introduction about Pseudo differential operators, and the definition of that type of operator was:
$$|\partial_x^\alpha \partial_\xi^\beta f(x,\xi) | \le C_{\alpha,\beta}(1 + |\xi|)^{...
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Solution operator to scalar hyperbolic equation on Sobolev space $H^s$
If i have the solution operator $e^{itA}$ to the following equation $$\frac{du}{dt}=iA(x,D)u$$ where $A\in OPS^1$ is a peudo differential operator of order 1. Then how could one show that the solution ...
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22 views
Proof of the Pseudolocal property of a pseudo differential operator
Let $a\in S^{\infty}, u\in\mathcal{S}', \Omega=\mathbb{R}^n-singsupp(u)$. Then $\phi u\in C_0^\infty, \forall \phi \in C_0^\infty(\Omega)$ and for any $\psi\in C_0^\infty(\Omega)$ one can find a $\phi\...
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42 views
Adjoint of a symbol of a pseudo differential operator
This is a passage from Hormanders book, the analysis of linear partial differential operators volume 3.
If the adjoint of the symbol $a(x,\xi)$ is $b(x,\xi)=(2\pi)^{-n}\int e^{-i y \cdot \eta}\bar{a}...
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0answers
43 views
Is this operator pseudodifferential or trace-class?
Let $M$ be a closed manifold and $D$ a Dirac operator on $M$. Then (the closure of) $D^2+1$ has a bounded inverse
$$(D^2+1)^{-1}:L^2\rightarrow H^2,$$
where $H^2$ is the second Sobolev space. In ...
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0answers
27 views
Parametrix for an operator on a cylinder
Suppose $T$ is a Dirac-type differential operator on a closed manifold $M$ (or $\mathbb{R}^n$). Let $Q$ be a parametrix for $T$, so that
$$QT-1,\qquad TQ-1$$
are with smooth Schwartz kernels. Consider ...
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0answers
27 views
Reference request: trace class operator
Suppose $D$ is an elliptic differential operator on a non-compact manifold $M$ of order $1$. Then $D$ has a pseudodifferential parametrix $Q$ of order $-1$ such that
$$DQ-1=S_0,\qquad QD-1=S_1$$
are ...
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0answers
47 views
Expressing an operator as a sum of the identity and a compact operator
My problem concerns with the unique solvability of a linear system of integral equations. In my problem, I was able to write the system in matrix form:
$$
M \begin{align}
\begin{bmatrix}
...
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22 views
Continuous extension for PDO
Let $R \in \Psi^{m}(\mathbb{R}^{d})$ be a compactly supported pseudodifferential operator with $m<0$ and $H^{s}_{K}=\{ u \in H^s: \hbox{supp } u \subset K\}$. Is it true that the operator $R$ ...
1
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2answers
42 views
Existence of solutions of $u_t = Lu$ for pseudo-differential operators $L$?
I'm wondering if it is known if existence results for the equation $$\left\{\begin{array}{ll}
\partial_t u = Lu & \text{on }(0,T)\times\mathbb{R}^n\\
u(0,-) = \varphi & \text{on }\mathbb{R}^n\...
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0answers
29 views
Difference of two pseudodifferential operators of the same order
Let $M$ be a manifold. Does there exist a theorem that says something like: "given two pseudodifferential operators $P$ and $Q$ on $M$ of the same order $k$, there exists an $L^2$-bounded operator $B$ ...
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1answer
124 views
Dense subspaces of $L^\infty(\Omega\times\Omega)$
Let $\Omega\subset\mathbb R$ be open and bounded. For continuous functions $C(\Omega\times\Omega)$, the Stone-Weierstrass theorem shows that the products $a(x)b(y)$ of univariate continuous functions ...
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0answers
38 views
The inverse of a PsDO is a PsDO
The following question looks quite simple, but unfortunately I was not able to find an answer in the literature so far.
Let $A \in OPS^m(X)$, $m \in \mathbb R$, be a pseudodifferential operator on a ...
4
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0answers
33 views
Elliptic pseudodifferential operator has at most finite number of non-positive eigenvalues
Let $P \in OPS^m(M)$ ($m$ can be negative of positive) be a self-adjoint elliptic pseudodifferential operator on a compact manifold $M$. How to show that $P$ has only finitely many non-positive ...
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0answers
90 views
Is the Fourier transform of a compactly supported distribution a symbol?
I am learning some pseudo-differential operator and wondering is the Fourier transform of a compactly supported distribution a symbol, which gives a way to interpret the Fourier Inversion for those ...
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19 views
How can we prove that the exponential function shifts an arbitrary non-chaotic differential operator by a constant?
if $f(D)y$ is an arbitrary differential operator, of the basic differential $D \equiv \frac{d}{dx}$, acting on any generic infinitely differentiable function $y(x)$ of some real independent variable $...
1
vote
1answer
81 views
Is strongly elliptic operator necessarily properly elliptic?
Let us consider
$$
A(D)u:=\sum_{|p|\leq l} a_p D^p u,
$$
where $a_p$ may be complex, and define
$$
A_0(\xi)= \sum_{|p|=l} a_p \xi^p
$$
Def 1The operator $A$ is called strongly elliptic if
$$
...
2
votes
1answer
115 views
BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?
Let me illustrate my question by starting with the simplest possible example: Let us consider $P := - \mathrm{d}^2/\mathrm{d}x^2$, an elliptic partial differential operator on $\mathbb{R}$; let us ...
2
votes
0answers
245 views
Every Pseudo-Differential Operator is Pseudo-Local operator.
This is the theorem 8.9 of the book Introduction to PDE, Folland, G. B. I'm trying to complete the proof details. I would like to know how to justify the "Afirmation", with more rigor.
Every pseudo-...
2
votes
1answer
206 views
Extension of Pseudodifferential Operator to the real line
When defining Pseudodifferential-Operators on manifolds, there is the
subtlety that one theoretically has to consider all possible charts instead of just a chosen atlas. I am trying to understand how ...
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0answers
58 views
Pseudo differential operator in pde
I'm reading a paper (https://arxiv.org/pdf/1405.7565.pdf) and I don't quite understand the way the authors used for the pseudo differential operator inside after looking up some notes on pseudo ...
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0answers
154 views
Pseudodifferential operator on manifold
I have a question about the definition of pseudodifferential operator on a manifold $M$.
In most of the cases, assume $(M_i,\phi_i)$ is a chart for $M$, $P$ is defined as a linear operator from $C^\...
1
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0answers
147 views
Is the Index of a self adjoint elliptic operator zero?
Let $P:C^{\infty}(V) \to C^{\infty}(V)$ be an elliptic self adjoint pseudo differential operator of order $d$. Here $V$ is a vector bundle with a metric over a compact oriented riemannian manifold $M$ ...
3
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1answer
194 views
Relation between Fredholm Operators and Elliptic Operator
Let $P:C^{\infty}(M) \to C^{\infty}(M)$ be a pseudo differential operator of order $d$ on a compact manifold $M.$
Then $P$ is said to be elliptic if there exists an operator $Q:C^{\infty}(M) \to C^{\...
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0answers
60 views
Smoothing inside the null-space of a partial differential operator
Here is a fundamental question on PDEs whose answer must be known but is not easy to find:
Let $(Lu)(x) := \sum_{|\alpha|\leq m} c_\alpha(x)\partial_x^\alpha u(x)$ be a partial differential operator ...
3
votes
1answer
157 views
Concrete examples of elliptic pseudo-differential equations
Remember that $p \in S^{m}_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$ or that $p: \mathbb{R}^n \times \mathbb{R}^n \longrightarrow \mathbb{C}$ is a simbol if it is a smooth function such that
\begin{...
2
votes
1answer
70 views
Fourier Transform $\frac{e^{-ik\sqrt{(z-z_0)^2+a^2}}}{\sqrt{(z-z_0)^2+a^2}}$
I am trying to compute the 1D Fourier transform (in $z$) of
$$\frac{e^{-ik\sqrt{(z-z_0)^2+a^2}}}{\sqrt{(z-z_0)^2+a^2}}.$$
I tried to use the fact in 3D,
$$\frac{e^{-ik\sqrt{(z_0-z)^2+(y_0-y)^2+(x_0-...
1
vote
1answer
187 views
Wells 'Differential Analysis on Complex Manifolds' page 127
How does the first equation for $Qu(x)$on this page follow from the defining equation (3.10) on the previous page. This is from the section on pseudodifferential operators in chapter 4. I'm starting ...
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1answer
124 views
Show that a regularizing operator $K : C_c(\Omega) \to \mathcal{D}'(\Omega)$ has kernel $k \in C^\infty(\Omega \times \Omega)$.
I am reading Francois Treves' Introduction to pseudodifferential and Fourier integral operators, vol. I.
Let $\Omega \subseteq \mathbb{R}^n$ be open. On page 11, Treves defines what it means for a ...
3
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0answers
93 views
How far can we push the Schwartz kernel theorem?
The Schwartz kernel theorem works for operators defined on $C_ {c}(\mathbb{R}^n,E)$, as long as $E$ is finite-dimensional and we introduce the right notion of a generalised section. In every ...
2
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0answers
19 views
k-symbol differential opeartor L, and its independent of choices.
This material is in O. Well's Differential analysis on complex manifold, page 115.
Let, $(x,v) \in T'(X)$ ($T^*(X)$ with deleted zero section) and $e \in E_x$, Find $g\in \epsilon(X)$ and $f\in \...
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1answer
429 views
Gårding's inequality for $\mathbb R^n$ implies that for bounded smooth domains?
We are given (weak) Gårding's inequality for elliptic pseudodifferential operators:
Given $a\in S^m$ such that $\operatorname{Op}(a)$ is an elliptic operator, namely $\exists c,R>0$ such that ...
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0answers
166 views
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, https://en.wikipedia.org/wiki/...
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0answers
173 views
Inverse Fourier Transform involving inverse square root
I'm currently working on this paper: http://web.calstatela.edu/faculty/rcooper2/article.pdf and I want to proof Lemma 3.0 in the case of $n=2$, on page 441. It seems that the Ph.D. thesis the author ...
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0answers
55 views
Ellipticity of an operator in Gunther's proof of the isometric embedding
In Deane Yang's notes about Gunther's proof of the celebrated isometric embedding theorem,
at the end it is stated that $v$ inherits the regularity of $h$ because the operator $I-Q_0(v,\cdot)$ is ...
2
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1answer
281 views
Basic examples of functions in Hörmander class
The Hörmander class $S_{\rho,\delta}^m$ (with $\rho,\delta\in[0,1]$) consists of smooth functions $p(x,\xi)$ with
$$|D_x^\beta D_\xi^\alpha p(x,\xi)|\leq C_{\alpha\beta}(1+|\xi|^2)^{(m-\rho|\alpha|+\...
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0answers
36 views
Elliptic regularity of second order pseudos
Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that ...
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184 views
Problem with double integral using Fourier transform?
I have a doubt concearning the convergence of an integral.
Let $\mathscr{S}(\mathbb R^n)$ be the Schwartz space on $\mathbb R^n$. Given $u\in \mathscr{S}(\mathbb R^n)$ we have an well defined ...
0
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1answer
49 views
How to show this inequality involving derivatives of a smooth function?
I must use the following:
Lemma: Let $f\in C^2([-1, 1], \mathbb R)$ and $\displaystyle C_j:=\max_{t\in [0, 1]}|f^{(j)}|$ with $j=0, 2$. Then $$|f^\prime(0)|\leq 4C_0(C_0+C_2).$$
to prove:
...
0
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1answer
190 views
Composition of Pseudodifferential Operators - Remainder term of Asymptotic Expansion
first off this is my first time posting here so I am only learning how to format questions. Please bear with me.
I am trying to prove the asymptotic expansion for the symbol of the composition of ...
2
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2answers
174 views
$\displaystyle g(x):=\int_{\mathbb R^n} e^{2\pi ix\cdot \xi} e^{-\frac{|\xi|^2}{2}} \widehat{f}(\xi)\ d\xi$ is not compact supported?
Which function $f\in C^\infty_0(\mathbb R^n)$ should I choose to show that $$\displaystyle g(x):=\int_{\mathbb R^n} e^{2\pi ix\cdot \xi} e^{-\frac{|\xi|^2}{2}} \widehat{f}(\xi)\ d\xi$$ is not ...
1
vote
0answers
78 views
Compactness of Pseudo-differential Operators on $H^s(\mathbb R^n)$?
The Sobolev space of order $s\in\mathbb R$ in $\mathbb R^n$, denoted by $H^s(\mathbb R^n)$, is defined as follows:
$$H^{s}(\mathbb R^n):=\{u\in\mathscr{S}^{'}(\mathbb R^n): \exists f\in L^1_{\textrm{...
1
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0answers
63 views
Quantitative estimate of smoothing versions of pseudodifferential operators
Let $\phi, \theta$ be two cutoff functions on a torus $\mathbb{T}$ in $\mathbb{R}^d$ such that their supports are disjoint. Let $P$ be a pseudodifferential operator on $\mathbb{T}$ and its symbol is ...
