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

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Definition of formal adjoint of covariant derivative

I read in Einstein Manifolds, L. Besse that the covariant derivative $D: \mathcal{J}^{(r,s)}(M)\to \Omega^1(M)\otimes\mathcal{J}^{(r,s)}(M)$ admit an formal adjoint $D^*:\Omega^1(M)\otimes\mathcal{J}^{...
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58 views

why $\varphi''\in L^{2}(R)$ [on hold]

I have the following question: Let $T_{c}$ be an unbounded operator with domain $D(T_{c})=\{u\in L^{2}(R), T_{c}(u)\in L^{2}(R)\}$. If $\forall \varphi \in \mathcal{C}^{\infty}_{0}(R): \|\varphi''\...
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1answer
199 views

The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]

In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...
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47 views

Domain of operator

Let be $\lambda\in C^{*}$. Consider the following operator: $ T_{\lambda}=-\Delta_{R^{2}}++\frac{\lambda^{2} }{4} (x^{2}+y^{2})+i\lambda N$, where $N=(x \frac{d }{dy} -y \frac{d }{dx})$ , ...
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1answer
121 views

a condition for Laplacien

Let $u\in L^{2}(R^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(R^{2})$ where $c>0$. Is true $-\Delta u \in L^{2}(R^{2})$? Thank you in advance.
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123 views

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 $S$ $\Big<\psi,T_{a}\varphi\...
1
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1answer
163 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 $Hf(x)=(-\frac{d^{2}}{dx^{2}}+cx^{2})...
2
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1answer
228 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 X\|^...
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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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
127 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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0answers
75 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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86 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
121 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 ...
3
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1answer
82 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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83 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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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 $\...
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1answer
83 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 $y=\ln(2)-2\ln(\cosh(cx+d))+2\ln(c)$, $c,d \in R$...
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1answer
156 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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208 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 $\mathcal{I} = \ker \triangle^{\flat}$. Definition:...
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96 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
105 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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787 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) ...
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102 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 $z=x+ i y \in \mathbb C$ ...
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89 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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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 ($n>1$...
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161 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. ...
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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?
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2answers
121 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 $p(x,\xi):\mathbb{R}^n_x\times\mathbb{R}^n_\xi\...
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74 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 ...
5
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127 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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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 ...
4
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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, $$Y(t_1)...
8
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2answers
701 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 $d+d^*$ (...
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1answer
96 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 ...
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36 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 ...
5
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90 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) ...
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87 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 ...
5
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103 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. ...
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1answer
182 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 ...
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48 views

Equivalence of fractional power of second-order positive differential operator as pseudodifferential operator and a fractional definition

Let $A$ be a second-order differential operator on a closed manifold $M$ satisfying $$(Au,u) \geq 0$$ with $A=-\Delta$ the Laplace-Beltrami the model case. One can define for $s \in (0,1)$ the ...
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385 views

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 ...
6
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1answer
260 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 ...
5
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1answer
150 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 ...
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 ...