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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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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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245 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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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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Operators in polar coordinates in n-dimensions

I want help on converting differential operators such as the reduced wave operator (L=Δ+c) and the biharmonic operator (L=Δ^2) from Cartesian to spherical coordinates in n-dimensions. For example I ...
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Given $g$ find an $f$ which is solution for $L f = g$. How do I do this?

I am learning about Stochastic processes. To characterize uniqueness of solutions to a given Stochastic differential equation, I need to find for each continuous function $g :\Bbb{R}^2_+ \to \Bbb{R}$ ...
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Resolvent operator

Let's consider the following operator on $L^2(\mathbb{R}^3)$ $$A(t)=\Delta+b(t,x)\cdot\nabla$$ where $\Delta$ is the Laplace operator and $b(\cdot,\cdot)$ a smooth vector field. How to compute the ...
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Prove two operators commute and a further question

Let $\psi,\varphi\in L^2(-1,1),$ and $\lambda,\chi$ be constants to be determined. First we got a integral equation $$\int_{-1}^1\frac{\sin c(x-y)}{\pi(x-y)}\psi(y)dy=\lambda\psi(x),\qquad|x|\le1.\...
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Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 0 \...
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Exercise about the Sturm-Liouville problems

Under what condition on the constants $c$ and $c'$ are the boundary conditions $$f(b)=cf(a)$$ and $$f'(b)=c'f'(a)$$ self-adjoint for the operator the operator $$\mathcal{L}f=\frac{d}{dx}\left(p_0(x)\...
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Formula for the application of a linear differential operator to the product of exponential and polynomial functions

In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients: $$ P\left(\frac{d}{dx}\...
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Differential operators on quaternions

If $\Bbb H$ denotes the quaternions one can write every element $a+bi+cj+dk\in \Bbb H$ as $1(a+bi)+j(c-di)$, so $\{1,j\}$ is a $\Bbb C$-basis of $\Bbb H$. Now let $w_1,\ldots, w_n$ be the standard ...
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The algebra generated by derivations

This may be a boring question. Suppose everything in the following is on a field $k$ (of characteristic $0$) without specification otherwise. Consider a commutative ring $A$ and the Lie algebra of ...
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$Af=f'$, $D(A)=\{f\in C^1[0,1]: \text{$f$ vanish near $0$}\}$. Show that $A$ is not closed.

As in title, $A$ is a differential operator with the domain $D(A)=\{f\in C^1[0,1]: \text{there is $\epsilon>0$ such that $f(x)=0$ for $x\in [0,\epsilon)$}\}$ This came up as an example of non-...
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Rotation of an exponential operator acting in Hilbert space? Interpretation in a certain subspace.

I have a self-adjoint operator $h$ acting in the Hilbert space $L^2(R^3)$ that has the following eigenvalues $$ h \varphi_s= \varepsilon_s \varphi_s\;,\quad s=1,2,... , $$ where $\varphi_s$ are ...
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Understanding Operators in context of Green's function derivation

I am trying to understand what operators actually mean when deriving the definition of green's function. What is the interagl representation of an operator? Is this correct? $$ D = <x|\int D|x> ...
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Different solutions to the Hermite equation

The Hermite differential equation is given as such $$ y'' - 2xy'+2\lambda y=0 $$ writing this in strum-liouville form you get $$-(\exp(-x^2)y')'= 2\exp(-x^2)\lambda y $$ However, in order for it to ...
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Sobolev spaces and symmetric operators

I am slightly confused with regards to the way one obtains self - adjoint differential operators in spectral theory. The aspect that I'd like to understand better is the following: Suppose we are ...
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Metric tensors and linear (differential) operators defined on a manifold and its osculating sphere

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
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Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
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Proving that a certain differential operator is self-adjoint

Consider the differential operator $T:u\mapsto -iu'$ for any $u\in D(T):=\{f\in AC[-\pi,\pi]~|~f'\in L^2(-\pi,\pi),f(-\pi)=f(\pi)\}$; we consider $T$ as a densely-defined operator on $L^2(-\pi,\pi)$. ...
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When does a differential operator restrict to a subvariety?

I recently encountered the following nice fact, and I'm wondering if it's part of a more general story. Let $x\in \mathbb{C}^n$ satisfy $$x^2:=\sum_i x_i^2 = 0,$$ and consider functions $f(x)$ ...
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elliptic operator and wave front set

Let us $f(x) \in C^\infty $ on $\mathbb{R}^n$, and the pseudo-diff. operator $ Q$ is defined by: $(Qu)(x)=(2\pi)^{-n}\int_{\mathbb{R}^{n}}e^{ix\xi }f(x)\left | \xi \right |\hat{u}(\xi) d\xi$ Where ...
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Inverse of a certain differential operator (resolvent)

I am doing a research on a certain type of operator, and in the course of it I need to determine the following: Given the operator $D$ below, and identity operator $I$, $$ D=\begin{pmatrix} ...
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Finding a potential function for $\vec F=\frac{-y}{x^2+y^2}\vec i+\frac{x}{x^2+y^2}\vec j$ on $\Omega=\mathbb{R}^2- \{(x,y)\,|\,y=0,\,x\geq0 \}$

In one of my Calculus III classes the professor presented the following vector field, defined on the set $S$ of all points $(x,y) \neq(0,0)$ : $$ \bbox[6px,border:1px solid black] { \vec{F}=\frac{-y}{...
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Smoothness of fundamental solution of a hypoelliptic operator.

Let $p$ be a polynomial such that $p(D)$ is a hypoelliptic operator, i.e. if $u$ is a distribution satisfying $p(D)u = 0$ then $u$ is smooth. Here, $D = -i\partial$. Let $E$ be a fundamental solution ...
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Elliptic boundary condition eigenvalue problem

In Chapter 1, section 1.5 of "The Dirac Spectrum" by Nicolas Ginoux, different elliptic boundary conditions for Dirac operators are introduced. On page 24, there is the following theorem Theorem 1....
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Symbol map for sheaves of differential operators

For a complex manifold $X$, its sheaf of differential operators $\mathcal{D}_X$ is a sheaf of filtered algebras, and there is an isomorphism of sheaves of graded algebras $$\text{gr } \mathcal{D}_X \...
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How to compute gradient of complicated scalar function ( limit and iteration)?

I have a function which gives scalar potential: $$P(c) = \lim_{n \to \infty} \frac{1}{2^n} \ln|f^{n}_c(0)|$$ where: $c$ is complex variable $f$ is the complex quadratic polynomial $$f_c(z) = z^2 ...
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Why de Rham complex is a complex of D-modules?

Usually, by a complex of $R$-modules, where $R$ is a ring (or sheaf of rings), one means a sequences of $R$-modules $C_n$ and $R$-homomorphisms $d_n:C_n\to C_{n-1}$ such that $d_{n-1}\circ d_{n}=0$. ...
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When an Hilbert–Schmidt operator is invertible? How to construct the kernel of the inverse?

This question is motivated by the following example. I have the following integral operator defined on $L^2(\mathbb{T}^n)\to L^2(\mathbb{T}^n) $ $$ T f(x) = \sum_{k \in \mathbb{Z}^n} \sigma(k,x) \ \...
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Differential operators and rules for Ore polynomials

When dealing with (nonlinear) dynamical systems, one often deals with state space representation, i.e. systems of the form $$\dot{x}=f(x),\quad x(t)\in\mathbb{R}^n.$$ Let $x^*$ be a solution of this ...
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Is a Sturm-Liouville operator the only 2nd order linear differential operator that is self-adjoint/Hermitian?

When learning about Sturm-Liouville operators and their properties, we also learned that any second order linear differential operator can be written in Sturm Liouville form after multiplying by an ...
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When do differential operators pull back?

Given smooth vector bundles $E \to M$ and $F \to M$ over a smooth manifold $M$, a linear differential operator from $E$ to $F$ is an $\mathbb{R}$-linear sheaf homomorphism $D: \mathcal{E} \to \mathcal{...
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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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Eigenfunction and Eigenvalues of an Integral Operator

Given the following integral operator: $(Au)(y)=\int \limits_{\Gamma}u^*(x,y) u(x) \,ds_x$ with $\Gamma=\{x\in\mathbb{R}^2:x=r\begin{pmatrix}\cos(2\pi t) \\ \sin(2\pi t) \end{pmatrix},0 \leq t < ...
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Operator that kill the wave function

I have the following function in $x$. $\sum_{d=0}^{\infty} \frac{1}{\hbar^d}\frac{1}{d!}\left(\prod_{i=1}^{d-1}(1+i\hbar)^{m}\right)x^d$ I need a differential operator involving $(x,\frac{d}{dx},h,...
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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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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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157 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-...
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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) ...
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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 ...