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18
votes
2answers
690 views

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, ...
9
votes
2answers
226 views

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 ...
8
votes
4answers
326 views

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 ...
8
votes
1answer
185 views

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 ...
8
votes
1answer
206 views

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., ...
7
votes
1answer
437 views

Differential operator and kernel

Let $P$ a polynomial of two variables, say over the field of real numbers. We define $\partial P$ as $P(\partial_x,\partial_y)$. In this question, it has been shown that if $P_0(x,y)=x^2+y^2$ and ...
5
votes
1answer
770 views

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 ...
5
votes
1answer
126 views

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 $[A,B]$ ...
5
votes
1answer
142 views

Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem

From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves ...
5
votes
0answers
57 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
2answers
104 views

Linear operators on $C^\infty[a,b]$

I do not know too much about linear operators so forgive me if this doesn't make much sense, but what would the space of linear operators on $C^\infty[a,b]$ be defined as? If we denote this space as ...
4
votes
1answer
171 views

differential operator on manifold

I am currently trying to understand the local expression of a (pseudo)differential operator $$ \int_{R^n} e^{(x - y)\cdot \xi} \sigma(x,\xi) \, d \xi $$ on a manifold $M$ (compact and boundaryless, ...
4
votes
1answer
120 views

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 ...
4
votes
1answer
60 views

Pseudo-differential operators

What is the meaning of the formula $\sigma (PQ)=\sum \frac{1}{\alpha!}\partial _{\xi }^{\alpha}pD_{x}^{\alpha}q\; ;\;\; \sigma (Q)=q,\;\;\; \sigma (P)=p$ if the series on right side is infinite? ...
4
votes
1answer
169 views

What does it mean for the leading symbol of a differential operator to be scalar?

I would like to better understand what it means for the leading symbol of a differential operator to be scalar. Concretely, I am currently looking at the Laplace - Beltrami operator on an ...
4
votes
0answers
48 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 ...
4
votes
0answers
65 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 ...
3
votes
2answers
125 views

How to prove convergent function imply its derivative equals to zero?

Let $f\colon (0,\infty) \to\Bbb R$ be differentiable and let $A$ and $B$ be real numbers. Prove that if $f(t) \to A$ and $f′(t) \to B$ as $t \to \infty$ then $B = 0$.
3
votes
2answers
75 views

Selfadjointness of Coulomb Hamiltonian in d>=3 dimensions

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $dom(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic form a form ...
3
votes
2answers
150 views

Composition of pseudo-differential operators

Denote by $S^m$ the set of functions $p: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ such that $p \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ and $$\left| ...
3
votes
1answer
132 views

Subset of differential operators is a finitely generated module?

I was reading about differential operators, and there is a small claim I don't understand. First, let $A$ be a commutative algebra over $k$, a field. We have the recursive definition for the algebra ...
3
votes
0answers
61 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 ...
2
votes
2answers
624 views

The minus Laplacian operator is positive definite

In a textbook of functional analysis I found this equation derived from Green's first identity $$\int _{ \Omega }^{ }{ u{ \nabla }^{ 2 }ud\tau } =\int _{ \partial \Omega }^{ }{ u\frac { ...
2
votes
1answer
54 views

Space of Differential Operators

Is is possible to talk about a "space of differential operators"? If one defined such a space would it be possible to talk about limits? I don't really have much background so I'm really just looking ...
2
votes
1answer
90 views

How do you calculate an exterior derivative on forms in $\mathbb{R}^3$?

If we have a form, say, $\omega = f(x,y,z) \, dx + g(x,y,z) \, dy + h(x,y,z) \, dz$, what is the formula for the exterior derivative $d \omega$?
2
votes
1answer
205 views

Error in proof of self-adjointness of 1D Laplacian

I have successfully checked self-adjointness of simple and classic differential operator - 1D Laplacian $$D = \frac {d^2}{dx^2}: L_2(0,\infty) \rightarrow L_2(0,\infty)$$ defined on $$\{f(x) | f'' ...
2
votes
1answer
3k views

Transforming the Laplace operator from Polar to Cartesian coordinates

I'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates: $$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + ...
2
votes
1answer
25 views

Question about differential operators

Say $N = ab$. How can I express $\frac{d}{dN}$ in terms of $\frac{d}{da}$ and $\frac{d}{db}$?
2
votes
1answer
97 views

Lie algebra (su(1,1)) from legendre polynomials; question regarding http://arxiv.org/abs/1205.6353

Apologies if this question is a duplicate. OK, so my question heavily involves the paper http://arxiv.org/abs/1205.6353 which nicely details the Lie algebra su(1,1) coming from the Laguerre $L_n$, ...
2
votes
1answer
118 views

Simple proof of Chain Rule through $\frac{\Delta y}{\Delta x} = \frac{dy}{dx}\biggr|_{x=x_1} + k$

In an online lecture (link to Youtube), the professor proves the Chain Rule using the following statement: $$\frac{\Delta y}{\Delta x} = \frac{dy}{dx}\biggr|_{x=x_1} + k$$ $$\Delta y = ...
2
votes
2answers
95 views

Function of a differential operator.

Friends of mine who study Quantum Field Theory asked me about the following problem. The task is to simplify the expression $$ f_1(\frac{d}{dx})f_2(x) $$ so that it doesn't contain derivatives, but ...
2
votes
1answer
40 views

Homogenous Polynomial Functions and the Symbol of a Differential Operator

I have a trivial question concerning Lawson/Michelsohn's "Spin Geometry", Chapter III.§1. There, the symbol of a differential operator $P$ is defined to be a section $\sigma(P)$ in the bundle ...
2
votes
1answer
219 views

Justification behind changing coordinates of a differential operator

On many websites focused on physics, (say http://skisickness.com/2009/11/20/ ) they like to represent differential operators in different coordinates. I.e. going from the standard basis to polar ...
2
votes
0answers
66 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
62 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 $L^2(-\pi,\pi)$. ...
2
votes
0answers
33 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 $f(x)$ ...
2
votes
0answers
34 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
46 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
1answer
71 views

Handling matrix of differential operator when using the Ritz method for an extremum problem

The internal energy (or strain energy) of a typical structural member under a displacement field $\{u\}$ can be represented as: $$ U = \frac{1}{2}\int_{V}[u]^T[B]^T[F]^T[B][u]dV $$ where $V$ is the ...
2
votes
0answers
46 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
53 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
1answer
68 views

Differential operator on a manifold in Geometric Calculus

In the context of Geometric Calculus, as stated in book Clifford Algebra to Geometric Calculus (pag. 142), let $M$ be a differentiable vector manifold, $F$ be a field on $M$ and $a$ be a tangent ...
1
vote
2answers
36 views

Write out the operator (A)^2 for A = (d/dx + x)

I am having difficulties understanding how this operator is multiplied out. I have the answer, but do not know why (see below) it is what it is. Imagine there is a carat (^) above the "A" for correct ...
1
vote
2answers
95 views

What does the symbol $\Delta$ stands for?

While studying Landau-Lifshitz equation following term appears, $-m \times (m \times \Delta m) = \Delta m + |\nabla m|^2 m$ In above equation m is a vector quantity. It will be great if someone can ...
1
vote
2answers
336 views

Transpose of the differential operator

Is the differential operator ${d\over dx}$ antisymmetric? If so, what does it even mean to take it's transpose? Thank you.
1
vote
1answer
518 views

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 ...
1
vote
2answers
82 views

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 = ...
1
vote
1answer
852 views

Laplacian Operator Represented as a Matrix - Problem Finding the Hermiticity

I'm trying to discretize the Laplacian operator, and represent it with a matrix, but I'm running into a problem: my result is not hermitian when it should be. Here are my calculations: In one ...
1
vote
2answers
132 views

What does ad$f$ mean, for $f$ a smooth function?

I am currently reading Nicole Berline "Heat Kernels and Dirac Operators". On page 64 Differential Operators are introduced that are generalized from operators acting on scalar functions to vector ...
1
vote
1answer
12 views

del operator question regarding commutativity with a scalar

Is the following true if $f$ is a scalar? $f\,(\nabla\circ\textbf{B})=f\frac{\partial B}{\partial x}+f\frac{\partial B}{\partial y}+f\frac{\partial B}{\partial z}=\frac{\partial B}{\partial ...