# Questions tagged [differential-operators]

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

172 questions
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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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### 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 ...
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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 ...
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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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### 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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### 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 ...
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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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### 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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### Neumann series as an integral operator

Let $D$ be the derivative operator, and set $$T:=\sum_{n=0}^\infty D^n.$$ It is known that under certain conditions $$T=(Id-D)^{-1} .$$ Set $v=Tu$, applying $I-D$ to both sides, in this case, gives ...
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### Treating the derivative as a linear operator for a system modeled by a differential equation?

I have a system with input $x(t)$ and output $y(t)$. It is modeled by this differential equation: $\dfrac{d^2y(t)}{dt^2} + y(t) = \dfrac{dx(t)}{dt} + x^2(t)$ I need to see if this system is linear ...
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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 $f\in \... 0answers 69 views ### Approximating an element in the domain of an unbounded operator by a sequence in a dense subset of the domain. Let$T$be a closed unbounded (in my case also symmetric) operator on a Hilbert space$\mathcal{H}$with dense domain$\mathcal{D}(T)$, and let$f\in \mathcal{D}(T)$. Suppose there is a dense ... 0answers 144 views ### self-adjoint differential operator on$C^{0}([a,b])$? I've got problems in understanding the way a special (self-adjoint) differential operator is acting on the domain and the range. So, I try to explain my difficulties: The differential operator is ... 0answers 155 views ### Show that the Hilbert transform is a pseudo-differential operator of order 0 I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform$Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$is a pseudo-... 0answers 217 views ### Change of variables for linear differential operators I am trying to find an expression for a change of variables (invertible and$C^{\infty}$) of a linear differential operator in$\mathbb R ^d \newcommand{\t}{\tilde} \newcommand{\p}{\partial}$. i) ... 0answers 44 views ### Closed form for a binomial containing a differential operator Is there a closed form for$(x + D)^n$where D is the differential operator with respect to x? Avi's comment in this post helped me understand this expression a bit better, but I'm still curious if ... 0answers 190 views ### Eigenvalues of differential operator If$L : C^2[a,b] \rightarrow C^0[a,b] $is s.t.$L y(t) = \ddot y(t) +p \dot y + q y(t) $and$L$is invertible then$L^{-1}$has at most countable eigenvalues and they accumulate in$0$. Why ... 0answers 210 views ###$b$uniformly elliptic and bounded$x\mapsto (b(x))^{-1}$uniformly elliptic and bounded? I am not able to prove or find a counter example for the following statement. Let$b:\mathbb{R}^n\rightarrow GL(n,\mathbb{R})$be such that for$C>0\frac{1}{C}\vert \xi\vert^2\leq \xi^Tb(x)\...
Let $\mathcal{A}(-a,a)$ be the vector space of functions that are analytic on the interval $(-a,a)$ Is there a common topology to place on this space, if yes what is the topology and is it induced ...