In mathematics, a differential operator is an operator defined as a function of the differentiation operator. (Def: http://en.m.wikipedia.org/wiki/Differential_operator)

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Finding the polynomial [on hold]

Find a nontrivial polynomial function $p(x)$ such that $p(2x)=p'(x)p''(x)\not=0$
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Closure of an operator

I am wondering what is the closure of the domain of the operator $A_0:D(A_0)(\subset H)\to H$in $H=L^2(0,1)$ $$A_0= f^{(4)}-f^{(6)}$$ $$D(A_0)=\big\{ f\in H^6(0,1)\cap H_0^3(0,1) |f^{(3)}(1)=f^{(4)}(...
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Operator that kill the wave function

I have the following function in $x$. $\sum_{d=0}^{\infty} \frac{1}{\hbar^d}\frac{1}{d!}\left(\prod_{i=1}^{d-1}(1+i\hbar)^{m}\right)x^d$ I need a differential operator involving $(x,\frac{d}{dx},h,...
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1answer
481 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 $ ...
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1answer
21 views

Divergence of Material Derivative

Let $u : \Bbb{R}^n\times \Bbb{R} \to \Bbb{R}^n\times \Bbb{R} $ be a divergence free vector field. Then the material derivative $D $ is given by: $$ \frac { \partial u_j}{\partial t}+\sum_{i=1}^{n} ...
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1answer
48 views

Definition of formal adjoint of covariant derivative

I read in Einstein Manifolds, L. Besse that the covariant derivative $D: \mathcal{J}^{(r,s)}(M)\to \Omega^1(M)\otimes\mathcal{J}^{(r,s)}(M)$ admit an formal adjoint $D^*:\Omega^1(M)\otimes\mathcal{J}^{...
2
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2answers
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Where to learn the algebra behind the use of differential operators in calculus

Algebra with differential operators is often used as a shortcut in calculus problems. I have previously asked about manipulations such as these: $$\int x^5e^xdx =\frac1Dx^5e^x=e^x\frac{1}{1+D}x^5=e^x(...
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How to find operator with Fibonacci eigenvalues?

How can I find the operator that satisfies this equation? $$F_nx^n=Dx^n$$ Summing over $n$ we can rewrite this as $$\frac1{1-x-x^2}=D\frac1{1-x}$$ I am unsure whether this can be solved. I am ...
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21 views

Differential Equation Invariant under Isometric Mapping

I started reading a book on Finite Elements ("Finite Elements", Braess) and one section describes elliptic PDE's of the form $Lu = f$. The author goes on to say "If a differential equation is ...
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What is the meaning of $1/(D+a)$, where $D$ is the derivative operator?

Today I read the answer to this post, in which the poster integrates $x^5e^x$ by making these manipulations with the differential operator $D$: $$\frac1Dx^5e^x=e^x\frac{1}{1+D}x^5=e^x(1-D+D^2+...)x^5$$...
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Differential equations with inverse trigonometrical functions

$(x^2+y^2)^(1/2)$=$e^(asin(y/(x^2+y^2)^1/2))$ Prove that $\frac{d^2 y}{dx^2}$=$2((x^2+y^2))/(x-y)^3$, x>0 I started with taking the natural log of the given equation and differentiating it, I ended ...
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convolution of Schwartz functions with $f(x) = (1+\|x\|)^{-\frac{1}{2}}$

Let $f(x) = (1+\|x\|)^{-\frac{1}{2}}$ for $x \in \mathbb{R}^n$. This is clear that $f\star g \notin \mathcal S$ where $\mathcal S$ is algebra of Schwartz functions on $\mathbb{R}^n$ and $ g \in \...
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Complex Air operator

Help me to do this exrcice Consider the differential operator $A=-\partial^{2}_{x}-ix$ on $\mathbb{R}$ with $D(A)=\{f\in L^{2}(\mathbb{R},dx), Au\in L^{2}(\mathbb{R},dx)\}$ Check that $A$ is colsed ...
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36 views

Adjoint of differential operator

I would like to find the adjoint of the operator $T_a$ ($a\in \mathbb{C}$) on $ \mathcal{H}=L^{2}(\mathbb{R}^{2},dxdy)$ with $(u,v)=\int \int u(x,y)\overline{v(x,y)} dx dy$ $$T_a(u)=ia(y\partial_xu-x\...
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23 views

A estimate about constant coefficient partial differential operator on $C_0^{\infty}(\Omega)$

This problem is from Stein Real Analysis,Chapter 5,exercise 12. Problem: We consider whether the inequality $||u||_{L^2(\Omega)} \le c||Lu||_{L^2(\Omega)}$ can hold for open sets $\Omega$ that are ...
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inverse of operator

I want to calculate the inverse of the operator $T=-\frac{\partial^{2} }{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-(x^{2}+y^{2})+2( x\frac{\partial }{\partial y}-y\frac{\partial }{\...
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1answer
39 views

How do solve this pde problem?

EDIT: I know somehow, we end up with an equation relating the derivative of some coefficients to the rest of the stuff. I'm not sure where this equation, or even the constant that we use to get it, ...
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2answers
48 views

Self adjoint and symmetric operator

I am wondering whether for an operator defined on a real Hilbert space to be positive we need to show that it is self-adjoint at first. It seems to me that they are two different property and can be ...
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1answer
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Stoke's Theorem Application

Problem: By using Stoke's Theorem, deduce that $\int_{C} \mathbf{r} ( \mathbf{r} \cdot d\mathbf{r}) = \int \int _{S} \mathbf{r} \wedge d \mathbf{S}$. Where $C$ is the simple closed curve bounding the ...
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Find the eigenvalues of the operator defined by $(Lu)(x)=u'(x)$

Let $$L : D(L) \to L^2([0, \pi] ; \mathbb{C}$$ be a linear differential operator defined by $$(Lu)(x)=u'(x)$$ where $x \in [0, \pi]$ The domain $D(L)$ of $L$ is given by $$D(L)=\{ u \in L^2([0,\pi]): ...
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Operator in spherical coordinates

The background is physics but I think the problem is pure mathematical. If $$\mathbf{L}=i\hbar\left(\frac{\hat{\theta}}{\sin \theta}\frac{\partial}{\partial\phi} - \hat{\phi}\frac{\partial}{\partial\...
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When can I Taylor expand a function of an operator?

1-) Is the expression $f(A) = \sum_n \frac{f'(0)}{n!}(A)^n$ always meaningful for any diagonalizable linear operator $A$ and for any analytic function $f$? This seems strange to me because then I ...
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Counting zeroes of global sections

Let $X$ be a compact connected Riemann surface and let $\Phi:M\rightarrow N$ be an elliptic differential operator where $M$ and $N$ are two complex line bundles on $X$. Let $f$ be a $C^\infty$-global ...
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Is there any connection between these two definitions of coercivity (ellipticity) in PDE and bilinear form?

In the field of partial differential equation, we say that the following operator \begin{equation} Lu=-\sum\limits_{i,j=1}^n (a^{ij}u_{x_i})_{x_j}+\sum\limits_{i=1}^n b^i u_{x_i}+cu \end{equation} is (...
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Find the extended form of the group generated by an operator?

I tried to find the extended form of the group generated by the following operators. (I): The first operator $$A=z\frac{\partial }{\partial z}+1$$ To find the extended form of the group ...
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1answer
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Sturm-Liouville operator with Dirichlet BC

I am trying to understand why Sturm-Liouville operator $$L(f)(x)=f''(x)-p(x)f(x)$$ with Dirichlet boundary conditions on $[a,b]$ is unbounded. $f$ is twice continuously differentiable, $p(x)>0$ is ...
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Is $\frac{d}{dx}$ the same as $\frac{d(1)}{dx}$?

I have a quantum mechanics problem asking me to prove that the commutator of $x$ and $p$ is equal to some value. In my computation I get something along the lines of $$ (a*x) \frac{d}{dx} - (b)\frac{...
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If a k-form vanishes in the neighborhood of p then it vanishes at p

Let w be a k-form defined in an open set A of $\mathbb R^n$. We say that w vanishes on x if w(x) is the zero tensor. Show that if w vanishes at each x in a neighborhood of $x_0$ then dw vanishes at $...
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Is it possible to decompose this expression?

Is it possible to factorize $$(-\partial^2+\phi^2(r))^2-\left(\frac{\partial\phi(r)}{\partial r}\right)^2,$$ where $\phi(r)$ is a function of $r\equiv\sqrt{x^2+y^2+z^2+\xi^2}$? Note that we can not ...
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Lie group of differential operators

I have the following three partial differential operators $$A=y \frac{\partial}{\partial y}$$ $$B=y^{-1}(z\frac{\partial}{\partial z}+y\frac{\partial}{\partial y}+c-1)$$ $$C=y((1-z)\frac{\...
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1answer
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Can this be written more succinctly (preferably as an eigenequation)?

I have the set of equations $$\pmatrix{L_x \\ L_y \\ L_z}=\pmatrix{\partial_xL_x &\partial_yL_x &\partial_zL_x \\ \partial_xL_y &\partial_yL_y &\partial_zL_y \\ \partial_xL_z &\...
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Partial Derivatives and Operator Commutivity

I have an operator $$L\psi=\frac{1}{r^2}\partial_z^2\psi+\frac{1}{r}\partial_r(\frac{1}{r}\partial_r\psi)$$ I am interested in taking $\partial_rL\psi$ and $\partial_zL\psi$. Do the partial ...
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Is $(Id-\Delta)\colon H^2(\Omega)\to L^2(\Omega)$ bijective?

Is $$(Id-\Delta)\colon H^2(\Omega)\to L^2(\Omega),$$ with Neumann or Dirichlet boundary conditions, bijective? I know that this holds, but why? ($\Omega$ is in $\mathbb{R}^n$ with $n=2,3$ and smooth ...
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What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?

Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the ...
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Show that the operator is NOT symmetric.

Show that the Sturm-Liouville operator $L$ in $L^2([a,b])$ given by $$L=\frac{1}{r(x)}\left(DpD+q\right)$$ is not symmetric. I'm assuming $p=p(x)>0$ and $q=q(x)\geq 0$, as described by the problem ...
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1answer
36 views

Show that $L$ is formally self-adjoint.

Consider the differential operator $$L=e^xD^2+e^xD,\;\;D=\frac{d}{dx},\;0\leq x\leq1,$$ $$u^\prime(0)=0,\;\;\; u(1)=0.$$ Show that $L$ is formally self-adjoint. I just don't really know how to start ...
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self-adjoint operators and linear dependence

Let $L$ be a self-adjoint differential operator given by $L=\frac{d}{dx}\left(a_2\frac{d}{dx}\right)+a_0$. If $u_1$ and $u_2$ are two solutions of $Lu=0$ and $J(u_1,u_2)=0$ for some $x$ for which $a_2(...
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concomitant and self-adjoint operator

If $Lu = u^{\prime\prime}+\omega^2u$, show that $L$ is formally self-adjoint and the concomitant is $J(u,v)=vu^\prime-uv^\prime$. Moreover, if $u$ is a solution of $Lu=0$ and $v$ is a solution of $L^*...
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What is the proof that linear operators can be treated as variables?

I understand what a linear operator is, but I don't understand why you can just treat it as a variable.
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Generalization to differential operators

There is a well know result: Suppose $ {f_n}$ is a sequence of functions, differentiable on $ [a, b]$, and such that $ {f_n(x_0)} $ converges for some point $ x_0$ on $ [a, b]$. If $ f'_n$ ...
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Understanding Operators in context of Green's function derivation

I am trying to understand what operators actually mean when deriving the definition of green's function. What is the interagl representation of an operator? Is this correct? $$ D = <x|\int D|x> ...
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About Second-order Linear Homogenous ODE

One way to solve second-order linear homogeneous ode with constant coefficients is to do the following things: $$a\left(\frac{\mathrm d^2}{\mathrm dx^2}\right)f+b\left(\frac{\mathrm d}{\mathrm dx}\...
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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.
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How to define properly the radial and angular dependence of a function?

Recently I've came across the following situation when studying Quantum Mechanics: suppose we have two operators $A,B$ on the space of functions $L^2(\mathbb{R}^3)$ and suppose they have the following ...
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How to calculate the derivative of logarithm of a matrix?

Given a square matrix $M$, we know the exponential of $M$ is $$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$ and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(I-M)^k}{k}$$ The derivative of $\...
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Link to question regarding treating differential operator as a ratio [duplicate]

I have attempted to find the post which provides an explanation as to the circumstances in which we can treat $\frac{dy}{dx}$ as a ratio which appears to be used in solving separable DE's, but I have ...
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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} + \frac{...
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1answer
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Doubts related to Differential Operator of Infinite order

Let $$f(s)=\sum_0^\infty c_vs^v$$ be some entire function. We say that the differential operator $f(d/dx)=\sum_0^\infty c_vd^v/dx^v$ is defined in some fundamental space $\varPhi$, if for any $\varphi ...
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1answer
69 views

Solving the Sturm-Liouville problem using Green's function and Spectral Theorem.

I am reading a paper that deals with the solution of the Sturm-Liouville problem: $u''(t) + \rho (t) u + \lambda ^{-1}u= -f $ $ u(0)=u(1)=0$ For $\rho(t) \leq 0 $. First it is solved the ...
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Elliptic operators for a Laplacian transform

I need to show that the operator: $L[u]=(1-x^2)\frac{\partial^2 u}{\partial x^2}+2xy\frac{\partial^2 u}{\partial x \partial y}+(1-y^2)\frac{\partial^2 u}{\partial y^2}$ Find the transformation of ...