# Questions tagged [differential-operators]

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

400 questions
11 views

### Bounding the symbol of a differential operator

Consider a bundle map $P$, between the bundles $\Bbb R^m \times \Bbb R^n \rightarrow \Bbb R^m$ and $\Bbb R^m \times \Bbb R^k \rightarrow \Bbb R^m$. On each fiber above $\xi \in \Bbb R^m$, $P$ acts as ...
33 views

### Solution of $\left(D^2+1\right)y=\sin x$ by partial fractions and direct formula

To find the particular integral of the equation $\left(D^2+1\right)y=\sin x$,I converted this equation to its inverse $y=\frac{1}{D^2+1}\sin x$ . The solution to this can be obtained by taking the ...
12 views

53 views

### 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 ...
117 views

### Green's function for $2^\text{nd}$ order PDE

Has anyone an idea how to find the Green's function of the operator $$\widehat{{\mathcal{O}}}=-\frac{1}{R} \partial_z^2 - \partial_R \frac{1}{R} \partial_R?$$ UPDATE1: For my own record (and ...
62 views

### High order derivatives on manifold

Suppose I have a Riemannian manifold $M$ and a smooth function $f:M \to \mathbb{R}$. I denote the gradient of $f$ with $\nabla f$. What is the meaning of $$\nabla^N f$$ with $N \ge 3$ integer? Is it ...
39 views

54 views

90 views

### Exponential differential operator

Consider the operator $D= e^{ax*d/dx}$ operating on an infinitely differentiable function f(x). My approach: $Df(x)= f(x) + ax*df(x)/dx + (ax)^2*d^2f(x)/dx^2 + ...$ $=f(x+ax)$ But this does not ...
18 views

### Convergence to the differential operator

This is not an accurate question. I am not so sure about unbounded operators on a Hilbert space. Let $D$ be the differential operator on $L^2(0,1)$. Well, to somewhat, we can extend $D$ to a normal ...
23 views

### $\exp(a^2\partial_x^2)f(x) = ?$

I can prove that $\exp(a\partial_x)f(x) = f(x+a)$, but what happens for second derivatives? To be more precise, what is the right-hand side of $\exp(a^2\partial_x^2)f(x)$? The above operator has an ...
36 views

96 views

### 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 \...
18 views

### suggest a course on differential operator and Weyl

Can someone suggest me a course in pdf or textbook about differential operator and Weyl algebra. let $R$ a commutative algebra over a field $k$ let M, N two R-modules. we define the set of ...
97 views

### Computing a derivative through Lie series

Consider the $N$-dimensional autonomous system of ODEs $$\dot{x}= f(x),$$ where a locally unique solution $x(t)$, starting from the initial condition $x$, is denoted as $x(t)=\phi(t,x)$. Assume ...
16 views

### Differential operator commuting with Euclidean transformations

Why is it true that a differential operator $S$ on $\mathbb{R}^n$ commuting with translations and rotations must be of the form $$S = \sum a_j \Delta^j$$ where the $a_j$ are constant coefficients ?
82 views

### Closed form/ meaning of sum of geometric series of operator exponentials

I'm taking my first class on quantum mechanics right now and we've been using various operators throughout it; one operator we derived was the displacement operator, $$e^{a\frac{d}{dx}}f(x)=f(x+a)$$ I ...
42 views

64 views

40 views

### Generalized linear transport equation

I stumbled upon a transport equation of the form $$u_t(x,t)=u_x(x,t) + u_x(1,t).$$ Since I can write it in the form $u_t(x,t) = Lu(x,t)$ where L is some linear operator I thought that there must be ...
63 views