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

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4
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404 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
231 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
347 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
206 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 ...
10
votes
1answer
489 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
157 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
1k 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
161 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 ...
2
votes
2answers
329 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
votes
1answer
364 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 ...
17
votes
5answers
2k 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
497 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
252 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
411 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
270 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 ...
5
votes
1answer
763 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 ...
19
votes
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 ...
0
votes
0answers
188 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
298 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
256 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 ...
6
votes
1answer
364 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 ...
10
votes
3answers
603 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
461 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
216 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 ...
4
votes
1answer
705 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
414 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
193 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 ...
7
votes
1answer
203 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 $x_1$, $x_2$, ..., ...
5
votes
0answers
181 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 ...
2
votes
1answer
350 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
733 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 ...
7
votes
3answers
355 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
266 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
934 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
95 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 ...
7
votes
2answers
839 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
751 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 ...
3
votes
0answers
334 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
449 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 ...
11
votes
1answer
450 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 ...
8
votes
1answer
542 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 ...
2
votes
0answers
183 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
488 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
364 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 $v$ ...
1
vote
1answer
288 views

Modules over rings of differential operators

If $M$ is a left $\mathbb{C}[t] \langle \partial _t \rangle$-module ( a left module over the Weyl algebra), then $\mathrm{Hom}(M,\mathbb{C}[t] \langle \partial _t \rangle)$ is equipped with a ...
2
votes
0answers
110 views

Invariant linear manifolds for multiplication by the independent variable in L^2 (R)

In general I am trying to determine when the self-adjoint operator $M$ of multiplication by the independent variable in $L^2 (\mathbb{R})$ has a symmetric restriction to a dense linear manifold ...
2
votes
1answer
426 views

What's an example of a commutative algebra over $\mathbb Q$ that fails to satisfy this version of the “PBW theorem”

In a recent question, I recalled the notion of differential operator, polyderivation, and principal symbol for a commutative algebra $A$ over some fixed commutative ring $k$. (I will not repeat those ...
5
votes
1answer
470 views

Localizability of differential operators a la Grothendieck

Hello, Maybe this question is trivial, so sorry Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1). Then we can define the module of differential operators $D^{\leq n} ...
7
votes
1answer
503 views

For which algebras does \{Differential Operators\} satisfy a PBW-like theorem?

Let $k$ be a commutative ring, $A$ a commutative $k$-algebra, and for some other part of why I'm asking this question I only care about the case when $k \supseteq \mathbb Q$. Recall the following ...
2
votes
1answer
315 views

differential operator power coefficients

Let $(F(x)\frac{d}{dx})^n=\sum_{i=1}^n H_{n,i}(F, F', F^{(2)}, \ldots , F^{(n)})\frac{d^i}{dx^i}$. I'm curious about the exact formula for $H_{n,i}(y_0, y_1, \ldots , y_n)$. What is known about it?