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Questions tagged [pseudo-differential-operators]

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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 ...
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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 ...
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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 ...
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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: ...
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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-...
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1answer
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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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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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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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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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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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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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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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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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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 $R$ ...
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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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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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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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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 ...
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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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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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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 $...
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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 $$ ...
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1answer
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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 ...
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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-...
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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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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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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^\...
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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 $M$ ...
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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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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 ...
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1answer
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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{...
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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-...
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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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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 ...
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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 ...
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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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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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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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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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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 ...
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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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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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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 ...
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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: ...
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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 ...
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$\displaystyle g(x):=\int_{\mathbb R^n} e^{2\pi ix\cdot \xi} e^{-\frac{|\xi|^2}{2}} \widehat{f}(\xi)\ d\xi$ 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 ...
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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{...
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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 ...