Questions tagged [differential-operators]
Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.
558 questions
2 votes
1 answer
99 views
Smoothness of the differential on the group of diffeomorphisms over a compact Riemannian manifold
Let $(M, \langle \cdot, \cdot \rangle)$ be a closed Riemannian manifold (compact without boundary) and consider $\mathcal{D}^s$ the group of diffeomorphisms from $M$ to $M$ of class $H^s$ ($s$-th ...
- 135
1 vote
1 answer
181 views
An exact sequence of jet space associated to a vector bundle
Suppose $L$ be a line bundle over Riemann surface $X$. Then show that $ 0 \longrightarrow J^2(L) \longrightarrow J^1(J^1(L)) \longrightarrow L\otimes K_X \longrightarrow 0 ,$ where $J^k(L)$ is the $k$-...
- 29
2 votes
3 answers
305 views
Reference for relation between $ \left( \frac{1}{x} D \right)^{n+1} $ and the Bessel coefficients
Background In the OEIS sequence A001498, the coefficients of the Bessel polynomials are described. They adhere to the formula $$ a(n, k) = \frac{(n+k)!}{\left(2^k \ (n-k)! \ k! \right)} \tag{1} \label{...
- 5,025
8 votes
1 answer
462 views
Tomonaga-Schwinger evolution equation: Rigorous setting?
Is there any rigorous setting for formulating and studying Tomonaga-Schwinger equations? In quantum field theory (QFT), in particular in quantum electrodynamics (QED), the dynamic evolution in time is ...
- 332
3 votes
0 answers
105 views
Length of a formally exact compatibility complex for linear differential operator
Let $\Delta:\Gamma(E_0)\rightarrow\Gamma(E_1)$ be a linear differential operator (LDO) between vector bundles $E_0,E_1\rightarrow M$ (everything is assumed smooth by default, $\dim M=m$, and LDOs are ...
- 3,402
5 votes
0 answers
152 views
On Schuricht and Schönherr approach for proving the divergence theorem on general Borel sets
This question stems from a ZBmath search I did yesterday evening, and it is somewhat related to the following MathOverflow question: "On which regions can Green's theorem not be applied? ". ...
- 6,830
0 votes
0 answers
62 views
Commutator estimates for non-local operator
I am looking for a commutator estimate for a non-local operator: namely, for functions $f,g :\mathbb{R}^2\rightarrow \mathbb{R}$ (nice enough functions) can we find a commutator estimate of the form $$...
- 101
1 vote
0 answers
126 views
Norm of the resolvent of the Laplacian acting on $L^p$
Let $M$ be a closed Riemannian manifold and let $\Delta$ be the positive Laplacian acting on functions on $M$. If $\lambda$ is not an eigenvalue of $\Delta$, we have the resolvent operator $R_\lambda=(...
- 374
-4 votes
1 answer
211 views
$L^p$ operator norm decay
Assume $f$ and $g$ are smooth compactly supported functions on the real line and $A$ and $B$ are linear self-adjoint operators on $L^2(\mathbb{R}^n)$. If the operators $A$ and $B$ commute is it ...
- 1
0 votes
0 answers
38 views
Decay of fundamental solution of differential operator
How can we find the decay of the fundamental solution of the 1D operator $L=D_x^{\alpha}+1$ for a real number $2>\alpha\geq 1$ where the differential operator $D_x^{\alpha}$ is defined by the ...
- 101
2 votes
0 answers
96 views
Inequality of Peetre for a positive operator
How can one prove the interpolation inequality of Peetre for a positive operator, stated as: $$ A^s \leq C A^t + C' I, \quad \text{for } 0 \leq s \leq t, $$ where ( A ) is a positive self adjoint ...
- 571
1 vote
0 answers
86 views
Estimation of Schatten norm
Let $L$ be the Laplacian operator on the Heisenberg group $\mathbb{H}^n$. The Fourier transform on this group is defined as follows: for $f \in L^1(\mathbb{H}^n)$, the Fourier transform $\widehat{f}(\...
- 571
1 vote
0 answers
83 views
Proof of codifferential identity [closed]
Let $(M,g)$ be a Riemannian manifold, $\omega \in \Omega^p(M)$ is a $p$-form and $D$ a linear connection in $T(M)$ such that $D(g)=0$ and is symmetric (this gives us the formula $D\wedge\omega=d\omega$...
- 111
9 votes
0 answers
1k views
About the operator $(x\mathrm{I})^n$ where $\mathrm{I}$ is integration
In the framework of operational calculus, Grunert's formula expresses powers of the operator $xD$ as: $$ (xD)^{n} = \sum_{k=0}^{n} S(n,k) \, x^{k} D^{k} \tag{1} $$ where $S(n, k)$ are the Stirling ...
2 votes
0 answers
139 views
Regarding Rankin-Cohen bracket of two modular forms
How does U-operator act on Rankin-Cohen brackets of two modular forms $[f_{1},f_{2}]$ where $f_{1}, f_{2} \in M_{k_{i}}(N, \chi)$ for $i \in \{1,2\}$? Here, $U_{m}(f) = \frac{1}{m} \displaystyle{\...
- 95
4 votes
0 answers
110 views
A differential operator on the unit sphere bundle of an Einstein Kähler manifold
I am interested in a certain differential operator on the unit sphere bundle of a Riemannian manifold, mentioned in the seminal paper by Alfred Gray. $\square_\psi$ is defined below: I am interested ...
- 1,146
0 votes
0 answers
62 views
Can fractional Laplace operators and divergence operators be commuted?
If $u$ is a smooth function, by direct calculation, we obtain that if $\nabla\cdot u=0$, then $\nabla\cdot (\Delta u)=\Delta (\nabla\cdot u)=0$. A natural question arises: If $\nabla\cdot u=0$, then ...
- 11
7 votes
1 answer
293 views
Ground states of Schrödinger operators via the Fokker–Planck equation
I will introduce the problem set-up, but I assume most of this is likely well-known by the reader. Apologies if this is elementary, the main questions can be found at the bottom. The Fokker–Planck ...
- 171
5 votes
0 answers
73 views
Generators and relations for invariants of the Weyl algebra
Let $k$ be an algebraically closed field of zero characteristic, and $A_n=k\langle x_1, \ldots, x_n, \partial_1, \ldots, \partial_n \rangle$ the rank n Weyl algebra, which can also be described as a ...
- 3,758
2 votes
0 answers
122 views
Reference request on Lindblad equation
Suggested by Willie Wong, I reformulate my question posed Reference request : differential equation with operator valued unknowns to a concrete case. I wish to emphasise that I look for the references ...
- 1,701
2 votes
1 answer
159 views
Boundness of purely imaginary powers of Bessel potential
Let $s\in \mathbb{C}$ with $0\leq \operatorname{Re}s<\infty$, $f\in \mathcal{S}'(\mathbb{R}^n)$ and $\xi\in \mathbb{R}^n$. We define the Bessel potential $\Lambda_s$ of order $s$ of $f$ as $$\...
3 votes
0 answers
101 views
When is a surface with a metric $g$ of modified covariant Hessian type?
A surface $(M^2,g)$ is said to be of (locally) Hessian type when there exists coordinates $(x^1,x^2)$ such that there exists a function $f(x^1,x^2)$ satisfying $$g = \frac{\partial^2 f}{dx^i dx^j} dx^...
- 193
6 votes
1 answer
821 views
Birth of "crystals" - Grothendieck's quote: “I’m reading Manin’s ... I think one should introduce the notion of slope, and Newton polygon”
The must-read article on Grothendieck's epoch: "Reminiscences of Grothendieck and His School" by Luc Illusie ("chatted by the fireside, recalling memories of his days with Grothendieck&...
- 26.3k
1 vote
0 answers
104 views
Reference request - Fourier multiplier of vector valued function
I would like to understand the concept of multiplier for vector valued functions and find appropriate references for the multiplier theorems out there. For instance say that we would like to express $\...
- 133
0 votes
0 answers
130 views
Curl-Div equation with singular matrix
I want to solve the equation: $$ \begin{cases} \nabla \times (A \mathbf v)=f, \quad x\in \Omega \\ \operatorname{div}(\mathbf v)=0, \end{cases} $$ where $\Omega \subset\mathbb{R}^n$, is an open set, $...
- 617
2 votes
0 answers
79 views
Can one explicitly define a right inverse for a convolution operator on the space of entire functions?
A result of Meise and Taylor in 1988 shows that every non-zero convolution operator on the Frechet space $H(\mathbb{C})$ of all entire functions on $\mathbb{C}$ has a continuous linear right inverse $...
0 votes
0 answers
205 views
Generalized Laplacian
I was wondering if any of you had ever encountered operators on $L^2(\mathbb{R}^2)$ of the form $$ \nabla \cdot (A(x)\nabla) $$ where $A(x)$ is some symmetric matrix field (viewed as $L^2(\mathbb{R}^{...
- 41
0 votes
0 answers
87 views
In what sense is a change of boundary conditions a finite rank perturbation?
Also asked on MSE (https://math.stackexchange.com/questions/4987654/in-what-sense-is-a-change-of-boundary-conditions-a-finite-rank-perturbation and https://math.stackexchange.com/questions/4875398/how-...
- 121
5 votes
2 answers
266 views
Showing an operator is (or not) closed on $L^2(\mathbb{R})$
I am linearizing nonlinear waves and get operators of the form below. Everything is considered in $L^2(\mathbb{R})$. Consider the operator $L_1=\frac{d}{dx}$. The domain is $H^1(\mathbb{R})$ and it is ...
6 votes
0 answers
157 views
Schwartz kernel of spectral projection of Laplacian and integrated density of states
I'm reposting here a question I asked on MSE which did not receive an answer. I am considering the Dirichlet Laplacian $\Delta$ on some smooth domain $U$. For now assume that $U$ is bounded, and later ...
- 251
1 vote
0 answers
214 views
The tensor product of two Fredholm operators
What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of ...
- 302
6 votes
1 answer
428 views
Hochschild cohomology and differential operators
The Hochschild-Kostant-Rosenberg theorem says, that for a commutative algebra $R$ over a field $k$ with certain smoothness and finiteness, we have an identification $\mathrm{HH}^\bullet(R)\cong \...
- 1,095
5 votes
0 answers
270 views
Relations between two Schwartz kernels in dimensions $n$ and $n+1$
Let $(M,g)$ be an $n$-dimensional pseudo-Riemannian manifold and $\Box_g$ be the Laplace-Beltrami operator on $M$. Consider $z \in \mathbb{C}$ such that $\mathrm{ Im}(z)>0$, and we define $P_0 := \...
- 339
4 votes
0 answers
148 views
Mean-value type property for eigenfunctions of Casimir operator on $\operatorname{SL}(2,\mathbb{R})$
The purpose of this post is to ask for the community's opinion on a conjecture about a certain mean-value property for functions on $\operatorname{SL}(2,\mathbb{R})$. This conjecture appears at the ...
- 1,769
5 votes
0 answers
305 views
Differential equation involving Casimir operator on $\operatorname{SL}(2,\mathbb{R})$
Throughout this post, we let $G$ denote the Lie group $\mathrm{SL}(2,\mathbb{R})$. For $t,\theta \in \mathbb{R}$ we define the following elements of $G$: \begin{align*} A(t) &= \begin{bmatrix}e^t &...
- 1,769
3 votes
0 answers
175 views
Analytic analogue of implicit functions for differential operators
Let $p\colon \mathbb{R}^2 \to \mathbb{R}$ be a polynomial with a non-vanishing gradient at $p^{-1}(0)$. Then, the implicit function theorem says that $S = \{(x,y) \in \mathbb{R}^2 \mid p(x,y) = 0\}$ ...
- 515
0 votes
0 answers
134 views
Question on notation for definition of symbol of differential operator
I was looking at this definition of the symbol of a differential operator, and am unsure what "$T^*X\otimes_XE$" means. I couldn’t find an explanation anywhere on nlab either. My main ...
3 votes
1 answer
201 views
Deriving differential equation from difference of PDE solutions
This is an edited cross-post from Math SE because after several days it's received no good answer. I think it's less appropriate for a general QA Math site and is likely better for Overflow with ...
- 33
15 votes
2 answers
1k views
Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?
Fix a compact Riemannian manifold $M$ (leaving the metric implicit). What I'd like to know is if the corresponding Hodge decomposition of smooth $n$-forms $$ \Omega^n(M) \simeq \mathcal{H}^n(M)\oplus ...
- 36.8k
1 vote
0 answers
99 views
Quantisation of shifted cotangent bundles
The cotangent bundle $T^*X$ of a smooth space $X$ quantises (e.g. in the deformation quantisation sense) to the sheaf $D_X$ of differential operators on $X$. What is the analogous quantisation of the ...
- 6,153
2 votes
1 answer
185 views
Non-holonomic modules for $\mathcal{D}(X)$, where $X$ is an affine open subspace of the affine space
Let $k$ be any algebraically closed field of zero characteristic. Let $A_n$ be the n rank Weyl algebra, and $M$ a finitely generated module. We have that $GK(M)$ is always a positive integer and (...
- 3,758
0 votes
0 answers
122 views
A naive looking question about Gelfand-Kirillov dimension
Let $A$ and $B$ two affine algebras, $A$ a subalgebra of $B$. If we have a left $A$-module $M$ we can extend the scalars: $B \otimes_A M$. I will denote the resulting $B$-module by $N$ How are $\...
- 3,758
2 votes
0 answers
82 views
Transform connecting powers of integration and differentiation operators
Just by a chance, I found the following power series identity, which holds for any analytic function $F(\cdot)$, nonnegative integer $m$, and constants $u,v$ not depending on indeterminates $z,t$: $$\...
- 38.3k
3 votes
0 answers
171 views
Fixed point formula of Atiyah and Singer applied to a Dirac operator on a spin manifold
Let $G$ be a compact Lie group acting by orientation-preserving isometries on a compact even-dimensional spin manifold $X$, and assume that the $G$-action preserves the spin structure of $X$, so that ...
- 427
1 vote
0 answers
86 views
Finding thin plate spline subjected to boundary conditions
I am trying to formulate a problem as a PDE. What I want to know is if my formulation is correct, if it admits solution and what am I missing. This question is related to : Thin-Plate-Spline ...
- 129
1 vote
2 answers
243 views
Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold
I've already asked this question in: https://math.stackexchange.com/questions/4899825/question-about-the-index-of-two-elliptic-operators-over-a-4-dimensional-riemanni, and I've been suggested to ask ...
- 427
2 votes
0 answers
126 views
Differential operators and iterations of tangent bundle
Is there a relationship between higher order differential operators and higher tangent bundle viewed as bundle on the base manifold?
user522448
1 vote
1 answer
114 views
Germs of left invariant differential operators on a group
Are there germs at the identity of linear differential operators on a group which are not germs at the identity of left invariant differential operators? I feel like the answer is no but the statement ...
user525442
4 votes
1 answer
204 views
Analogue of vector for differential operators
A differential operators of order one is a vector field which is defined pointwise . Differential operators of order greater than one are not. The closest analogue to a vector is given by a germ of a ...
user522448
4 votes
3 answers
567 views
Generalized Fuchsian-type PDE
Consider $$ \big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0 $$ with the initial condition $A(x,0)=1$. In a small $t$...
- 149