Questions tagged [differential-operators]

Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.

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Index of a family of operators

In the usual setting of the Atiyah-Singer index theorem the situation is as follows: we have a closed smooth manifold $M$ without boundary and $D$ is some elliptic differential operator acting on ...
truebaran's user avatar
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Positivity of semiclassical pseudodifferential operators

Let me first give some background. (My reference is Martinez's book An introduction to semiclassical and microlocal analysis) Let $m\in\mathbb{R}$, and $p(x,\xi):\mathbb{R}^n_x\times\mathbb{R}^n_\xi\...
Liren Lin's user avatar
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12 votes
0 answers
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Holomorphic natural bundles and operators

I am wondering up to what extent the classical theory of (smooth) natural bundles and natural operations extends to the holomorphic setting. After a quick thought, I've gone through the standard ...
Quidam_t's user avatar
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Differential operators acting on the Schwartz space

I asked a similar question on math stack exchange but didn't get an answer so I will try to ask it here. Any help/suggestion is most than welcome! Let $D$ be a linear differential operator with ...
Pea's user avatar
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2 votes
0 answers
122 views

Mean value operator on Riemannian manifold

Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$) Consider the mean value operator, ...
Richard's user avatar
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2 votes
1 answer
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Rellich Embedding Theorem for the $2$-Sphere

I'm trying to understand the Rellich-Embedding Theorem in the non-flat case by looking at the $2$-sphere. To be precise, for $S$ the spinor bundle of $S^2$; $L^2(S^2)$ the space of square integrable ...
Pavel Katzo's user avatar
5 votes
1 answer
177 views

Getting out a system of linear ODEs by knowing the Magnus expansion

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients, $$Y(t_1)...
tobias's user avatar
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10 votes
2 answers
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Atiyah Singer index theorem and Hodge de Rham operator

When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ (...
truebaran's user avatar
  • 9,140
1 vote
1 answer
137 views

Estimate for differential of inverse map

Let $f: M \to N$ be a diffeomorphism between two riemannian Manifolds. Suppose there exist constants $0 < c \leq C$ such that for all $p \in M$, we have $c \leq |df_p| \leq C$. Here, $df$ denotes ...
H1ghfiv3's user avatar
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5 votes
0 answers
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The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$. We consider the following polynomial vector field on ...
Ali Taghavi's user avatar
4 votes
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Ring of SO(n)-invariant differential operators on M_n,m

I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/). There comes a point in the paper (Lemma 2.8) ...
Ken Schefers's user avatar
2 votes
0 answers
183 views

$\eta$-invariants of Riemann Surface

I am curious about a concrete computation of $\eta$-invariants for Riemann surface, e.g. Torus. Is there any nice review or notes talking about the computation? Or is it possible to express it as ...
Yingfei Gu's user avatar
7 votes
0 answers
181 views

$\eta$ invariants of Pin+ manifolds $\mathbb{RP}^{8k}$

In general, I am curious about the 'quantization' of $\eta$-invariants on Pin+ manifold, i.e., what is the 'minimal unit' of $\eta$-invariants on a manifold with certain choice of Pin+ structure. ...
Yingfei Gu's user avatar
4 votes
1 answer
325 views

Differential Operators On A Curve And On Osculating Circle

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
Vlad's user avatar
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Differential and pre-differential of Jacobi identity

Let M be a manifold. To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied? That is a Lie algebra structure for which $[X,fY]=f[X,Y]$. (For every ...
Ali Taghavi's user avatar
7 votes
1 answer
1k views

Chain rule for weakly differentiable functions

Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap ...
Mark Peletier's user avatar
6 votes
3 answers
652 views

A possible trace (inequality) defined under negative Sobolev scale

Why is that not possible (if it is not) to define the trace of a function in a very weak regularity space $H^{-s}(\partial \Omega)$? We usually encounter trace theorem as $$Tr: H^{s}(\Omega) \...
mediocre's user avatar
3 votes
1 answer
331 views

derivative of the adiabatic limit of the eta invariant

To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ...
Ho Man-Ho's user avatar
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3 votes
2 answers
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Density of Laplace image

Let $D$ define the differential operator $-\frac{\partial^2}{\partial x^2}$ on $\mathbb R$. Let $\xi\notin\mathbb R$ be a complex number. Is it true that $$ (D-\xi)C_c^\infty({\mathbb R}) $$ is dense ...
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1 vote
1 answer
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Ask the validity of Tauberian lemma in Sogge's book

In C.D.Sogge's Fourier Integrals in Classical Analysis pp.128-129, he proved Lemma4.2.3(Tauberian Lemma): Lemma. Let$g(\lambda)$ be a piece-wise continuous tempered function of $\mathbb{R}$. Assume ...
Henry.L's user avatar
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2 votes
0 answers
62 views

what and where is Dubrovin's tri-differential operator?

IFF memory serves, Dubrovin made good use of a tridiffereential operator long before H & R fluxes. What did he use it for and where?
Jim Stasheff's user avatar
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2 votes
1 answer
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Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed]

I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over $\mathbb{R}^3$...
user74375's user avatar
26 votes
2 answers
2k views

Harmonic spinors on closed hyperbolic manifolds

Does anyone know an example of a closed spin hyperbolic manifold of dimension 3 or greater such that the kernel of the Dirac operator is non-trivial? I'm mainly interested in the 3-dimensional case ...
Danny Ruberman's user avatar
1 vote
1 answer
1k views

Operator theory of the Hessian

How can I learn more about the operator theory of the Hessian? The Hessian of a function $u : \Omega \rightarrow \mathbb R$ over a domain $\Omega \subseteq \mathbb R^n$ is the matrix of second ...
shuhalo's user avatar
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1 vote
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74 views

Recursive formula for symbol of resolvent on noncompact manifold

On a compact Riemannian manifold $(M,g)$ without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator $A$ of positive order $m$: ...
harlekin's user avatar
  • 313
3 votes
1 answer
358 views

A Poincare-Type Inequality and its generalization

Let $f(\theta)$ be a fixed positive $2\pi-$periodic $C^1$ function on $\mathbb{R}$ with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$ Does for any $2\pi-$...
Alfred Chern's user avatar
22 votes
2 answers
1k views

Are there any natural differential operators besides $d$?

Let $\lambda = (\lambda_1, \ldots, \lambda_r)$ and $\mu = (\mu_1, \ldots, \mu_r)$ be partitions such that $\mu_j = \lambda_j +1$ for one index $j$ and $\mu_i = \lambda_i$ for all other $i$. Then there ...
David E Speyer's user avatar
1 vote
2 answers
327 views

Motivating the Bessel translation operator

In a paper I am reading on the Hankel transform (this paper to be exact), I've come across a somewhat peculiar definition for a generalized translation operator. The operator is designed with a ...
Cameron Williams's user avatar
3 votes
2 answers
567 views

Diffusion on a semi-Riemannian manifold?

A great deal of literature exists on the heat equation and heat kernel for a Riemannian manifold. The Laplace-Beltrami operator in the given metric replaces the flat Laplacian in the heat equation, ...
Idempotent's user avatar
2 votes
0 answers
586 views

Estimates of eigenvalues of elliptic operators on compact manifolds

The classical Weyl law says that if $\Delta$ is the Laplace operator on functions on a compact Riemannian manifold $(M^n,g)$, $n>2$, then its $k$th eigenvalue satisfies the asymptotic formula $$\...
asv's user avatar
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2 votes
0 answers
455 views

Discrete p-Laplacian

One of the definitions of the discrete (weighted) $p$-Laplacian is the following: $$\Delta_{p,w}u(x):=\sum_y |u(y)-u(x)|^{p-2}(u(y)-u(x))w(x,y).$$ Consider the one dimensional case. Then the free ...
scouser's user avatar
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3 votes
1 answer
272 views

adjoint of this closed (?) operator

I am currently dealing with an unbounded operator $T:\{f \in L^2(-2\pi,2\pi); f \in AC((-2\pi,2\pi)), T(f) \in L^2, \lim_{x \rightarrow \pm 2 \pi} f(x)g(x)=0\} \subset L^2(-2\pi,2\pi)\rightarrow L^2(...
Antonio Rapallino's user avatar
2 votes
1 answer
136 views

Proper domain for operators

in this paper on arxiv in equation 27, two operators $$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$ and $$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + \frac{mx}{\sqrt{1-x^2}...
Sergei Damyan Zhivkov's user avatar
1 vote
1 answer
203 views

Functional Calculus and Fredholm index

Let $-\Delta: W^{2,2} \subset L^2(\mathbb{S}^2) \rightarrow L^2(\mathbb{S}^2)$. Then it is "easy" to show that $-\Delta $ is self-adjoint. Now, I am looking for closed operators $T$ and $T^*$ of order ...
Ethan Co's user avatar
19 votes
2 answers
4k views

Exact Definition of Dirac Operator

Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator ...
Jjm's user avatar
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1 vote
1 answer
425 views

Cauchy problem for an overdetermined system of PDE

This question is strictly related to this one. Let us consider the differential system with constant coefficients $$\left(\begin{array}{ccc} B_{11} & B_{12} & 0\\ ...
guido giuliani's user avatar
2 votes
1 answer
208 views

Heat Kernel estimate at the level of the form

Let $(M,g)$ be a compact Riemannian manifold. It is known there exist Gaussian estimates of the heat kernel and its derivatives acting on functions on $M$. The kind of estimate I'm looking for could ...
user avatar
0 votes
1 answer
555 views

How to solve this differential equation with an infinite sum?

I would like to find solutions of the following differential equation: $ \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x)$ For example in space of function from $\mathbb R^*$ to $\mathbb ...
Bertrand's user avatar
  • 1,121
2 votes
1 answer
233 views

Local fractional Sobolev inequality

If $\mathcal{X}$ is a smooth cutoff near 0 in $\mathbb{R}^n$, then $M_0 = \mathcal{X}(-\Delta+Id)\mathcal{X}$ is a self-adjoint operator in $L^2(\mathbb{R}^n)$. Because $M_0$ is semi-positive and the ...
LI Jialun's user avatar
16 votes
4 answers
3k views

Green's operator of elliptic differential operator

Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
mdg's user avatar
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2 votes
0 answers
110 views

"simulteneous eigenvectors" under the full set of weighted Laplacians on a $g$-fold product of the Poincare half plane

This question is closely related to the following MO question Characterizing the real analytic Eisenstein series Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed ...
Hugo Chapdelaine's user avatar
8 votes
1 answer
2k views

Killing vector fields on sphere

Let $u$ be a smooth function on $\mathbb S^2$, and assume that for every killing vector field $V$ on $\mathbb S^2$. $$\int_{\mathbb S^2} V(u) x_j dS=0\text{,}\forall j=1,2,3$$ Is $u$ necessarily ...
Matchmaticians's user avatar
10 votes
1 answer
328 views

When are the Dolbeault and de Rham dgas homotopy equivalent?

Let $M$ be a compact Kahler manifold. Then the Hodge decomposition says that the Dolbeault dga (of forms of all bidegree) and the de Rham dga on $\Omega_{\mathbb C}^\bullet(M)$ have isomorphic ...
Eric O. Korman's user avatar
1 vote
0 answers
170 views

Existence of solution?

I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here. Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf Let $\mathcal{...
Nguyen's user avatar
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2 votes
0 answers
108 views

Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...
Ali Taghavi's user avatar
4 votes
1 answer
658 views

Horizontal lift of differential operator

On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that $X^{\mathrm{hor}}$ is a ...
Matthias Ludewig's user avatar
2 votes
0 answers
593 views

Is Laplacian a surjective operator?

For a closed manifold the laplacian is almost surjective operator since the index of $\Delta$ is zero and there is no a non constant harmonic function. So the codimension of the image ...
Ali Taghavi's user avatar
5 votes
1 answer
615 views

The "Rolle theorem" for sections of a vector bundle

1) Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
Ali Taghavi's user avatar
11 votes
2 answers
1k views

Elliptic operators corresponds to non vanishing vector fields

Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting ...
Ali Taghavi's user avatar
8 votes
0 answers
355 views

Noncommutative geometry and line length

I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds ...
Jon's user avatar
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