In mathematics, a differential operator is an operator defined as a function of the differentiation operator. (Def: http://en.m.wikipedia.org/wiki/Differential_operator)

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right definition of correct space of domain and range for a self-adjoint Operator

at first, I'm quite sure it's not necessary to pay too much attention to the way the Operator is defined, it's rather important which spaces to choose to obtain a self-adjoint operator I've got a ...
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Question about differential operators

Say $N = ab$. How can I express $\frac{d}{dN}$ in terms of $\frac{d}{da}$ and $\frac{d}{db}$?
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del operator question regarding commutativity with a scalar

Is the following true if $f$ is a scalar? $f\,(\nabla\circ\textbf{B})=f\frac{\partial B}{\partial x}+f\frac{\partial B}{\partial y}+f\frac{\partial B}{\partial z}=\frac{\partial B}{\partial ...
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Squaring an operator

There is an excercise of squaring an operator in my book of quantun mechanics. The operator is $$\hat{A}=\frac{\mathrm{d}}{\mathrm{d}x}+x$$ And I should compute $\hat{A}^2$. He gives me a result ...
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Lie algebra (su(1,1)) from legendre polynomials; question regarding http://arxiv.org/abs/1205.6353

Apologies if this question is a duplicate. OK, so my question heavily involves the paper http://arxiv.org/abs/1205.6353 which nicely details the Lie algebra su(1,1) coming from the Laguerre $L_n$, ...
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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 = ...
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Continuity domain for momentum operator

The momentum operator in one dimension in quantum mechanics is $P=-i\frac{d}{dx}$ (with $\hbar=1$). Consider it as an operator on $L_2(0,2\pi)$, the space of square-integrable functions on $(0,2\pi)$. ...
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Questions concerning the differential operator

Consider the differential equation:- $a \phi + (bD^3 - cD)w =0$, where $a, b$ and $c$ are constants, $D$ denotes the differential operator $\dfrac{d}{dx}$, and $w$ is a function of $x$. I'm ...
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106 views

Selfadjointness of Coulomb Hamiltonian in d>=3 dimensions

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $dom(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic form a form ...
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Differential operator on a manifold in Geometric Calculus

In the context of Geometric Calculus, as stated in book Clifford Algebra to Geometric Calculus (pag. 142), let $M$ be a differentiable vector manifold, $F$ be a field on $M$ and $a$ be a tangent ...
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Function of a differential operator.

Friends of mine who study Quantum Field Theory asked me about the following problem. The task is to simplify the expression $$ f_1(\frac{d}{dx})f_2(x) $$ so that it doesn't contain derivatives, but ...
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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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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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Convert an eigenvalue equation to ODE/s

For example define: $K=-i\frac{d}{dx}$ (non-discrete spectrum), so: $$Kf(x)=-i\frac{df}{dx}=kf(x)$$ Define $g(x,k)=kf(x)$, so: $$\frac{-i}{k}\frac{\partial{g}}{\partial{x}}=g(x,k)$$ ...
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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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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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Composition of Differential Operators

If I have: $A=\partial_x^2+u(x)$ $B=u(x)\partial_x$ How do I compose: $AB$ and $BA$?
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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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Differentation Operator

having trouble completing the proof for this question Let $D:\mathbb{R}[X] \to \mathbb{R}[X]$ be the differentiation operator $D(f(X))=f'(X) .$ Prove that $e^{tD}(f(X)) = f(X+t)$ for $t \in ...
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Exponential Operator Representing Solution to Autonomous First Order Differential Equations

I am studying Dominic Edelen's Applied Exterior Calculus Section 1-4 as a start on understanding derivatives in differential geometry. He uses an exponential function containing a derivative operator ...
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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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Distance between differential operators

Given two differential operators say $D_1$ and $D_2$ is there any meaningful way to define distance between them, does there exist some metric $d(D_1,D_2)$ that satisfies all the necessary properties? ...
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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 ...
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$b$ 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 ...
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Differential Operators over the space of Analytic Functions

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 ...
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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 ...
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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 ...
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Differential Maping in Elementary Analysis

The problem I stuck was : Let $ f = R^{n} \rightarrow R^{m} $ and suppose there is a constant $M$ such that for $ x \in R^{n} $, $ || f(x) || \leq M || x ||^{2} $. Prove that $ f $ is differentable ...
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Changing variables in a differential operator

Given the following differential operator, i am asked to rewrite it in polar coordinates $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + 2\frac{\partial^2 f}{\partial ...
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Change of variables in a differential operator.

I would like to know how could I change the coordinates to cilindrical coordinates of the following differential operator. $y\frac{\partial f}{\partial x} + xy^2z^5\frac{\partial f}{\partial y} + ...
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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 ...
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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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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? ...
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Poisson's equation under translation and scaling

Let $u\in C^\infty (\Omega,\mathbb R)$ be a solution of $$\begin{array}{cccl} -\Delta u & = & 0 & \mathrm{in}\ \Omega \\ u & = & g & \mathrm{on}\ \partial\Omega\end{array}$$ ...
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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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Handling matrix of differential operator when using the Ritz method for an extremum problem

The internal energy (or strain energy) of a typical structural member under a displacement field $\{u\}$ can be represented as: $$ U = \frac{1}{2}\int_{V}[u]^T[B]^T[F]^T[B][u]dV $$ where $V$ is the ...
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Prove that D (the differential operator) maps V (a vector space) into V.

I'm quite confused about what "into" means here and, more importantly, how I am supposed to prove that something maps a vector space into (not onto) another vector space. Here's some of the ...
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Composition of pseudo-differential operators

Denote by $S^m$ the set of functions $p: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ such that $p \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ and $$\left| ...
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Doubt about the spectrum of an operator

I consider the Laplacian operator $$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
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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 ...
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Homogenous Polynomial Functions and the Symbol of a Differential Operator

I have a trivial question concerning Lawson/Michelsohn's "Spin Geometry", Chapter III.§1. There, the symbol of a differential operator $P$ is defined to be a section $\sigma(P)$ in the bundle ...
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Justification behind changing coordinates of a differential operator

On many websites focused on physics, (say http://skisickness.com/2009/11/20/ ) they like to represent differential operators in different coordinates. I.e. going from the standard basis to polar ...
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Partial derivative of an integral operator functional

Suppose $f(x) \equiv f_0(x) + \epsilon t(x)$, where $x,\epsilon \in{\mathbb{R}}$. And let $\mathcal{L}[f(x)] \equiv \int_a^b f(x')dx'$. I want to differentiate $\mathcal{L}$ w.r.t. $\epsilon$. So I ...
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What does the symbol $\Delta$ stands for?

While studying Landau-Lifshitz equation following term appears, $-m \times (m \times \Delta m) = \Delta m + |\nabla m|^2 m$ In above equation m is a vector quantity. It will be great if someone can ...
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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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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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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, ...
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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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Differential operators: elliptic vs strongly elliptic

This morning a collegue of mine came to me with the following question: does there exist any elliptic operator of order $2m$ with real (variable) coefficients that is not strongly elliptic? After ...