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)

learn more… | top users | synonyms

8
votes
0answers
122 views

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, ...
5
votes
0answers
71 views

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 ...
5
votes
0answers
85 views

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 ...
5
votes
0answers
90 views

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 ...
5
votes
0answers
86 views

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 ...
5
votes
0answers
75 views

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 ...
4
votes
0answers
120 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 $g :\Bbb{R}^2_+ \to \Bbb{R}$ ...
4
votes
0answers
38 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 ...
4
votes
0answers
101 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 ...
4
votes
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 ...
4
votes
0answers
117 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 ...
3
votes
0answers
36 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 ...
3
votes
0answers
155 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 $L^2(-\pi,\pi)$. ...
3
votes
0answers
46 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 $f(x)$ ...
3
votes
0answers
72 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 ...
2
votes
0answers
20 views

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 ...
2
votes
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 ...
2
votes
0answers
32 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 ...
2
votes
0answers
55 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 ...
2
votes
0answers
101 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 ...
2
votes
0answers
101 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 ...
2
votes
0answers
92 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 ...
2
votes
0answers
147 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) ...
2
votes
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 ...
2
votes
0answers
179 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 ...
2
votes
0answers
130 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 ...
2
votes
0answers
57 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 ...
2
votes
0answers
55 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 ...
2
votes
0answers
57 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 $A$ ...
2
votes
0answers
149 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} ...
2
votes
0answers
56 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 ...
1
vote
0answers
28 views

Kronecker delta representation of a matrix (Quantum raising / lowering operators)

The Kronecker Delta is commonly used to represent a diagonal matrix: $$ a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right) $$ ...
1
vote
0answers
16 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 ...
1
vote
0answers
44 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 ...
1
vote
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 ...
1
vote
0answers
15 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 ...
1
vote
0answers
54 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 ...
1
vote
0answers
16 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 \! : ...
1
vote
0answers
52 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 ...
1
vote
0answers
66 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 ...
1
vote
0answers
42 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 ...
1
vote
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 ...
1
vote
0answers
48 views

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 $(0,2\pi)$. ...
1
vote
0answers
51 views

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: $$\frac{-i}{k}\frac{\partial{g}}{\partial{x}}=g(x,k)$$ ...
1
vote
0answers
64 views

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 ...
1
vote
0answers
26 views

How to prove this equivalence?

Consider the general elliptic operator $$M=\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j},$$ where $a_{ij}$ are continous functions. The function $u$ satisfies $$|Mu|\leq A(|\nabla ...
1
vote
0answers
70 views

Non-hypoelliptic operator

Consider the following operator $$\partial_t^2-\partial_x^2+\lambda(x,t)$$ where $\lambda$ is a $C^{\infty}$ function on $\mathbb{R}^2$ and the operators above are second partial derivatives. How can ...
1
vote
0answers
101 views

Cayley Transform (PDE)

I really need your help in solving the following problem: Prove that a unitary operator $U$ acting on a Hilbert space $H$ is the Cayley transform of some self-adjoint operator if and only if $1$ is ...
0
votes
0answers
7 views

Elliptic operators for a Laplacian transform

I need to show that the operator: $L[u]=(1-x^2)\frac{\partial^2 u}{\partial x^2}+2xy\frac{\partial^2 u}{\partial x \partial y}+(1-y^2)\frac{\partial^2 u}{\partial y^2}$ Find the transformation of ...
0
votes
0answers
18 views

Reference for Fractional calculus and Differential Operators

I`ve been struggling with Fractional calculus and differential operators while studying special functions, and got to the conclusion that I need some references for them. So I ask for as many ...