# Questions tagged [differential-operators]

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

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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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### Beach-ball like differential operators of a two-dimensional function

I'm looking for references, known names of, and other useful pointers and insight about (pairs of) differential operators that are "beach-ball like" because they sample a 2-dimensional function in ...
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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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### 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 ...
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### 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 ...
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### 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-...
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### 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) ...
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### 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 ...
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### 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 ...