Questions tagged [differential-operators]
In mathematics, a differential operator is an operator defined as a function of the differentiation operator.
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How does the canonical commutator works on differential operators?
I'm having trouble understanding how the canonical commutator works on differential operators: in this article, page 2, they define the multiplication-by-$x$ operator, that is a linear operator acting ...
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Determining whether there is a missing $i$ in Zworski's example of a pseudodifferential operator corresponding to a given symbol
In Zworki's book Semiclassical analysis, he gives examples of quantized symbols on page 57 with the following definition: $h > 0$ and $a\in\mathscr{S}(\mathbb{R}^{2n})$ is called a symbol. A ...
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Integral representation for the operator $\frac{\log{\Delta}}{\Delta}$ in two dimensions
Recently, I have stumbled upon the following non-local operator in $\mathbb{R}^{2}$
$$
\frac{\log{\Delta}}{\Delta} \ ,
$$
where $\Delta=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^...
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Matrix with partial differential terms: how to use in MAPLE
Anyone who knows how I could use a matrix in this form in Maple? I have the 2022.2 release. When I try defining such a matrix, or multiplying it with another one I get an error like: "internal ...
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Willmore extension theorem for tensor derivations
I came across the following theorem in Foundations of mechanics by Ralph Abraham
Let $M$ be a smooth manifold. Suppose for each open $U\subset M $, we have maps $E_U: C^\infty(U) \to C^\infty(U)$ ...
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Understanding $a\vec b * \nabla c$ [closed]
I have some difficulties when working with nabla-operators and I might need advice.
Is it correct to say that
$$a\vec b \cdot \nabla c = a\nabla \cdot (\vec bc) + ac\nabla \cdot \vec b~~~?$$
If so, ...
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Induced differential operator in long exact sequence of cohomology groups of differential complexes is well defined
In Differential Forms in Algebraic Topology, Bott & Tu they claim that given a short exact sequence of differential complexes:
$$0\rightarrow A \overset{f} \rightarrow B \overset{g} \rightarrow C \...
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Differential Operator squared
If I have the differential operator $$L = \dfrac{d}{dx} + c v(x)$$ what will be equal to $L^2$?
To $L^2 v(x)?$
What is the meaning of $L^2$? What are my issues to compute it?
In my task I have the ...
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Is it always okay to substitute $a^2$ in place of $D^2$ in the inverse differential operator of cosh ax?
In class, we were told that it was okay to make the following substitution while trying to solve a particular integral of a nonhomogeneous linear ODE, if $f(D)$ contains only even powers of $D$, and $...
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Can the eigenvalue of an differential operator be a function itself?
My issue is this:
Let's say I have the operator $\hat{X} = \frac{d}{dx}$ and I want to find complex-valued functions $f:\mathbb{C}\rightarrow\mathbb{C}$ which satisfy $\hat{X} f = \lambda f,$ with $\...
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What does $e^{\cos(D)}$ do?
We know for example, $$e^D f(x) = f(x+1)$$ and for example, $$e^{-D^2} x^n = H_n(x/2)$$ But what if the exponent is also an exponential, like $$e^{-\cos(D)}$$ or $$e^{-e^D}\ \ \ ?$$ Do we at least ...
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Why does this substitution give an incomplete particular integral solution for this linear ODE?
Trying to solve the following linear nonhomogeneous ODE
$$ (D + 2)^2y = \cosh 2x $$
We can easily find the complementary function
$$ y_c = c_1 e^{-2x} + c_2 {x} e^{-2x} $$
Using the inverse ...
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Adjoint of a differential operator with curl
I am looking for the way to find adjoint of the operator $L = (\boldsymbol{v} \times \nabla \times)$ where $\boldsymbol{v}$ is a vector in $\mathbb{R}^3$. Can someone help me with the correct steps or ...
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Does $\det(AB) = \det(A) \det(B)$ hold for differential operators $A$ and $B$?
Following the inconsistency I was facing in a previous post, I was wondering if the following property
$$ \det(A) \det(B) = \det(AB) $$
holds for generic $A$ and $B$ differential operators. It looks ...
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Finding characteristic curves for second order differential linear operator.
I've been reading Shubin's "Invitation to partial differential equations" and I've recently struck out on section $1.6$. We start with a differential operator of the form $$A = a\frac{\...
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Functional determinant inconsistency
While trying to compute functional determinants, I faced an inconsistency, which I can exemplify by defining the following matrix
$$ M = \begin{pmatrix} i \frac{d}{dt} + t & 0 \\\ 0 & i \frac{...
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Kernel and Differential Operator
I hope your day is great so far.
I have a question. I am given a Cauchy-Euler ODE
$$\big((x+2)^3\mathbf{D}^3+4(x+2)^2\mathbf{D}^2+3(x+2)\mathbf{D}+1\big)y=\frac{1}{x+2},$$
where $\mathbf{D}^n=d^n/dx^n$...
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Norm of the gradient of a function between Riemannian manifolds.
Let $(M, g), (N, h)$ be two Riemannian manifolds and $u: M \to N$ a smooth function. I would like to know how to show that, for $x \in M$,
$$|\nabla u|^2(x) = g^{\alpha \beta}(x)h_{ij}(u(x)) \frac{\...
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Coefficients of invariant differential operators an coefficients of bilinear forms
Let $\Delta$ be the Laplace operator on $\mathbb{R}^3$: $ \Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} +\frac{\partial^2 f}{\partial z^2}.$
A calculation shows that ...
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When considering matrices (or differential operators), $A > B \implies \exp(A) > \exp (B)$?
First, I would like to ask what is the conventionnal definition of $ A > B $ for matrices (or differential operators in my case of interest).
I guess that a natural definition would be that $A-B$ ...
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Maximal domain of unbounded linear differential operator
Let's consider the following (unbounded) linear Operator. (So called Transport-Operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
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Differential operators question
I am getting confused on differential operators, can some please help explain the following.
Suppose I have two differential operators $B = D_x^2 + \alpha A I$ where $I$ is the identity operator and $...
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Concerning a proof that the Sturm-Liouville operator have bounded below eigenvalues
I am currently studying Sturm-Liouville theory and I have a doubt about a proof I found in the book Sturm-Liouville Theory and Its Applications by M. A. Al-Gwaiz, about the eigenvalues of the SL ...
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Time derivative of the commutator of two operators $P,L$
I am reading a textbook Mathematics for Physics. In page 126, the author defines two operators:
$$L=-\partial_x^2+q(x)$$
$$P=\partial_x^3+a(x)\partial_x+\partial_x a(x)$$
Then the author continues to ...
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Math on operators in differential equations.
I watched this Michael Penn video and it lead me to try a couple of things and they ended up not working.
Specifically, he solves the following problem like so
Starting with $y'' = y$
$y'' - y = 0$
$\...
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The operator $(Xf)(x)=-if'(x)$ has Fourier series as eigenfunctions. Which operator $ (Af)(x)=??$ has the Taylor series as its eigenfunctions?
The operator $$ (Xf)(x)=-if'(x)$$ has fourier series as its eigenfunctions. I was wondering what kind of operator $$ (Af)(x)=??$$ has the taylor series as its eigenfunctions?
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What is a bidifferential operator?
Let $M$ be a smooth manifold. Then what is meant by saying that a map $c : C^{\infty} (M) \times C^{\infty} (M) \longrightarrow C^{\infty} (M)$ is a bidifferential operator?
Such terminology has been ...
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Pulling out a distribution ? $u\circ f$?
In an attempt to generalize solutions with discontinuities to a PDE, we want to consider distributions.
We have a continuous function $g\in C(Y)$ and a smooth function $f:\mathbb{R}^n\rightarrow Y,\ f$...
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Commutator estimates between a contraction evolution group and a differential operator
In the Appendix of "Commutator Estimates and the Euler and Navier-Stokes equations", Kato proves the following inequality
\begin{equation}\tag{1}\label{1}
\|J^s(fg)-f(J^sg)\|_{L^p}\lesssim \|...
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How to inverse a gradient?
Let $\alpha:\mathbb R^n\longrightarrow \mathbb R,\,\, X:\mathbb R^n\longrightarrow \mathbb R^n,\,\,Y :\mathbb R^n\longrightarrow \mathbb R^n\,\,$ be three $\mathcal C^\infty$ maps.
We define for all ...
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How to invert linear operators with functional coefficients?
So in this episode we described a way to invert linear operators with constant coefficients, that is operators of the form $O[f] = c_0f + c_1 f' + c_2 f'' ... = \sum_{n=0}^{\infty} c_n f^{(n)}$.
Now ...
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Is it feasible to use Operator Calculus to solve for $f(t)=\underbrace{\exp(\exp(\dots\exp}_{t}(0)\dots))$ over $\mathbb{R}$
Consider the function $f(t)=\exp^{(t)}(0)$ where $\exp^{(0)}(0)=0$ and $\exp^{(t+1)}(0)=\exp(\exp^{(t)}(0))$. That is,
$$f(t)=\underbrace{\exp(\exp(\dots\exp}_{t}(0)\dots)).$$
Such a function is not ...
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Does the zeta-regularized functional determinant coincide with its formal power series?
Let's say I have a differential operator $D=-\partial^2-A\cdot \partial+m^2$, acting on scalar fields $\mathbb{R}^d \stackrel{\phi}{\longrightarrow} \mathbb{R}$. Does the following equality hold? And ...
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What is the resolvent of the Laplacian operator?
If $\Delta$ is the Laplacian operator then $\Delta u=\mathcal{F}^{-1}(-|\xi|^2\widehat{u}(\xi)),\, \xi\in\mathbb{R}^n$
Question 1. For what functions is this representation valid? (I know that it is ...
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Is it okay to relabel tensor indices as long as the overall contravariant and covariant indices on each term in a tensor expression are the same?
I wanted to ask this in my previous question as I thought it was more appropriately placed given that all the context is present there, but a fellow user has suggested that it is better to ask it as a ...
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On the definition of the symbol of a differential operator
Let $E, F \to M$ be two vector bundles based at $M$ and $\pi: T^*M \to M$ the cotangent bundle. For any $\ell \in \mathbb Z$, we set
$$\text{Smb}_\ell (E, F) = \{\sigma \in \text{Hom}(\pi^*E, \pi^*F)~...
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Infinitesimal generator's domain
I am reading a book on diffusion process and I noticed that there is a concept about semi group approach and I feel unclear about this: what is the domain of the generator $A$? Or how to describe it ...
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obtaining general formula for permutation and shift operator $P$
I want to find some general formula for the following procedure
\begin{align}
A_x \mapsto A_x + \frac{\partial}{\partial A_x}
\end{align}
For example
\begin{align}
&A_{x} B_y = A_{x} B_{y} + B_{y,...
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Is $\{\text{polynomials}\}\ni p\mapsto p'(S)\in B(\ell^{2})$ a bounded operator, where $S$ is the forward shift on $\ell^{2}$?
Let $\mathcal{P}$ be the space of all polynomials endowed with the norm
$$\|p\|:=\sup_{|z|\leq1}|p(z)|=\sup_{|z|=1}|p(z)|.$$
Let $S$ be the forward shift operator on $\ell^{2}$, that is,
$$S(x_0,x_1,\...
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Convergence of closed formula for Moyal product
It is often stated that the Moyal product obeys the following identity
$$
\left(f_{1} \star_{\hbar} f_{2}\right) = m \circ e^{\frac{i \hbar}{2} \Pi}(f_1 \otimes f_2)= \sum_{k=0}^{\infty} \frac{(-i \...
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The formal adjoint of higher-order covariant derivatives
Let $\nabla$ be a Levi-Civita covariant derivative on a riemannian manifold $(M,g)$ and let $\nabla^E$ be a covariant derivative on a vector bundle $E$ over $M$. We can understand $\nabla^E$ as a map
\...
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Issues working with higher order Gâteaux derivatives
For normed real vector spaces $(X,\|\cdot\|_X),(Y,\|\cdot\|_Y)$, a function $f : X \to Y$, and $a \in X$, consider the following definitions.
Directional and Gâteaux Derivative: Let $h \in X$. If the ...
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Basic Definition of Diff Operator
I'm looking at up at how to properly define a differential operator, that is, explicitly write it as a mapping between function spaces.
Well, Wikipedia defines it by setting a mapping $A$ between ...
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Relation between the Laplacian, Dirac Distribution, and the Hessian Matrix [duplicate]
Edit: I've tried to make the question a lot simpler.
The Laplacian of the function $1/r$ is $-4\pi\delta(r)$. If I take the trace of the Hessian matrix of the function $1/r$, I find that the trace is ...
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Justifying a Numerical Technique for PDEs with Far-Field Boundary Conditions
I’m currently working on a problem which is governed by a PDE of the form $D_{r,t}f = F$ where $F$ is known, $D_{r,t}$ is a linear partial differential operator which is first-order in $t$ and second-...
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Deficiency indices of a differential operator
Consider the unbounded operator $D_t \equiv -i\frac d{dt}$ acting on $C^\infty_0(\mathbb R_+) \subset L^2(\mathbb R_+)$. Let $P = \sum_{k = 0}^m p_k \tau^k \in \mathbb R[\tau]$. Let $A = P(D_t)$. I ...
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Positive definite matrix and Hormander Theory
Let $\varphi \in C_0^\infty$, $\varphi \neq 0$.
We'll consider the inner product in $L^2.$
Let $\alpha ,\beta$ be multi-indices, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set$$\varphi _{...
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Formal adjoint of the exterior derivative on a manifold with boundary
Let $(M,g)$ a compact Riemmanian manifold with boundary. I want to define the formal adjoint of the exterior derivative $\mathrm d\colon C^\infty(M)\to \Omega^1(M)$.
If $\partial M=\emptyset$ and $\xi$...
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Is $x+\frac{d}{dx}$ differential operator diagonalizable?
I would like to prove or disprove the following statement:
Let $T=x+\frac{d}{dx}$ be a differential operator. Then $T$ is diagonalizable under some initial/boundary condition.
Here is my attemp to ...
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Elliptic operators on bounded open sets
Let $L$ be a differential operator of order $k$ defined on a bounded open subset $U$ of a Riemannian manifold $M$ that is elliptic when restricted to a smaller open subset $V$ such that $\bar V\...
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