# Tagged Questions

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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### 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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### 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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### 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 $r\equiv\sqrt{x^2+y^2+z^2+\xi^2}$? Note that we can not ...
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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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### 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 ...
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### Abstract definition of a differential operator

In 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 fibred manifold $Y' \to M'$. Is this is a ...
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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. On the other hand it seems that the following operator is fredholm(At least,...
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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 ...
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 $T:\mathcal{D}'(\Omega)\... 0answers 123 views ### Given$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}$... 0answers 50 views ### Why is n-th Fréchet derivative symmetric? Let$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)$is ... 0answers 105 views ### Metric tensors and linear (differential) operators defined on a manifold and its osculating sphere Given a 1D Riemannian manifold$\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 ... 0answers 45 views ### 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 ... 0answers 129 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 ... 0answers 18 views ### 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? $$D = <x|\int D|x> ... 0answers 46 views ### Sobolev spaces and symmetric operators I am slightly confused with regards to the way one obtains self - adjoint differential operators in spectral theory. The aspect that I'd like to understand better is the following: Suppose we are ... 0answers 163 views ### Proving that a certain differential operator is self-adjoint Consider the differential operator T:u\mapsto -iu' for any u\in D(T):=\{f\in AC[-\pi,\pi]~|~f'\in L^2(-\pi,\pi),f(-\pi)=f(\pi)\}; we consider T as a densely-defined operator on L^2(-\pi,\pi). ... 0answers 50 views ### When does a differential operator restrict to a subvariety? I recently encountered the following nice fact, and I'm wondering if it's part of a more general story. Let x\in \mathbb{C}^n satisfy$$x^2:=\sum_i x_i^2 = 0,$$and consider functions f(x) ... 0answers 74 views ### elliptic operator and wave front set Let us f(x) \in C^\infty on \mathbb{R}^n, and the pseudo-diff. operator Q is defined by: (Qu)(x)=(2\pi)^{-n}\int_{\mathbb{R}^{n}}e^{ix\xi }f(x)\left | \xi \right |\hat{u}(\xi) d\xi Where ... 0answers 14 views ### k-symbol differential opeartor L, and its independent of choices. This material is in O. Well's Differential analysis on complex manifold, page 115. Let, (x,v) \in T'(X) (T^*(X) with deleted zero section) and e \in E_x, Find g\in \epsilon(X) and f\in \... 0answers 34 views ### Different solutions to the Hermite equation The Hermite differential equation is given as such$$ y'' - 2xy'+2\lambda y=0 $$writing this in strum-liouville form you get$$-(\exp(-x^2)y')'= 2\exp(-x^2)\lambda y $$However, in order for it to ... 0answers 34 views ### Operators in polar coordinates in n-dimensions I want help on converting differential operators such as the reduced wave operator (L=Δ+c) and the biharmonic operator (L=Δ^2) from Cartesian to spherical coordinates in n-dimensions. For example I ... 0answers 66 views ### Resolvent operator Let's consider the following operator on L^2(\mathbb{R}^3)$$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 ... 0answers 123 views ### 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 ... 0answers 105 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 95 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 163 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 41 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 181 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 137 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)\... 0answers 58 views ### 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 ... 0answers 56 views ### 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 x\... 0answers 61 views ### Doubt about the spectrum of an operator I consider the Laplacian operator$$A=-\Delta$$in the domain$$H^2(\mathbb{R}^3)$$where it is selfadjoint. We know that its spectrum is [0,+\infty). Now I want to consider the restriction of A ... 0answers 158 views ### Inverse of a certain differential operator (resolvent) I am doing a research on a certain type of operator, and in the course of it I need to determine the following: Given the operator D below, and identity operator I,$$ D=\begin{pmatrix} ... 0answers 58 views ### Choosing boundary conditions for$(\frac{-d^2}{dx^2})^m$on$H^m((0,1))$? Consider the differential operator$D:$$$Du:=\frac{-d^2}{dx^2}u$$ on the function space $$C=\{u\in C^2([0,1]):u(0)=u(1)=0\}.$$ It's not hard to find the eigenvalues and eigenvectors(... 0answers 13 views ### Divergence of Material Derivative Let$u : \Bbb{R}^n\times \Bbb{R} \to \Bbb{R}^n\times \Bbb{R} $be a divergence free vector field. Then the material derivative$D $is given by: $$\frac { \partial u_j}{\partial t}+\sum_{i=1}^{n} ... 0answers 27 views ### inverse of operator I want to calculate the inverse of the operator T=-\frac{\partial^{2} }{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-(x^{2}+y^{2})+2( x\frac{\partial }{\partial y}-y\frac{\partial }{\... 0answers 16 views ### Counting zeroes of global sections Let X be a compact connected Riemann surface and let \Phi:M\rightarrow N be an elliptic differential operator where M and N are two complex line bundles on X. Let f be a C^\infty-global ... 0answers 24 views ### concomitant and self-adjoint operator If Lu = u^{\prime\prime}+\omega^2u, show that L is formally self-adjoint and the concomitant is J(u,v)=vu^\prime-uv^\prime. Moreover, if u is a solution of Lu=0 and v is a solution of L^*... 0answers 32 views ### Kronecker delta representation of a matrix (Quantum raising / lowering operators) The Kronecker Delta is commonly used to represent a diagonal matrix:$$ a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right) $$... 0answers 46 views ### Approximating an element in the domain of an unbounded operator by a sequence in a dense subset of the domain. Let T be a closed unbounded (in my case also symmetric) operator on a Hilbert space \mathcal{H} with dense domain \mathcal{D}(T), and let f\in \mathcal{D}(T). Suppose there is a dense ... 0answers 12 views ### Fundamental solution of the frozen opearator Let L be some differential operator of the form$$ Ly = y^{(n)}(x) + a_{n-1}(x) y^{(n-1)}(x) + \dots + a_0(x) y(x) $$with all a_k(x) being smooth. Let also M be the frozen at x=0 operator L... 0answers 16 views ### Decomposition of a differential operator Let \mathcal{O} be the ring of holomorphic functions on the unit disk deprived of the non-negative real numbers. Let \mathcal{D} be the ring of differential operators on the same space, \alpha a ... 0answers 61 views ### Find eigenfunctions of the integral operator with kernel \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny) Find the eigenvalue and eigenfunctions of the integral operator Ku=\int_0^\pi k(x,y)u(y)dy. k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny). This is how I ... 0answers 20 views ### Linear operator differentiation on a torus I'm trying to analyze this article about area-preserving diffeomorphisms and don't quite understand a sentence. 4.1. Linear involutions. We start characterizing the linear involutions R \! : \... 0answers 56 views ### Linear algebra references explaining matrix form of linear differential and integral operators Some years ago I was in a lecture where I met for the first time the matrix representation of some differential and integral operators (once discretized). Back then, someone mentioned me that every ... 0answers 69 views ### Selfadjointness of the differential operator in a singular potential The free Dirac operator is the differential operator of the following form$$ T_0 = i \alpha \nabla + \beta,$$where$\alpha$and$\beta$are Hermitian$4 \times 4$matrices, and$T_0$is selfadjoint ... 0answers 44 views ### Meaning of (generalized?) differential operator I am currently reading this paper which makes use of generalized differential operators. As I understood it, the operator$D_x$works like this: If$F$is a continuous function on$[a,b]$and$G$an (... 0answers 37 views ### Is there any significance to this matrix/operator? I am working on a problem involving the the polarized Hessian covariant in Cartesian coordinates on$\mathbb{R}^2[a,b] = \frac{1}{2} \frac{\partial ^2 a}{\partial x ^2} \frac{\partial ^2 b}{\...
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 $(0,2\pi)$. ...