Articles of differential operators

Computing $e^{isD}$ for a differential operator D

I’m trying to understand functional calculus of unbounded operators and everywhere I see proofs of its existence, but it seems that no one ever dares to compute some easy example. Lets take $D = i\tfrac{d}{dx}$ for example. I know that $e^{isD}$ is the operator on $L^2(\mathbb{R})$ given by translation by s but I’m completely lost […]

Differential Operator Issue

Let us consider the differential operator $H:= x \frac{d}{dx}$ and let us define \begin{equation} \hat{\mathcal{O}_n}= \frac{1}{n!}(H+n)(H+n-1)\cdots(H+2)(H+1). \end{equation} I proved – by induction – that \begin{equation} \hat{\mathcal{O}_n}\left ( \frac{\log x}{x} \right )=\frac{1}{nx} \end{equation} I notice that something similar happens with the following: \begin{gather*} \frac{\log(x)}{x^2}; \\ \frac{\log(x)}{x^3}. \end{gather*} In particular: \begin{equation} \hat{\mathcal{O}_n}\left ( \frac{\log x}{x^2} \right )=-\frac{1}{n(n-1)x^2} […]

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 allowed to have values in $\mathbb{R}^m$, via working with the components) Let $r=l+\alpha$ with $l \in \mathbb{N}$ and $0\le \alpha <1$. Case 1) If $\alpha >0$, $C^r_{*}(\Omega)$ is […]

Symbol of a (non linear) differential operator

I am interested in knowing whether there is a definition for the symbol of a PDO which is NOT linear. In Wikipedia and in the book I am reading (An Introduction to Partial Differential Equations by Renardy-Rogers) I only found the definition for linear PDOs. Here is the Wikipedia link:

What is the meaning of $1/(D+a)$, where $D$ is the derivative operator?

Today I read the answer to this post, in which the poster integrates $x^5e^x$ by making these manipulations with the differential operator $D$: $$\frac1Dx^5e^x=e^x\frac{1}{1+D}x^5=e^x(1-D+D^2+…)x^5$$ which I was amazed by but yet suspicious of. After reading a bit on differential operators, I know a few properties. For example, $D+a$ (where $a$ is constant) is a polynomial […]

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 defining $w = Lw’$ and $x=Lx’$, where $L$ is a constant. I’m trying to obtain $\phi$ in terms of $x’$. But I’ve two […]

When do Harmonic polynomials constitute the kernel of a differential operator?

Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., polynomials $h$ such that $\partial_{x^2+y^2}h=0$. Is it true that if the kernel of $\partial_f$ contains $H$ then $x^2+y^2|f(x,y)$? I would appreciate an elementary proof/disproof.

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 and Michelsohn’s Spin Geometry page 113 it says: “Racall that the principal symbol of a differential […]

How to derive these Lie Series formulas

Relates issues: How to properly apply the Lie Series Exponential of a function times derivative In my old notes about Lie groups and/or operator calculus, I’ve encountered the following formulas: $$ e^{\lambda\,x^2\,\frac{d}{dx}}\,f(x) = f\left(\frac{x}{1-\lambda\,x}\right) \\ e^{\lambda\,\frac{1}{x}\,\frac{d}{dx}}\,f(x) = f\left(\sqrt{x^2+2\lambda}\right) \\ e^{\lambda\,x^3\,\frac{d}{dx}}\,f(x) = f\left(\frac{x}{\sqrt{1-2\lambda\,x^2}}\right) $$ I know how to derive the first one, but have no idea […]

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 far that explains where they could be applied, in a mathematical physics […]