# Tagged Questions

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

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### 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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### 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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### How to calculate the derivative of logarithm of a matrix? [migrated]

Given a square matrix $M$, we know the exponential of $M$ is $$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$ and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(1-M)^k}{k}$$ The derivative of ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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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 ...
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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 ...
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### 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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### 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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### 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, ...
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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 $d+d^*$ ...
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### 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 ...
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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 ...
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### 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 ...
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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) ...
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### $\eta$-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 ...
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### $\eta$ 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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### 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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### 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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### 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 ...
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 $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap ... 1answer 140 views ### [This might be a easy question]: A possible trace (inequality) defined under negative Sobolev scale I have a stupid question: Why is that not possible (if it is not) to define the trace of a function in a very weak regularity space$H^{-s}(\partial \Omega)$? We usually encounter trace theorem as ... 1answer 149 views ### derivative of the adiabatic limit of the eta invariant To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ... 2answers 72 views ### Density of Laplace image Let$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 ... 1answer 138 views ### 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 ... 0answers 50 views ### comparison principle for viscosity solution to linear nonlocal equation with drift I met a problem about comparison principle for nonlocal equation when I study SDEs driven by Levy noise. Since, I have no background about PDEs, it may be a stupid question: $$Iu=\lambda ... 0answers 42 views ### what and where is Dubrovin's tri-differential operator? IFF memory serves, Dubrovin made good use of a tridiffereential operator long before H & R fluxes. What did he use it for and where? 1answer 71 views ### Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed] I have a real symmetric 3\times3 matrix \mathbf{M}(\mathbf{r}) which depends on \mathbf{r} \in \mathbb{R}^3. Each eigenvalue can be considered a scalar field e_i(\mathbf{r}) over ... 1answer 553 views ### Harmonic spinors on closed hyperbolic manifolds Does anyone know an example of a closed spin hyperbolic manifold of dimension 3 or greater such that the kernel of the Dirac operator is non-trivial? I'm mainly interested in the 3-dimensional case ... 1answer 369 views ### Operator theory of the Hessian How can I learn more about the operator theory of the Hessian? The Hessian of a function u : \Omega \rightarrow \mathbb R over a domain \Omega \subseteq \mathbb R^n is the matrix of second ... 0answers 141 views ### Intuition behind Stokes operator? I know that the definition of the Stokes operator (which appears in the functional form of the Navier-Stokes equations) is$$A = -P_L Δ$$where Δ is the Laplacian, and P_L is the Leray ... 0answers 39 views ### Recursive formula for symbol of resolvent on noncompact manifold On a compact Riemannian manifold (M,g) without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator A of positive order m: ... 1answer 206 views ### A Poincare-Type Inequality and its generalization Let f(\theta) be a fixed positive 2\pi-periodic C^1 function on \mathbb{R} with$$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$Does for any ... 2answers 737 views ### Are there any natural differential operators besides d? Let \lambda = (\lambda_1, \ldots, \lambda_r) and \mu = (\mu_1, \ldots, \mu_r) be partitions such that \mu_j = \lambda_j +1 for one index j and \mu_i = \lambda_i for all other i. Then there ... 2answers 148 views ### Motivating the Bessel translation operator In a paper I am reading on the Hankel transform (this paper to be exact), I've come across a somewhat peculiar definition for a generalized translation operator. The operator is designed with a ... 2answers 200 views ### Diffusion on a semi-Riemannian manifold? A great deal of literature exists on the heat equation and heat kernel for a Riemannian manifold. The Laplace-Beltrami operator in the given metric replaces the flat Laplacian in the heat equation, ... 0answers 163 views ### Estimates of eigenvalues of elliptic operators on compact manifolds The classical Weyl law says that if \Delta is the Laplace operator on functions on a compact Riemannian manifold (M^n,g), n>2, then its kth eigenvalue satisfies the asymptotic formula ... 0answers 141 views ### Discrete p-Laplacian One of the definitions of the discrete (weighted) p-Laplacian is the following:$$\Delta_{p,w}u(x):=\sum_y |u(y)-u(x)|^{p-2}(u(y)-u(x))w(x,y).$$Consider the one dimensional case. Then the free ... 1answer 203 views ### adjoint of this closed (?) operator I am currently dealing with an unbounded operator T:\{f \in L^2(-2\pi,2\pi); f \in AC((-2\pi,2\pi)), T(f) \in L^2, \lim_{x \rightarrow \pm 2 \pi} f(x)g(x)=0\} \subset L^2(-2\pi,2\pi)\rightarrow ... 1answer 118 views ### Proper domain for operators in this paper on arxiv in equation 27, two operators$$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}} + ... 1answer 150 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 ... 2answers 1k 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 ... 1answer 204 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\\ ... 1answer 119 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 ... 1answer 185 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 ...
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 ...