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12 views

Selfadjointness of the differential operator in a singular potential

The free Dirac operator is the differential operator of the following form T_0 = i\alpha grad + \beta, where \alpha and \beta are Hermitian 4 x 4 matrices, and T_0 is selfadjoint on some domain ...
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11 views

Show two quantum differential operators are the same $\hat{A} \psi(x)$ = $\psi(x+b)$

Consider an operator $\hat{A}$ = $e^{b*d/dx}$, where b is a constant. Show that $\hat{A}$$\psi(x)$ = $\psi(x+b)$ I'm guessing taylor exp
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17 views

Is $L^{*}L$ a real operator?

let $(M,h)$ be a compact complex manifold with a hermitian metric $h$. Let $L$ be a $\mathbb{C}$-linear differential operator with smooth coefficients \begin{equation} ...
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24 views

Derivative of a function which is treated as a variable

I have got a function $f=f(x)$. The derivative is $\partial_xf$. There are applications in which it is reasonable to treat $f$ as another variable in a larger context. In my application I now need an ...
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10 views

put $-\frac{d^2 u}{d x^2} +\frac{d u}{d x}=\exp (x)$ on $[0,1]$ and BC $B_0u = u(0) = 5$, $B_1u=u_x(1)+u(1)=2$ in bilinear form $a(w,u) = F(w)$

Consider the elliptic problem $Lu=\exp (x)$ on $[0,1]$ with $Lu = -\frac{d^2 u}{d x^2} +\frac{d u}{d x} $ and boundary contidions $B_0u = u(0) = 5$ and $B_1u= \frac{du}{dx}(1)+u(1)=2$. Show that the ...
6
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1answer
120 views

What is the mathematical truth behind the Leibniz notation in differentiating twice or more?

So $f: \mathbb{R} \to \mathbb{R}$ is $n>1$ (or more) times differentiable. The notation of the first derivative makes perfect "sense" with regard to what's going on: $$\lim_{h \to 0} ...
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21 views

Meaning of (generalized?) differential operator

I am currently reading this paper which makes use of generalized differential operators. As I understood it, the operator $D_x$ works like this: If $F$ is a continuous function on $[a,b]$ and $G$ an ...
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31 views

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 ...
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1answer
19 views

Is there any difference between formally symmetric and formally self-adjoint differential operators?

I work with the well known book of Dunford/Schwartz "Linear Operators (Part II)". At first I should mention that the general difference between self-adjoint and symmetric operators is obvious to me. ...
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43 views

Formal definition of “node” with respect to eigenvalues and functional analysis.

I'm concerned with a special problem of spectral analysis for a certain Sturm-Liouville-differential-operator, that is to say $L:=\frac{d^2}{dx^2}-q(x)$ and the spectrum $\sigma(L)$. While reading an ...
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1answer
119 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 ...
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59 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 ...
2
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81 views

self-adjoint differential operator on $C^{0}([a,b])$?

I've got problems in understanding the way a special (self-adjoint) differential operator is acting on the domain and the range. So, I try to explain my difficulties: The differential operator is ...
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0answers
31 views

Is there any significance to this matrix/operator?

I am working on a problem involving the the polarized Hessian covariant in Cartesian coordinates on $\mathbb{R}^2$ $[a,b] = \frac{1}{2} \frac{\partial ^2 a}{\partial x ^2} \frac{\partial ^2 ...
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64 views

Show that the Hilbert transform is a pseudo-differential operator of order 0

I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform $Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$ is a ...
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20 views

Matrix representation of quadratic partial differential equations

For a particular problem, I have two following quadratic differential equations: $f_{uu}g_{u} - g_{uu}f_{u}$ = 0 $(f_{uu}g_{v} - g_{uu}f_{v}) + 2(f_{uv}g_{u} - g_{uv}f_{v})$ = 0 here, $f$ and $g$ ...
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1answer
79 views

right definition of correct space of domain and range for a self-adjoint Operator

at first, I'm quite sure it's not necessary to pay too much attention to the way the Operator is defined, it's rather important which spaces to choose to obtain a self-adjoint operator I've got a ...
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25 views

example of Pseudo-differential operators?

I am a biggner in the area of Pseudo-differential operator. Please provide some example of Pseudo-differential operators and the application of Pseudo-differential operators in science & ...
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1answer
29 views

Question about differential operators

Say $N = ab$. How can I express $\frac{d}{dN}$ in terms of $\frac{d}{da}$ and $\frac{d}{db}$?
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1answer
20 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 ...
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1answer
119 views

Squaring an operator

There is an excercise of squaring an operator in my book of quantun mechanics. The operator is $$\hat{A}=\frac{\mathrm{d}}{\mathrm{d}x}+x$$ And I should compute $\hat{A}^2$. He gives me a result ...
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23 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 ...
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1answer
123 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
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1answer
144 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 = ...
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24 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 ...
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18 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 ...
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0answers
42 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 $(0,2\pi)$. ...
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1answer
90 views

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 ...
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87 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 ...
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30 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 ...
1
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1answer
79 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 ...
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2answers
106 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 ...
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0answers
75 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) ...
5
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70 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 ...
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0answers
42 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: $$\frac{-i}{k}\frac{\partial{g}}{\partial{x}}=g(x,k)$$ ...
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171 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 ...
3
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0answers
78 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 $L^2(-\pi,\pi)$. ...
3
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36 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
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1answer
90 views

Composition of Differential Operators

If I have: $A=\partial_x^2+u(x)$ $B=u(x)\partial_x$ How do I compose: $AB$ and $BA$?
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69 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 ...
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2answers
200 views

Differentation Operator

having trouble completing the proof for this question Let $D:\mathbb{R}[X] \to \mathbb{R}[X]$ be the differentiation operator $D(f(X))=f'(X) .$ Prove that $e^{tD}(f(X)) = f(X+t)$ for $t \in ...
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48 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 ...
2
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0answers
31 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 ...
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0answers
24 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? ...
2
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139 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 ...
2
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0answers
84 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 ...
2
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0answers
52 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 ...
5
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2answers
106 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 ...
3
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1answer
64 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 ...
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1answer
27 views

Differential Maping in Elementary Analysis

The problem I stuck was : Let $ f = R^{n} \rightarrow R^{m} $ and suppose there is a constant $M$ such that for $ x \in R^{n} $, $ || f(x) || \leq M || x ||^{2} $. Prove that $ f $ is differentable ...