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5
votes
1answer
116 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
129 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 ...
0
votes
1answer
238 views

Is this second order differential operator Hermitian (2 variables)?

Let $ T= \frac{ \partial^{2}}{\partial _{x} \partial _{y}}+iay\frac{ \partial}{\partial _{y}}+i(1-a)x \frac{ \partial}{\partial _{x}}+ \frac{i}{2}$ be the second order differential operator, where $ ...
5
votes
0answers
56 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
57 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
0answers
50 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
0answers
57 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
43 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
30 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
33 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
41 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
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
52 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
46 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 $L_n$, ...
1
vote
0answers
40 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
27 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 ...
1
vote
0answers
116 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 ...
1
vote
0answers
58 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 ...
1
vote
0answers
47 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 ...
1
vote
0answers
80 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} ...
1
vote
0answers
24 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
52 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
69 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
16 views

Naive definition of the principal symbol of a PsDO

Let $A$ be a properly supported pseudo-differential operator acting on functions on $\mathbb R^n$. Then its complete symbol is the function $$ \sigma_A(x,\xi) = e^{-i\langle \xi, x\rangle} ...
0
votes
0answers
20 views

Differential operators in the language of modules

I am reading articles about differential operators. Authors try to treat differential systems , by studying the differential operator on a vector space in the language of modules. In this manner ...
0
votes
0answers
20 views

Hydrogenhamiltonian self-adjoint in one or two dimensions

let $d\in\{1,2\}$. I'd like to know if the operator $H=-\Delta - \frac{1}{|x|}$ is self-adjoint as an operator acting on a dense subset of $L^2(\mathbb R^d)$. In particular I'd like to know how its ...
0
votes
0answers
14 views

partial differential operators

As we know, there exists a semigroup for partial differential operators $A = \sum_{i,j=1}^N D_i(a_{ij}(\cdot)D_j)$, see (Klaus-Jochen Engel, Rainer Nagel, one-parameter semigroups). My question is ...
0
votes
0answers
21 views

Time-space operator order in Green's Function

When solving the heat equation, and therefore using Green's functions in general, does the operator ordering matter? It seems to me (and this is where I'm confused) that the Green's function ...
0
votes
0answers
36 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: $$\frac{-i}{k}\frac{\partial{g}}{\partial{x}}=g(x,k)$$ ...
0
votes
0answers
39 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 ...
0
votes
0answers
23 views

Distance between differential operators

Given two differential operators say $D_1$ and $D_2$ is there any meaningful way to define distance between them, does there exist some metric $d(D_1,D_2)$ that satisfies all the necessary properties? ...