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

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Weyl algebra acting on a polynomial ring

Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots, x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl algebra. As usual $W$ ...
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21 views

how can we extend this result [duplicate]

Let $T_{a},a\in C$ be a closed operator defined on $D$ subspace of $L^{2}(R)$ onto $L^{2}(R)$ $(T_{a}: D\rightarrow L^{2}(R) )$ with $D$ contains a Schawrz space $S$ ...
1
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0answers
266 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 ...
1
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1answer
112 views

analytic continuation argument

In "Pseudo-spectra, the harmonic oscillator and complex resonances" (login required), the author says Sections $2$ and $3$ of this paper concern the operator ...
21
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1answer
744 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 ...
2
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1answer
222 views

Relation between Harmonic vector field and Harmonic 1-form

Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function $$E(X)=\frac{1}{2}\int_M\|dX\|^2dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla ...
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0answers
58 views

inverse of operator [migrated]

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 ...
4
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1answer
80 views

Zeroes of global sections killed by differential operators

I asked this question some two weeks ago on StackExchange, but received no feedback of any sort ... Let $X$ be a compact connected Riemann surface and let $\Phi:M\rightarrow N$ be an elliptic ...
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0answers
80 views

A specific type of first-order nonlinear ordinary differential equation

I am trying to divide $\mathbb{R}^+\times \mathbb{R}^+$ into some curves so that the integration of the function $ h(x)h(y) $(where $h(x)$ is a $C^1$ function from $\mathbb{R}^+\to \mathbb{R}^+$ that ...
8
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1answer
121 views

Generalized Dirac operators

So far I met three definitions of the so called generalized Dirac operator(or Dirac type operators. Everything takes place over Riemannian manifols $M$ and we have smooth hermitian vector bundle $S ...
2
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74 views

How can I prove that this D-module is free?

I have the following setup, I expect that it is studied in the theory of $D$-modules, and I apologize in advance if I am wrong. First, I have an algebra $A$ of differential operators on $n$ ...
5
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83 views

Spectra of Dirac operators

1. Suppose that $M$ is a spin manifold. The spin structure of $M$ is not uniquely defined: in other words, $M$ may have many nonequivalent spin structures. For each choice of spin structure there is ...
2
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1answer
120 views

Local form of Dirac operator

I have the following question. Provided that $E\to M$ is vector bundle and that a Clifford module $Cl(T^*M)$ acts on $\Gamma(E)$ via Clifford multiplication $c$, the Dirac operator of this Clifford ...
2
votes
1answer
77 views

Strongly continuous semigroups and symbols of pseudo differential operators

I am considering the Cauchy IVP for the evolution equation $$u_t + \Psi u =0$$ where $\Psi$ is a linear pseudo differential operator with symbol $\widehat{\Psi}\left(\underline{\xi}\right)$. The ...
2
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2answers
136 views

Finding a specific Global Smooth Function

Any help with this problem would be appreciated. Thanks Suppose $(M^3,g)$ is a smooth compact Riemannian manifold with smooth boundary and $\gamma$ is a simple smooth orientable curve in $M$. Does ...
2
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0answers
82 views

Lie algebra (co)homology of the Lie algebra of differential operators

Let $X$ be an algebraic variety (or a complex manifold) over $\mathbb C$. Let $D(X)$ be its algebra of differential operators. Mostly I am interested in algebraic differential operators, but the case ...
5
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0answers
82 views

Analytic continuation of $\mathfrak{so}(n)$ algebras to real $n$?

In a 1988 paper "The Lie algebras $\mathfrak{gl}(\lambda)$ and cohomologies of Lie algebras of differential operators", Feigin defined the analytic continuation of $\mathfrak{sl}(n)$ algebras (over ...
0
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1answer
81 views

Explicit solution for one-dimensional Gelfand problem

I wonder if the ODE $y''+e^{y}=a$ can be solved explicitly. For $a=0$, it is well-known that there is a two-parameter family of explicit solutions $y=\ln(2)-2\ln(\cosh(cx+d))+2\ln(c)$, $c,d \in ...
4
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1answer
152 views

Elliptic Operators on Vector Bundles

I know the kernel of an elliptic operator on a compact manifold has finite dimension. Is the kernel of an elliptic operator on sections of a vector bundle a finite dimensional space?
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773 views

Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are: Definition 1 ("naive"): Let $X$ be a (real) ...
6
votes
1answer
255 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 ...
8
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205 views

Is the formal neighborhood of the diagonal a generalization of the Jet bundle?

Let $f: X \to S$ be a morphism of locally ringed spaces and $\triangle: X \to X \times_S X$ the corresponding diagonal morphism with kernel sheaf $\mathcal{I} = \ker \triangle^{\flat}$. ...
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0answers
94 views

Eigenvalues of the D'Alembertian operator

My question about the spectral theory of the D'Alembertian operator on a Lorentzian manifolds (say the spacetime $M^{3+1}$) given by $$\square = -\partial_{t}^2 + \Delta$$ for the metric $g=(-+++)$. ...
4
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1answer
103 views

Lifting a differential operator

Let $D$ be a differential operator acting between the spaces of smooth sections of two vector bundles $E,F$ over compact manifold $M$. If $M$ is not simply connected one can construct the universal ...
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2answers
653 views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...
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98 views

How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
5
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88 views

Differentially closed fields

Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$. Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...
3
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0answers
106 views

Spectrum of the Laplace-Beltrami operator on a domain of finite volume in the hyperbolic space $H^n$

What is known about the ($L^2$) spectrum of the minus Laplace-Beltrami operator ($- \Delta$) with zero boundary conditions on $B =H^n/\Gamma$, where $H^n$ is $n$-dimensional hyperbolic space ...
7
votes
2answers
158 views

Existance of Integrating Factors, a Constructive Proof

Being a novice with differential equations, I recently learned that if $\mu$ is an integrating factor for $\frac{dy}{dx}f(x,y)+ g(x,y)=0$, then the corresponding 1-form, $\mu fdy+\mu g dx$, is exact. ...
7
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1answer
204 views

Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$?

Let $n\in\mathbb N$. Let $k$ be a commutative ring in which $1,2,3,\ldots$ are invertible. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., ...
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81 views

Spectrum of the Grassmannian Laplacian

The spectrum of the Laplacian (with respect to the Fubini--Study metric) was addressed in this old question. Does anyone know if these results have been extended to the Grassmannians?
4
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2answers
118 views

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 ...
4
votes
1answer
713 views

Pochhammer symbol of a differential, and hypergeometric polynomials

I have a minor result which I'm sure has come up somewhere before but I can't seem to find it. Consider a confluent hypergeometric function of the form $$\newcommand{\ff}{{}_1F_1} ...
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2answers
216 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, ...
2
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2answers
332 views

How to compute the index of such operator?

Let $M$ be a compact Riemannian manifold, with $R$ nowhere-vanishing vector field on $M$(whose orbit may be closed/ not closed). $E$ and $F$ are two vector bundle (Edit: which are sub-bundles of ...
4
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2answers
596 views

Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...
8
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73 views

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 ...
4
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1answer
179 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 ...
8
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2answers
696 views

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 $d+d^*$ ...
5
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125 views

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 ...
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0answers
83 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, ...
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35 views

Deciding whether linear equations are solvable over specific subrings of $K(x_1,..,x_n)$

The definition of 'linear equations are solvable' which is meant here is Let $R$ be a commutative ring (associative and with unity). For given $m \in \mathbb{N}$ and $b \in R$, it is decideable ...
2
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0answers
89 views

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 ...
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1answer
71 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, ...
2
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1answer
157 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 ...
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1answer
95 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 ...
5
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227 views

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 ...
19
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3answers
1k views

How we do actually compute the topological index in Atiyah-Singer?

This is migrated by math.stackexchange as I did not receive an answer. I do not know if it is too naive for this site. I am taking a lectured class in Atiyah-Singer this semester. While the class is ...
5
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1answer
149 views

[This might be a easy question]: A possible trace (inequality) defined under negative Sobolev scale

I have a stupid question: 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 ...
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88 views

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) ...