Tagged Questions

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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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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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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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 ...
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Differential Operators and Coefficients

First question on Math StackExchange here. I have been staring at this for a bit, but wasn't quite able to get the hang of it. Here it goes. We are given \begin{align} \frac{\partial}{\partial x} = \...
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Operators in polar coordinates in n-dimensions

I want help on converting differential operators such as the reduced wave operator (L=Δ+c) and the biharmonic operator (L=Δ^2) from Cartesian to spherical coordinates in n-dimensions. For example I ...
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Divergence in Definition of Laplace-Beltrami Operator

I am trying to derive an explicit formula for Laplace-Beltrami operator in global Cartesian coordinates for a special case of plane curve. I have found this article, and I would like to match their ...
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Consider a first-order operator

Consider the first-order linear differential operator L = D + 5 a) do all solutions of lx(t)=0 a subspace of F b) find these solutions c) do all solutions of lx(t) = 1 form a subspace of F I'm no ...
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Trial Solutions to Non-homogeneous Differential Equations

I'm having trouble finding information on something in my Differential Equations & Linear Algebra class. When you're trying to find the general solution to an nth order linear non-homogeneous ...
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The domain of a root of a self-adjoint operator associated with an interpolation space

We now that $V$, $H$ are separable Hilbert spaces such that $V$ is dense in $H$ and $V\hookrightarrow H$ continuous, by representation theorem exists $A: D(A)\subset V\rightarrow H$ self adjoint e ...
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Integral of the Laplace-Beltrami Operator multiplied by a function

I have the following problem: Let $\mathcal{M}$ be a $2D$-manifold in $\mathbb{R}^3$ and let $g$ denote its metric. Furthermore it is known that $\mathcal{M}$ is a closed manifold (i.e. it has no ...
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What's an example of algebra where differential operators aren't generated by order 1?

For any commutative algebra $A$ over a field $k$, one can define its algebra of differential operators $\def\Diff{\operatorname{Diff}} \Diff_*(A)$, which has a filtration by order. In many cases the ...
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Taylor expansion of $\frac{1}{1-D}$, where $D$ is the differential operator

I understand that we can represent $e^D$ simply as a power series of D. But what about functions of D which are not entire on the complex plane? What if the function has no taylor expansion, or if it ...
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Continuous dependence on initial conditions for second order eigenvalue problem

Consider the Schrödinger eigenvalue problem in one dimension $$\phi'' - V\phi + \mu \phi = 0$$ on $[0,a]$ with boundary $\phi(a) = c$. Suppose that I already have the existence of solution/...
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Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
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Manifold,affine connection,vector field

An affine connection on $M$ is a differential operator, sending smooth vector fields $X$ and $Y$ to a smooth vector field $∇_X Y$ , which satisfies the some conditions.I would like to know the ...