Questions tagged [differential-operators]

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

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

It is helpful to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science).

555 questions
28
votes
2 answers

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, but I cant find anything in the literature thus…
18
votes
2 answers

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 of the principal symbol. Specifically, In Lawson…
16
votes
0 answers

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, given by…
16
votes
7 answers

Notation of the differential operator

I see the differential operator both with upright and italic d in different books/articles. So I'm curious about $$ \int x^2 \, dx \quad \text{vs.} \quad \int x^2\, \mathrm{d}x,$$ and $$\frac{d}{dx}f(x) \quad \text{vs.} \quad…
molarmass
  • 1,874
  • 10
  • 16
13
votes
0 answers

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 boundary and outside of some compact region). 1 -…
12
votes
3 answers

How to look at the Lie derivative as a partial differential operator?

In Gromov's book Partial Differential Relations, a partial differential operator (PDO) is defined as follows. Given a fibration $X\to V$ and a vector bundle $G\to V$ over a manifold $V$, a PDO of order $k$ is a map $D:\Gamma(X)\to \Gamma(G)$,…
11
votes
1 answer

Quadratic P.S.D. differential operator that is invariant under $\textrm{SL}(2, \mathbb{R})$

Given some function $f \in L^2(\mathbb{R}^2)$, I'm interested in finding a positive semi-definite differential operator $\mathcal P: L^2(\mathbb{R}^2) \rightarrow L^2(\mathbb{R}^2)$ that is quadratic in $f$ and invariant under the the action of…
11
votes
4 answers

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dz^k}f(z)$$ one can formally express the…
10
votes
2 answers

"Universal" differential identities

I now asked this at MO. Let $f:\mathbb{R}^d \to \mathbb{R}$ be smooth. The mixed derivatives commute: $f_{xy}=f_{yx}$. This identity is "universal" in the sense that it holds for any smooth map. Question: Are there any universal identities which…
9
votes
1 answer

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 $[A,B]$ is given by $A\circ B-B\circ A$. For any $f\in…
Aris
  • 405
  • 2
  • 9
9
votes
4 answers

"Natural" proof of $P\left(\frac{d}{dx}\right)\bigl(e^{xy}Q(x)\bigr)=Q\left(\frac{d}{dy}\right)\bigl(e^{xy}P(y)\bigr)$.

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}\right)\bigl(e^{xy}Q(x)\bigr) …
9
votes
2 answers

Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a theory surronding the $\sqrt[n]{D_y}$-idea?
9
votes
1 answer

How to prove $(0,1)$ form is not $\overline\partial$-exact

On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is closed, but gives a nonzero integral for your…
9
votes
0 answers

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 hypoelliptic operators can be Fredholm but…
9
votes
1 answer

Linear transformations in infinite dimensional vector spaces

If we look at an $n$ - dimensional vector space $V$ and a linear transformation \begin{equation} T : V \to V, \quad x \mapsto Tx \quad \forall \, x \in V \end{equation} then given a choice of basis for $V$ one can represent $T$ in terms of a $n…
harlekin
  • 8,060
  • 5
  • 34
  • 70
1
2 3
36 37