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Questions tagged [differential-operators]

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

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Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
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Notation of the differential operator

I see the differential operator both with upright and italic d in different books/articles. So I'm curious about $$ \int x^2 \, dx \quad \text{vs.} \quad \int x^2\, \mathrm{d}x,$$ and $$\frac{d}{dx}f(...
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0answers
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Mean value operator on Riemannian manifold

Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$) Consider the mean value operator, ...
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Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
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3answers
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How to look at the Lie derivative as a partial differential operator?

In Gromov's book Partial Differential Relations, a partial differential operator (PDO) is defined as follows. Given a fibration $X\to V$ and a vector bundle $G\to V$ over a manifold $V$, a PDO of ...
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2answers
291 views

Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
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4answers
490 views

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\...
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2answers
226 views

“Universal” differential identities

I now asked this at MO. Let $f:\mathbb{R}^d \to \mathbb{R}$ be smooth. The mixed derivatives commute: $f_{xy}=f_{yx}$. This identity is "universal" in the sense that it holds for any smooth map. ...
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What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?

Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the ...
9
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1answer
359 views

How to prove $(0,1)$ form is not $\overline\partial$-exact

On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is ...
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1answer
745 views

How to show that differential operator can be defined in terms of certain commutator operators

Let $U$ be any open subset of $\mathbb{R}^n$ (or, more general, of some smooth manifold). Define $\mathcal{D}_{-1}(U):=\{0\}$. For any two linear operators $A$ and $B$, the commutator operator $[A,B]$ ...
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Elliptic and Fredholm partial differential operators

As I learn from the comments to this question a non elliptic operator on a compact manifold can not be a fredholm operator. However, I learn also from the conversations in the same post that the ...
8
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1answer
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Linear transformations in infinite dimensional vector spaces

If we look at an $n$ - dimensional vector space $V$ and a linear transformation \begin{equation} T : V \to V, \quad x \mapsto Tx \quad \forall \, x \in V \end{equation} then given a choice of basis ...
8
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1answer
830 views

Differential operator and kernel

Let $P$ a polynomial of two variables, say over the field of real numbers. We define $\partial P$ as $P(\partial_x,\partial_y)$. In this question, it has been shown that if $P_0(x,y)=x^2+y^2$ and $\...
8
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1answer
271 views

When do Harmonic polynomials constitute the kernel of a differential operator?

Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., ...
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1answer
519 views

differential operator on manifold

I am currently trying to understand the local expression of a (pseudo)differential operator $$ \int_{R^n} e^{(x - y)\cdot \xi} \sigma(x,\xi) \, d \xi $$ on a manifold $M$ (compact and boundaryless, ...
8
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1answer
151 views

Abstract definition of a differential operator

In Natural Operations in Differential Geometry by Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another ...
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2answers
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What is the meaning of $1/(D+a)$, where $D$ is the derivative operator?

Today I read the answer to this post, in which the poster integrates $x^5e^x$ by making these manipulations with the differential operator $D$: $$\frac1Dx^5e^x=e^x\frac{1}{1+D}x^5=e^x(1-D+D^2+...)x^5$$...
7
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2answers
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How to properly apply the Lie Series

I am trying to solve this problem from Symmetry Methods for Differential Equations A Beginner's Guide (Peter E. Hydon). Use the Lie Series $$F(\hat{x},\hat{y})=\sum_{j=0}^{\infty}\frac{\varepsilon^j}{...
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Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
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2answers
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Linear operators on $C^\infty[a,b]$

I do not know too much about linear operators so forgive me if this doesn't make much sense, but what would the space of linear operators on $C^\infty[a,b]$ be defined as? If we denote this space as ...
6
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1answer
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Is there a simple classification of vector fields $\mathbf{F}$ on $\mathbb{R}^3$ that satisfy $\mathbf{F}\cdot \nabla\times\mathbf{F}=0$?

In multivariable calculus, a common mnemonic to remember $\text{div}(\text{curl}(\mathbf{F}))=0$ for vector fields $\mathbf{F}$ on $\mathbb{R}^3$ is treating the gradient operator $\nabla=\langle\frac{...
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2answers
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How to compute one-parameter group and corresponding vector fields

I have two related questions to ask - $1)$ Let $\rho : \mathbb{R} \rightarrow G$ be a one-parameter group. ($\mathbb{R}$ and $G$ are Lie groups). If we take $G = S^1$ then the left invariant vector ...
6
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1answer
274 views

What is the mathematical truth behind the Leibniz notation in differentiating twice or more?

So $f: \mathbb{R} \to \mathbb{R}$ is $n>1$ (or more) times differentiable. The notation of the first derivative makes perfect "sense" with regard to what's going on: $$\lim_{h \to 0} \frac{f(x+...
6
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2answers
228 views

Is $x * d/dx$ closed on $L^2(\mathbb{R})$?

More formally, sticking to the one-dimensional case for this problem, let $M_x$ be the multiplication-by-$x$ operator, $$\begin{split} \mathfrak{D}(M_x) &= \lbrace f(x) \in L^2(\mathbb{R}): x f(x)...
6
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1answer
91 views

Find the extended form of the group generated by an operator?

I tried to find the extended form of the group generated by the following operators. (I): The first operator $$A=z\frac{\partial }{\partial z}+1$$ To find the extended form of the group generated ...
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0answers
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Is it possible to decompose this expression?

Is it possible to factorize $$(-\partial^2+\phi^2(r))^2-\left(\frac{\partial\phi(r)}{\partial r}\right)^2,$$ where $\phi(r)$ is a function of $r\equiv\sqrt{x^2+y^2+z^2+\xi^2}$ and $\partial^2$ is the ...
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0answers
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Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
6
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1answer
358 views

Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem

From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves ...
5
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2answers
183 views

Selfadjointness of Coulomb Hamiltonian in $d\geq3$ dimensions

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $\operatorname{dom}(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic ...
5
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1answer
119 views

Eigenvalues of the Divergence Operator

I am researching the spectrum of eigenvalues for the divergence operator on Riemannian manifolds and how they deform tensor fields. This is mainly motivated by trying to understand dynamical systems ...
5
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2answers
360 views

Is the kernel of the product of two commuting differential operators the sum of the kernels?

Consider the wave equation in one dimension: $$\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}=0.$$ The most general solution of this can be written as $F(x-t)+G(x+t)$ for ...
5
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1answer
637 views

Exponential of the Laplacian operator as diffusion equation

Let $u$ be a function on a domain $\Omega$ with some fixed boundary condition. I have recently seen a notation $e^{\tau \Delta}u$ as meaning the the time evolution of $u$ by diffusion for a time $\...
5
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1answer
290 views

Integral of the Laplace-Beltrami Operator multiplied by a function

I have the following problem: Let $\mathcal{M}$ be a $2D$-manifold in $\mathbb{R}^3$ and let $g$ denote its metric. Furthermore it is known that $\mathcal{M}$ is a closed manifold (i.e. it has no ...
5
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1answer
101 views

Pseudo-differential operators

What is the meaning of the formula $\sigma (PQ)=\sum \frac{1}{\alpha!}\partial _{\xi }^{\alpha}pD_{x}^{\alpha}q\; ;\;\; \sigma (Q)=q,\;\;\; \sigma (P)=p$ if the series on right side is infinite? ...
5
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2answers
135 views

About the closedness of $\frac d{dx}$ operator

The example section of the closed linear operator in the wikipedia page https://en.wikipedia.org/wiki/Unbounded_operator#Closed_linear_operators says that (1) "Consider the derivative operator $A =\...
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Why is n-th Fréchet derivative symmetric?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$. Let $E$ be open in $V$ and $f:E\rightarrow W$ be a twice Fréchet-differentiable function. Then, $D^2 f: E\rightarrow \mathscr{L}_2(V^2,W)$ is ...
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Is it possible to construct a 1-D linear differential operator with given spectrum $0\leq\lambda_0\leq \lambda_1\leq\dots\leq\lambda_n\le\dots$?

Suppose one is given with a sequence $S$ of non-negative real numbers $0\leq\lambda_0\leq \lambda_1\leq\dots\leq \lambda_n\leq\dots$. Under what conditions on $S$, is it possible to construct a Linear ...
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0answers
241 views

Applications of Microfunctions

Can anyone suggest good (a) uses/applications or (b) construction of micro-functions (introduced by Mikio Sato in 1971) in analysis? I am trying to understand the subject better. Suggestions of ...
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0answers
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Proving continuity on spaces of distributions?

Let $\mathcal{D}'(\Omega)$ be the space of distributions on an open set $\Omega$, and $\mathcal{E}'(\Omega)$ the compactly supported ones. When you have a linear operator $T:\mathcal{D}'(\Omega)\...
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2answers
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Find $a_{n,i,j}$ in the expansion $(x + D)^n = \sum\limits_{i,j} a_{n,i,j} x^i D^j.$

This is problem 47c. in Stanley's Enumerative Combinatorics Vol. 1. Background: Let $D$ be the operator $\frac{d}{dx}$. Part (a) asks to prove $$ (xD)^n = \sum\limits_{k = 0}^n S(n,k)x^k D^k $$ ...
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2answers
977 views

How to prove convergent function imply its derivative equals to zero?

Let $f\colon (0,\infty) \to\Bbb R$ be differentiable and let $A$ and $B$ be real numbers. Prove that if $f(t) \to A$ and $f′(t) \to B$ as $t \to \infty$ then $B = 0$.
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3answers
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Logarithm of differential operator

Using the Taylor series expansion we have (for a sufficiently regular function $f$): $$ f(x+a)=\sum_{k=0}^n \frac{f^{(k)}(x)a^k}{k!} $$ So, defining the differential operator $D=\frac{d}{dx}$ and ...
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2answers
2k views

Transpose of the differential operator

Is the differential operator ${d\over dx}$ antisymmetric? If so, what does it even mean to take it's transpose? Thank you.
4
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1answer
950 views

Simple proof of Chain Rule through $\frac{\Delta y}{\Delta x} = \frac{dy}{dx}\biggr|_{x=x_1} + k$

In an online lecture (link to Youtube), the professor proves the Chain Rule using the following statement: $$\frac{\Delta y}{\Delta x} = \frac{dy}{dx}\biggr|_{x=x_1} + k$$ $$\Delta y = \frac{dy}{dx}\...
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1answer
912 views

How do you calculate an exterior derivative on forms in $\mathbb{R}^3$?

If we have a form, say, $\omega = f(x,y,z) \, dx + g(x,y,z) \, dy + h(x,y,z) \, dz$, what is the formula for the exterior derivative $d \omega$?
4
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1answer
45 views

General formula for Differentiation Operator

I was considering the Operator $$ x\,\frac{\rm d}{{\rm d}x} $$ and applying it $n$ times to an arbitrary function $f(x)$. Is there a general formula for it? I started with the first few \begin{align} ...
4
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1answer
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$\frac{dx}{dt} = p, \frac{dy}{dt} = q$: Solution of these ODE imply the solution is constant along characteristics of the form $qx − py = constant$.

My lecture notes state the following: When we were dealing with first order equations we saw that a differential operator of the form, $$p\frac{\partial}{\partial{x}} + q\frac{\partial}{\...
4
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1answer
276 views

Space of Differential Operators

Is is possible to talk about a "space of differential operators"? If one defined such a space would it be possible to talk about limits? I don't really have much background so I'm really just looking ...
4
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1answer
6k views

Divergence of stress tensor in momentum transfer equation

Let suppose that we work in a 2D cartesian coordinates. what will be x and y components of $\nabla.\left[-p I+\mu \left(\nabla \text{u}+(\nabla \text{u})^T\right)-\frac{2}{3} \mu (\nabla.\text{u}) I \...