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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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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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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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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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 $(u,v)=\int \int u(x,y)\overline{v(x,y)} dx dy$ ...
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2answers
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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
38 views

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 ...
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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} - ...
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31 views

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
25 views

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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2answers
51 views

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} - ...
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29 views

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 $$A=y \frac{\partial}{\partial y}$$ $$B=y^{-1}(z\frac{\partial}{\partial z}+y\frac{\partial}{\partial y}+c-1)$$ ...
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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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1answer
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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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54 views

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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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 ...
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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 ...
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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 ...
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786 views

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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1answer
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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} + ...
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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
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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 ...
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1answer
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Sturm-Liouville operator, basis functions.

When dealing with a Sturm-Liouville operator: $$\hat A=\frac{1}{w(x)}\{\frac{d}{dx}(p(x)\frac{d}{dx})+q(x)\}$$ It is Hermitian if its functions satisfy the boundary conditions: $$[f^*p ...
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Reference for Fractional calculus and Differential Operators

I`ve been struggling with Fractional calculus and differential operators while studying special functions, and got to the conclusion that I need some references for them. So I ask for as many ...
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Fokker-Planck derivation. Path integral?

I am trying to understand the development of Fokker-Planck equation as is described here. Unfortunately, I cannot understand how the first equation on page 4, \begin{multline} \frac{1}{2} ...
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1answer
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Can anyone prove this equation? (Eq. with operators)

I am trying to understand the last equation from page 2 of this pdf http://physics.gu.se/~frtbm/joomla/media/mydocs/LennartSjogren/kap7.pdf but I am not being able to develop as here it says. Could ...
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Finding an matrix for an operator.

I was attempting to find a matrix for the function $x\frac{d}{dx}$ in the span of the set $\{1,x,x^2\}$ for the the standard dot product. Could someone guide me in how to do this?
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1answer
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How to find the differential with respect to the supremum norm and $L^1$ norm

I'm given a function $F:C([0,1])\rightarrow C([0,1])$, $F(f)=f^2$ (where $C([0,1])$ is given the supremum norm) and I want to find $D_g F(f)$ for any $f,g\in C([0,1])$. I find that $D_g F(f)= 2fg$, ...
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Closure of a differential operator

Consider $A:\mathcal{D}(A)\subset L^{2}[0,1]\to L^{2}[0,1]$ given by $$A:=-\frac{d^{2}}{dx^{2}},\qquad\mathcal{D}(A):=C^{2}_{0}(0,1)$$ Now, I assume that the closure of $A$ is its extension defined ...
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Kronecker delta representation of a matrix (Quantum raising / lowering operators)

The Kronecker Delta is commonly used to represent a diagonal matrix: $$ a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right) $$ ...
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Proving the principle symbol is globally defined

I want to prove the principle symbol is globally defined as an element \begin{align} \Gamma(T^* M / \{0\}, \textrm{Hom}(\pi^*E, \pi^* F) ) \end{align} To more specify, let me explain the definition ...
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$\forall$ $\epsilon > 0$, $\exists \ \delta > 0$, $\|s\|, \|t\| < \delta \implies$ $|f(x_0 + s) - f(x_0 + t) - Df(x_0)(s-t)| < \epsilon \|s - t \|$

$E$ is a normed vector space and $\Omega \subset E$ is open. Let $f: \Omega \to \Bbb R$ be Fr├ęchet differentiable on $\Omega$. Let $Df$ be the derivative map of $f$; $Df: \Omega \to \mathcal L(E, ...
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Linear Differential Operator

Let $\phi({D})$ be a differential operator defined by $$D^n+a_{n-1}D^{n-1}+a_{n-2}D^{n-2}+...+a_0=\sum_{j=0}^{n}a_jD^j$$ with constant coefficients $a_j$ (setting $a_n=1$) and the derivative ...
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k-symbol differential opeartor L, and its independent of choices.

This material is in O. Well's Differential analysis on complex manifold, page 115. Let, $(x,v) \in T'(X)$ ($T^*(X)$ with deleted zero section) and $e \in E_x$, Find $g\in \epsilon(X)$ and $f\in ...
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Different solutions to the Hermite equation

The Hermite differential equation is given as such $$ y'' - 2xy'+2\lambda y=0 $$ writing this in strum-liouville form you get $$-(\exp(-x^2)y')'= 2\exp(-x^2)\lambda y $$ However, in order for it to ...