Questions tagged [differential-operators]

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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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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}\...
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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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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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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 ...
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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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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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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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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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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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“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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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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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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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 ...
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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 ...
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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 ...
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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 $|...
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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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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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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 ...
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$ 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}+...
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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 $y=f(x)$ ...
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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 ...
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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 ...
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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 \...
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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 ...

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