Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.
4
votes
0answers
40 views
Euler number of the complex of basic forms
Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ...
1
vote
1answer
55 views
Solving Bessel-like equation by using Bessel Kernel [closed]
Consider the drifted-Bessel equation as follows.
\begin{equation}
x^2\ddot y + x\dot y + (x^2-n^2)y=f,
\end{equation}
where $n$ is an integer and $f$ is a known function. If $f\equiv 0$, the solution ...
0
votes
0answers
44 views
Parabolic partial differential equation, initial conditions
Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
13
votes
1answer
400 views
Derivation on real analytic manifolds
Let $M$ be a real analytic manifold. By $C^{\omega}(M)$ we mean the algebra of all analytic functions from $M$ to $\mathbb{R}$. Assume that $D$ is a derivation on $C^{\omega}(M)$ .
Is there a ...
7
votes
0answers
313 views
Why should the Laplacian in $\mathbb{C}^n$ act on a specific line bundle over the quadric $x^2=0$ in $\mathbb{P}^{n-1}$?
I recently encountered the following nice fact, and I'm wondering if it's part of a more general story.
Let $x\in \mathbb{C}^n$ satisfy
$$x^2:=\sum_i x_i^2 = 0,$$
and consider functions $f(x)$ ...
0
votes
1answer
48 views
Estimate the analytical wavefront set $WF_A(u)$ given $WF_A(A_K u)$
Let $X$ and $Y$ be two real-analytic manifolds and $Z \subset X \times Y$ be a real-analytic embedded closed submanifold. Suppose now that $K \in \mathscr D'(X \times Y)$ is a distribution conormal to ...
2
votes
1answer
94 views
Practical way to check whether a distribution is conormal
Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. Hörmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that
$$
L_1 ...
1
vote
0answers
82 views
Action of Landweber-Novikov algebra on infinite polynomial ring
Denote $\text{Aut}(\hat{{\mathbb A}}^1)$ be the affine group over ${\mathbb Z}$ that sends some ring $R$ to the strict automorphisms of $R[[t]]$, i.e. those of the form $X\mapsto X + r_1 X^2 + r_2 X^3 ...
2
votes
0answers
107 views
Radon-Nikodym derivatives as limits of ratios
Let $\mu_1$ and $\mu_2$ be measures with $\mu_1 \ll \mu_2$. Suppose we can characterize (a version of) their Radon-Nikodym derivative this way:
$$\frac{d\mu_1}{d\mu_2}(x) = \lim_{n \to \infty} ...
4
votes
1answer
201 views
Question about a (relatively simple looking) differential operator and its eigenvalues
A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form
$L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$
for ...
2
votes
2answers
174 views
Schrodinger's equation via Spectral Theorem
How do you prove basic facts on the Schrodinger equation using the spectral theorem? More precisely, here is what I have in mind.
The version of the Spectral Theorem I am familiar with is the ...
3
votes
1answer
123 views
Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis?
Let $K$ be a field of characteristic zero. Let $\Omega = K[x_1, \dots, x_n, dx_1, \dots, dx_n]$ be the differential ring of algebraic differential forms over $K[X_1, \dots, X_n]$.
Is there an ...
8
votes
1answer
300 views
Atiyah-Singer for pseudodifferential operators via heat kernel?
The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. ...
3
votes
2answers
120 views
Space of differential operators
Let $A$, $B$ be two smooth vector bundles of finite rank over a smooth manifold $M$. Let $Diff(A,B)$ be the space of differential operators from $A$ to $B$. Can I talk about "the space of smooth maps ...
1
vote
1answer
349 views
Precise versions of “differential operators are unbounded but closed linear operators”
I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...
-1
votes
1answer
126 views
Stone Cech compactification for exponential map
Recently I met with a problem related to Stone-Cech Compactification theorem
in Furstenberg's famous paper "non-commuting product."
I try my best to understand Stone-Cech compactification theorem by ...
4
votes
1answer
248 views
Poisson structure on the cotangent bundle
Let $X$ be a smooth variety over $\mathbb{C}$ and $\mathscr{A}$ a sheaf of twisted differential operators on $X$. The latter comes equipped with a natural filtration and the associated graded algebra ...
13
votes
5answers
901 views
Book Recommendation - PDE's for geometricians / topologists
I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
2
votes
1answer
256 views
Mellin transform between heat kernel and zeta-function
For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of ...
1
vote
0answers
207 views
Formula for the curvature of an induced connection
Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles ...
7
votes
2answers
214 views
Fredholm alternative result for general elliptic system?
Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
4
votes
1answer
154 views
Please recommend some literature on the systematical theory of the elliptic systems!
Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
3
votes
1answer
348 views
Algorithm to find exponential map of differential operators acting on function
I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator.
Examples:
$\exp(\varepsilon ...
0
votes
0answers
128 views
Equivariant integration (localization formula)
We consider the action of $S^{1}$ on $S^{2}$ by rotation respect the vertical axes. We want to integrate the $2-$ equivariant form
$$\alpha(X)= -X\cos(\phi) +\sin(\phi)\,d\phi\,d\theta.$$
We have ...
1
vote
1answer
279 views
Combinatorics: Product Rules.
I couldn't find a way to figure this out, though it is a somewhat basic question that came up when studying the stationary phase expansion of an integral. The abstract version is the following:
I ...
1
vote
1answer
150 views
Symmetric Operators Robin Boundary Conditions
How can you show that an operator is symmetric with robin boundary conditions?
I know I need to show that < Tf,g > = < f,Tg >; however, the robin boundary conditions are throwing me off.
This ...
5
votes
1answer
290 views
Index of a differential operator between trivial bundles.
Let $M$ be a closed parallelizable manifold and $D: \Gamma(E) \to \Gamma(F)$ an elliptic differential operator between trivial vector bundles $E,F \to M$. The Atiyah Singer index theorem implies that ...
7
votes
3answers
447 views
Is there any general index theorem for manifold with boundary?
My understanding is Atiyah-Patodi-Singer solved the index theorem for manifold with boundary only for certain types of Dirac operators, correct?
There is still no (or no hope to get) uniform theorem ...
4
votes
2answers
356 views
Surface Laplace-Beltrami without coordinates, exterior calculus?
Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator ...
2
votes
1answer
197 views
Fourier transform and spectrum of PDOs in $L^p$
Let $K$ be a compact subset in $\mathbb{R}^n$ with $m(K)=0$, Suppose $supp\hat{u}\subset K$ for some $u\in L^p$,where $2\leq p\leq \frac{2n}{n-1}$,can we get $u\equiv 0$ ?
Motivation: If $K$ is a ...
3
votes
1answer
461 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
$$\newcommand{\ff}{{}_1F_1}
...
2
votes
1answer
308 views
Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?
Consider a Riemannian manifold and let
$\mathrm{id}$ be the identity operator, let
$\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let
$t > 0$ be a parameter.
Does ...
1
vote
0answers
171 views
Non-linear Perturbation Operator Examples
Consider a non-linear operator $\cal H$ which maps a function to a function (e.g., a map from a starting wave function $f(x,y,z)$ to a later wave function according to some non-linear PDE) and an ...
5
votes
0answers
110 views
Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$ ?
Let $n\in\mathbb N$. Let $k$ be a commutative ring. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., $x_n$ over $k$. (The multiplication in ...
4
votes
0answers
153 views
Differential operators that preserve real-rootedness
Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of ...
1
vote
1answer
240 views
Leibniz rule for Pseudo-differential operators of negative order
Does anyone know of some good references for a fractional Leibniz rule for pseudo-differential operators of negative order? As a specific example, I would like to compute $\partial_{x}^{-1}(uv)$, ...
0
votes
1answer
670 views
Can we construct a Hilbert space where the operator following differencial operator is symmetric?
I'd like to know if one can define a pertinent Hilbert space where the operator
$$A_p v := -\frac{1}{2} v" + (vF + v\int_\mathbb{R} Sp + p\int_\mathbb{R} Sv )'$$ is symmetric. Here, $p$ satisfies ...
5
votes
3answers
274 views
Criteria for Positivity of Pseudoddifferential Operators on Manifolds
Let $(M,g)$ be a Riemannian Manifold and $L^2$ the Hilbert space given by the volume form associated to the metric. Let $L_0^2$ be the subspace which is orthogonal to the constant functions. When is ...
0
votes
1answer
221 views
A Version of Nullstellensatz for Rings of Dİfferential Operators
Here is one of the classical versions of the nullstellensatz: Let $K$ be a field and let $\mathfrak{m}$ be a maximal ideal of the polynomial ring $K[T_1,\ldots,T_n]$. Then ...
1
vote
3answers
829 views
book on PDE on manifolds
let $M$ be a Riemannian manifold and $\alpha$ be any some unknown form on $M$. I am interested in solutions or some references of the equation of type $(d + \delta) \alpha = 0$ where $\delta$ is the ...
1
vote
1answer
81 views
Choosing boundary conditions for $(\frac{-d^2}{dx^2})^m$ on $H^m((0,1))$?
Consider the differential operator $D:$
$$
Du:=\frac{-d^2}{dx^2}u
$$
on the function space $$C=\{ u\in C^2([0,1]):u(0)=u(1)=0\}.$$
It's not hard to find the eigenvalues and ...
5
votes
2answers
553 views
Relationship between double tangent bundle, exterior derivative and connection
I am totally new to the subject differential geometry, and that probably reflects itself in the naive question that I'm trying to formulate. I hope this question does not get closed because of this.
...
1
vote
1answer
434 views
Lie bracket of algebraic vector fields
Let $X$ be an algebraic variety (over any field). The definition of the Lie bracket of two vector fields on $X$ (i.e. sections of the tangent sheaf) which I know characterizes vector fields as ...
2
votes
0answers
232 views
Cyclic vector theorem
Could be proved the cyclic vector theorem for a differential module (over a differential field of characteristic zero) by using the fact that the ring of differential operators (over the same ...
11
votes
1answer
377 views
Can a PDE constrain the degree of a $C^\infty$ map germ?
Let $\pi:E\to M$ be a smooth vector bundle over a smooth manifold, with $\text{rank}(E)=\text{dim}(M)$. For a section $\sigma$ of $E$ with a zero at $p\in M$, define the degree of the zero at $p$ to ...
10
votes
1answer
379 views
Hochschild (co)homology of differential operators
I googled the title on the internet, and arrived at the following result
$$HH_k(D)\cong H_{DR}^{2n-k}(M).$$
Here $M$ is a smooth manifold of dimension $n$, and $D$ is the ring of differential ...
7
votes
1answer
441 views
Is there a really big ring of differential operators in characteristic p?
$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...
1
vote
0answers
146 views
root system generalizations of Sekiguchi-Debiard (aka Laplace-Beltrami) operators
For the root system $A_n$, taking the limit $q = t^\alpha$ and $t \to 1$, and letting $Y = (t-1) X -1$ one obtains from the Macdonald operator the so-called Sekiguchi-Debiard operator:
$$D_\alpha(X) ...
2
votes
0answers
477 views
What essential property justifies the name “derivative”?
Most, if not all, of the notions of derivative that I have so far seen have the property that they are locally defined -- meaning that the derivative of a map-type object at a point depends on the map ...
3
votes
1answer
328 views
eigenspinors of Dirac operator
$M$ compact manifold. Let $\lambda$ be an eigenvalue for the Dirac operator of multiplicity greater than 2. I'm interested in showing the existence of two linearly independant eigenspinors $u$ and $v$ ...
