Questions tagged [differential-operators]
In mathematics, a differential operator is an operator defined as a function of the differentiation operator.
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questions
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Scaling as a repeated shift
I was wondering if the following derivation of the scale operator starting from the shift operator is good.
Everybody knows that an operator $T_a$ acting on a function $f$ as
$$
T_af(x)=f(x+a)
$$
can ...
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0answers
24 views
Normal intersect the $xy$ plane
I am so stuck in this problem any help will be apreciated:
show that the point where the normal to the surface $\mathcal S:\:x^2 + y^2 + z^2 = xf(y/x)$ intersects the $xOy$ plane is at the same ...
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2answers
20 views
Differential Operator: Solving Simple ODEs
This will likely be an exceedingly simple question, but the text's use of the operator is confusing me.
To preface, the operator $D$ is defined as:
$$
\frac{d^{n} y}{d x^{n}}=D^{n} y
$$
and can be ...
1
vote
1answer
28 views
How is the $\nabla^n $ operator defined?
In Quantum Mechanics, the translation operator $\hat{T}$ can be written as
$$\hat{T}(\boldsymbol{x}) = 1 - \dfrac{ix\cdot \hat{p}}{\hbar} - \dfrac{i(x\cdot \hat{p})^2}{2\hbar^2} - \dfrac{i(x\cdot \...
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20 views
Invariant Subspaces of $(L^2(\Omega))^3$ under some constant matrix
For a bounded Lipshitz domain $\Omega\subset\mathbf{R}^3$and if we denote by $n$ the outward unit normal vector to the boundary $\partial\Omega$, lets consider the following orthogonal decomposition ...
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93 views
Just what *is* $D$ in Analysis
In my Analysis class we keep using the symbol $D$ to stand for differentiation in our analysis course, but what is $D$ itself, really?
nLab seems to say its a functor, but for some reason requires ...
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2answers
41 views
Method of variation of parameter
By method of variation of parameters find the PI of the differential equation
$$\frac{dy}{dx}+2y=4e^{2x}$$
I have learned how to do the find out the PI of second order differential equation. But, ...
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22 views
Linear Operator over the space of functions whose first derivative is continuous.
I came across a question regarding Linear transformation stating as follows:
Let $V$ denotes the space of all real valued functions whose first derivative is continuous. And $T$ is the linear ...
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0answers
40 views
Differentiation Operator as a Function Between Spaces of Function Spaces
The differentiation operator $D$ takes a function $f:A\to B$ between differentiable manifolds $A$ and $B$ and assigns to it a function $Df : A_0 \to \mathcal{L}(A,B)$ which in turn assigns, to each ...
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Relation between Resolvent and Green function
If we have the system
$$
(id-A)z=f,
$$
where $A$ is a differential operator.
What is the relation between the Green function associated to the system and the resolvent $(id-A))^{-1}$.
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2answers
53 views
Rotational invariance of Laplacian operator
I was reading in Wikipedia about Rotational invariance and noticed that the two-dimensional Laplacian operator $\nabla^2 = \frac{\partial^2 }{\partial x^2} + \frac{\partial^2 }{\partial y^2}$ is ...
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34 views
Invertibility of operator
Gives conditions under which $L$ is an invertible operator.
$L:u \rightarrow -u''+p(x)u'+q(x)u$
$u \in dom(L)= \{u\in C^{2}[a,b], u'(a)-\theta_{a}u(a)=0, u'(b)+\theta_{b}u(b)=0 \} $
$\theta_{a}, \...
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votes
1answer
27 views
Range of differential operator
Show that the range of the operator $L$ is the whole space $C[a,b]$, and hence the inverse $L^{-1}$ has domain C[a,b].
$L:u \rightarrow -u''+p(x)u'+q(x)u$
$u \in dom(L)= \{u\in C^{2}[a,b], u(a)=0, u'...
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1answer
32 views
Where do the constants in this formula come from? I do not exactly know what they are.
I am reading about inverse operators and the book is going over this one.
After proving some stuff about this operator it finally says if $$P(D)y=(a_nD^n+...+a_1D+a_0)y=bx^k$$
then
$$y_p=\frac{1}{P(D)}...
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Bi-differential operators are necessarily tensor products
When working with deformation quantisations and Moyal products, the notion of bi-differential operators comes up very often. For example, see here: https://en.m.wikipedia.org/wiki/Moyal_product
My ...
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1answer
22 views
I do not know what this series expansion is.
I am reading my differential equations book and it is going over the differential inverse operator and more specifically this case where $y_p=\frac{1}{D-a_0}(bx^k)$.
So then they do these two steps ...
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0answers
26 views
Grunwald fractional derivative [duplicate]
I came across this maths statement
\begin{align*}
f(t+h) &= f(t) + h\frac{d}{dt}f(t) + \frac{h^2}{2!}\frac{d^2}{dt^2}f(t) + \frac{h^3}{3!}\frac{d^3}{dt^3}f(t) + \cdots \\
&= f(t) + hDf(t) + \...
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0answers
8 views
Discretization of Differential Operators in Fourier Spectral Method
I am currently implementing a Fourier psuedospectral method to solve some rather complicated nonlinear PDEs where I have operators of the form $[y^3y''']'$ and $[D(x)y']'$ for some real unknown $y(x)$....
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1answer
31 views
A bound for a differential operator in Sobolev norms
Let $s$ be an integer and $L$ a periodic linear partial differential operator $L=\{L_{ij}\}$ of order $l$ on $\mathcal{P}$, the space of $2\pi$-periodic functions $R^n\longrightarrow C^m$. The sobolev ...
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votes
0answers
33 views
Differential operator as sum of Dirac delta functions
For $x \in [0,1]$, let
$$\tilde{G}(x) = \frac{1}{2(e - 1)}(e^x + e^{1-x})$$
And let $G(x)$ be the 1-periodic extension of $\tilde{G}(x)$ to all of $\mathbb{R}$.
Show that, in the sense of ...
1
vote
1answer
41 views
D-operator-methods
Solve the following differential equation:
$$(D^2-2D+1)y=x^2e^{3x}$$
I found the $C.F.=(c_1+c_2x)e^x$
$$\begin{align}
P.I. & =\frac{x^2e^{3x}}{(D^2-2D+1)}\\
& =e^{3x}\frac{x^2}{(D+3)^2-2(D+3)+...
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2answers
24 views
D operators method
Find P.I. of the following equation:
$$(D^2-1)y=(1+x^2)e^x$$
I have tried like this:
$$\begin{align}
P.I.& =\frac{(1+x^2)e^x}{D^2-1}\\
& =e^x\frac{(1+x^2)}{(D+1)^2-1}\\
& =e^x\frac{(1+x^2)}...
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votes
1answer
36 views
D operator method
Verify that $y=x^2-6$ is a solution of
$$y''+y'-2y=14+2x-2x^2$$
I have tried like this:
$$\begin{align}
P.I. &=\frac{14+2x-2x^2}{D^2+D-2}\\
& =\frac{-2(x^2-x-7)}{-2\left(1-\frac{D^2+D}{2}\...
0
votes
1answer
23 views
D-operator method
Evaluate a particular value of $$\frac{1}{D^2+4D}\sin 2x$$
I know the complementary function(C.F.) & particular integral(P.I.)....but I can't understand what is particular value, please explain
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1answer
40 views
Definition of an elliptic operator with measurable coefficients
Let $\mathbb{k}$ be one of $\mathbb{R}$ or $\mathbb{C}$. Say I'm given an $m$-th order linear partial differential operator $L$ in the form of a $\mathbb{k}$-linear operator
$$
L = \sum_{\substack{\...
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Adjoint differential operator
I am a little confused. Let $g(x)$ is a nontrivial solution of the equation $-y'' + q(x)y = \mu y$. Then define (in one book) the operators
$$A = g(d/dx) (1/g) \quad and \quad A^* = -(1/g) (d/dx)g$$
(...
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0answers
64 views
Exponential of the differential operator $x \frac{d}{dx}$
I am studying conformal field theory, and I have run into an problem with calculations involving the quantum-mechanical translation and dilatation operators. It actually boils down to an issue about ...
1
vote
1answer
49 views
Pullback of a Vector-valued form treated as Differential Operator
Suppose we are given a vector bundle $E\to M$ and a $E$-valued $p$-form, $$\omega\in\Gamma(E\otimes \Lambda^pT^*M)$$ For any smooth $f:\Sigma\to M$ then we have the pullback $p$-form, $$f^*\omega\in\...
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Linearization of operator in almost complex manifolds
In
Lectures on Holomorphic Curves in
Symplectic and Contact Geometry
by Wendl,
the linearization of the operator $\overline{\partial}_J$ is defined as
for $u: (\Sigma, j) \rightarrow (M,J)$ , $\...
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0answers
21 views
Reference for differential operators on homogenous spaces
Is there a simple reference for differential operators on homogenous spaces (i.e a space on the form $G/H$ where $G$ is a real Lie group and $H$ is a closed Lie group) ? I would be interested by ...
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0answers
31 views
Differential operator on modular forms
Let $z=x+iy \in \mathbb{H}$ and $f \in M_k(\Gamma)$ be a modular form and $\Gamma$ does not have to be the full modular group. Derivative of $f$ is given by: $f' = \frac{1}{2 \pi i} \frac{\partial}{\...
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0answers
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Finding the eigenpairs of an atypical differential operator.
I came across a problem that asked me to find the eigenpairs of the operator $M : C^2_D[0, \ell] \to C[0, \ell]$ defined by
$$
My = -c\frac{d^2y}{dx^2}+dy,\ \ \ c,d > 0
$$
subject to Dirichlet ...
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vote
0answers
28 views
Estimate with L^2 Norm of gradient
For given $u,v\in H_{0}^{1}$,$b_{i}\in L^{\infty}$ i have to show that
$$\sum \int |b_{i}uv_{x_{i}}|\leq \|b\|_{L^{\infty}}\|u\|_{L^{2}}\ \|\nabla v\|_{L^{2}}$$ holds. I think it is obvious that I ...
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votes
1answer
23 views
“real part” of complex differential operators on $Sl_{2}(\mathbb{C})$
Let $D$ be a complex differential operator on the complex Lie-Group $SL_{2}(\mathbb{C})$. How is the "real part" of $D$ with regards to $SL_{2}(\mathbb{R})$ defined?
Im currently reading a lecture ...
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1answer
82 views
Integration over a discontinuity involving its derivatives (as distributions)
Given a discontinuous complex function $\varphi_k$ on the real line written as
$\varphi_k(x) = \theta(-x)u_k(x) + \theta(x)v_k(x)$, where $u_k$ and $v_k$ are smooth and respectively defined on the ...
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0answers
42 views
nabla operator: is it standard?
I would like to know what does it mean $$\nabla f(t,\ldots,t)$$ on the page 554 $(4.3)$ here.
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0answers
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How to verify the following function is an eigen-function of the given operator
I have an operator given by $$K=\cosh ax\hspace{2pt}\partial_t-\frac{\sinh ax}{at}\partial_{x}$$ and my function is $$f(t,x)=e^{ikx}J_{\pm\frac{ik}{a}}(mt)$$ where $J_{\nu}(t)$ is the first kind ...
2
votes
1answer
80 views
How can I show the symmetry of an operator?
Given the operator $L$ and $p, q, V$ smooth functions:
\begin{equation}
L=\frac{1}{p(x)}\left[\frac{d}{dx}\left(q(x)\frac{d}{dx}\right)+V(x)\right]
\end{equation}
I should show that, whenever p is not ...
2
votes
1answer
42 views
Why is this resolvent trace-class?
I am trying to read Richard Froese's paper Asymptotic distribution of resonances in one dimension and I cannot follow the logic of Proposition 7.3. He defines a cutoff $\chi$ to be multiplication by ...
0
votes
1answer
25 views
What are the main properties of eigenvalues of normal unbounded operators?
I am interested in the properties of the eigenvalues of unbounded normal operators. For compact linear operators we have that for every $t >0$, the set of distinct eigenvalues $\lambda$ such that $|...
1
vote
1answer
49 views
Why is $\left(\frac{\partial}{\partial x_i}\right)f = \left(\frac{\partial f}{\partial x_i}\right)$
i'm currently reading An Introduction to Morse Theory by Yukio Matsumoto and on p.62 it says
A vector field itself is sort of a differential operator, since it
assigns to each point a "tangent ...
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0answers
40 views
Are there any applications of the “imaginary derivative” $D^i$ operator?
Where $D$ is the normal differential operator, $D f(x) = \frac{d}{dx} f(x)$, $D^n f(x) = (\frac{d}{dx})^n f(x)$, we can define, for example, the "half-derivative" $D^\frac{1}{2}$ such that $D^\frac{1}{...
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Differential operators of a principle G-bundle
I understand for a pair of smooth vector bundles $E$ and $F$ over a smooth manifold $M$ it makes sense to talk about the differential operators between the section spaces of $E$ and of $F$. The ...
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1answer
26 views
Derive squared angular momentum operator in spherical coordinates easily
there! I am studying tensor analysis and now try to apply it to solving a quantum physics problem.
Here I am trying to calculate angular momentum squared written in terms of the spherical coordinates ...
0
votes
1answer
15 views
$ w\in(\textrm{ker}(\tilde{L}))^\perp~\Longleftrightarrow~w\in(\textrm{ker}(L))^\perp $?
Consider
$$
u_t=u_{xx}+f(u)
$$
with some non-linearity. Making the ansatz $u(x,t)=U(z), z=x-ct$ gives the ODE
$$
U_{zz}+cU_z+f(U)=0.
$$
Linearizing in $U$, gives the linear operator
$$
L=\partial_{zz}+...
1
vote
3answers
82 views
Is $\int \frac{{dx}^2}{dy}$ valid?
It appears to me that
$\int \frac{{dx}^2}{dy}$ can be rewritten as
$\int \frac{dx}{dy}\cdot{}dx$ which in turn can be rewritten as
$\int f^{-1}{^\prime}(y)\cdot{}dx$. (it is assumed that $y=f(x)$ ...
0
votes
0answers
26 views
Is there a name for the differential operator $\nabla\otimes$
There seems to be quite a few names for differential operators involving $\nabla$:
gradient: $\nabla$
divergence: $\nabla\cdot$
curl: $\nabla\times$
laplacian: $\nabla^2$
I was wondering whether ...
4
votes
1answer
103 views
Example of an additive map $\mathcal{O}_X \to \mathcal{O}_X$ that is not a differential operator on a Scheme $X$.
I'm looking for a $S$-Scheme $X$ (with structural morphism $f$) and an additive (or rather $\newcommand{\O}{\mathcal{O}}f^{-1} O_S$-linear) endomorphism of sheaves $D: \O_X \to O_X$ which is not a ...
2
votes
1answer
107 views
Operator norm and Lipschitz continuous problem
Suppose that $f : \mathbb R^6 \rightarrow \mathbb R$ is a function with the following two properties: $f(0) = 0$, and at at any point $c \in \mathbb R^6$ and any increment $h$, $\Vert Df(c)(h)\Vert \...
3
votes
1answer
75 views
Mathematical operations order when using an operator
I am not very familiar with operators (as I do not study mathematics) and I have just started a Quantum Mechanics course in a university. However, I am not sure what should be the precise order of ...
