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

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688 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 ...
6
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3answers
287 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
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1answer
229 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 ...
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3answers
849 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 ...
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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
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2answers
596 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
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1answer
494 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
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0answers
247 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
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1answer
393 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
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1answer
393 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
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1answer
450 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 ...
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0answers
152 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
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0answers
481 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
333 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
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1answer
257 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
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0answers
100 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
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1answer
401 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 ...
4
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1answer
427 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
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1answer
454 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
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1answer
306 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?
2
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1answer
327 views

Symbol map in Getzler calculus

I hope someone can help me, although this question is rather specific. I am reading John Roe's chapter on Getzler symbols in "Elliptic operators, topology and asymptotic methods" to understand the ...
1
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1answer
496 views

Relating eigenvectors of two self-adjoints operators

Suppose I have a self-adjoint operator $\mathbf{L}$ which I seperate in two parts which are themselves self-adjoint. I write this in terms of their eigenvalues/eigenvectors: $\mathbf{v} \Lambda ...
4
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0answers
324 views

Monodromy of differential equations

Let $D$ an ordinary differential equation (let's assume $D$ only has regular singularities). Suppose $D$ = $D_0$$\partial$, where $\partial$ is $q\frac{d}{dq}$ My question is: Does there exist any ...
2
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2answers
231 views

Smooth dependence of the spectrum on the operator

I would like to know if there are theorems that state under which circumstances spectra of operator families depend smoothly on the parameter. To clarify, suppose I have a 1-parameter family $T_h$ of ...
5
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0answers
259 views

Between Being a Connection and Being an Elliptic Operator

Let $E$ be a smooth complex vector bundle over a smooth compact manifold $M$ and let $H$ be the $Z_{2}$-graded (with the $Z_{2}$-grading given by even/odd forms) Hilbert space of $L_{2}$ (with respect ...
6
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4answers
933 views

Symbol of pseudodiff operator

Hello, I am trying to understand the calculus of pseudodifferential operators on manifolds. All the textbooks I could put my hand on define the principal symbol of a pseudodifferential operator ...
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0answers
310 views

Definition of spectral gradient

Consider this differential operator $$ \mathcal{H}(\phi(\mathbf{x})) = -\triangle + V(\mathbf{x})H_\epsilon (\phi(\mathbf{x})) $$ where $\mathbf{x} \in \mathbb{R}^2$, $\phi : \mathbb{R}^2 \rightarrow ...
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0answers
187 views

Does regularity of a D-module for an unusual filtration imply regularity for the usual one?

One definition of regular D-modules on affine space is that a D-module is regular if it has a filtration compatible with the order filtration on differential operators whose associated graded is ...
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0answers
162 views

Spektrum of special Laplace type operator

How can I find Eigenvalues of the differential operator $-\frac{d^2}{dt^2} + \mathrm{sin} =\Delta + \mathrm{sin}$, defined on $M=S^1$, the 1-Sphere? It seems that the underlying ODE is pretty much ...
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2answers
925 views

Diagonalization of a matrix of differential operators

Dear community, i have a question regarding differential operators acting on vector valued functions and how to "diagonalize" them. To explain my question i will use an example: Let $V^k$ be the ...
3
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0answers
253 views

Decomposition of linear partial differential operators

I was wondering about the following: Let $M$ be a smooth, second-countable (possibly noncompact) manifold and let $E$ and $F$ be smooth vector bundles over $M$. Can every smooth linear partial ...
3
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0answers
360 views

PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field

Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases). Let $\nabla^{2}$ be the vector Laplacian. Let ...
3
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1answer
448 views

A name for PDE systems which are neither under- nor overdetermined?

The concepts of overdetermined and underdetermined PDE systems are well known. However, all sources I have so far looked into appear to avoid giving any name to PDE systems which are neither ...
4
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1answer
678 views

Growth of Laplacian eigenvalues on a compact domain?

Let $\mathcal{M}$ be a compact Riemannian manifold and let $\Delta$ be the (scalar) Laplace-Beltrami operator on $\\mathcal{M}$. Then $\Delta$ has a discrete spectrum and if we order its distinct ...
11
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2answers
1k views

The algebraic Version of Riemann Hilbert Correspondence

It is well known that if I have a differentialable manifold (holomorphic maniford) $M$, then I have a functor from the categroy of vector bundles on $M$ with flat connections to the categroy of local ...
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0answers
304 views

Some help with differential operators

Hello I am trying to follow the derivation of the Generalized Cornish Fisher Expansion from this paper. I dont see how the author goes from equation (21) to (22). For one to be able to do this ...
7
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1answer
866 views

What is the “correct” generalization of operator norms for nonlinear operators?

I have been recently wondering what is a (or even the) "correct" generalization of the notion of an operator norm to nonlinear operators? Please excuse the naivete of my question; if you think that ...
19
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1answer
791 views

What are some geometric reasons why a Dirac operator would have a gap in its spectrum?

My question is motivated by the following well-known computation. Let $M$ be an even dimensional Riemannian spin manifold and let $D$ be the spinor Dirac operator on $M$. Lichnerowicz showed that ...
4
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1answer
1k views

is $\nabla \cdot ( c^2 \nabla)$ a Laplace-Beltrami operator?

Someone mentioned, in passing, to me that $u \mapsto \nabla \cdot ( c^2 \nabla u)$ is a Laplace-Beltrami operator. Does anyone have some insight into this? From my understanding, the Laplace-operator ...
13
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1answer
1k views

Essential self-adjointness of differential operators on compact manifolds

Let $L$ be a linear differential operator (with smooth coefficients) on a compact differentiable manifold $M$ (without boundary). Suppose $M$ is endowed with a smooth volume form (actually, a smooth ...
20
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3answers
1k views

When are the eigenspaces of the Laplacian on a compact homogeneous space irreducible representations?

I was writing up some notes on harmonic analysis and I thought of a question that I felt I should know the answer to but didn't, and I hope someone here can help me. Suppose I have a compact ...
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0answers
115 views

Relation between different spatial derivatives of a random field (related to complex integral and/or bessel function)

2 random fields $b$ and $c$ are derived from random field $a$ by $b=\nabla^2a\equiv(\partial_{xx}+\partial_{yy})a $ and $c \equiv c_1+i c_2 = (\partial_{xx}-\partial_{yy}+2i \partial_{xy}) a$. ...
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0answers
102 views

Elliptic complexes for bundles with infinite dimensional fibers

I am looking for differential geometric treatment of bundles over finite dimensional manifolds which have infinite dimensional fibers. It would be nice to have theory for general locally convex ...
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2answers
492 views

index of a family of Dirac operators in $K^1$

Suppose I have a family of Dirac operators over a compact base space B. From the paper of Atiyah and Singer about skew adjoint Fredholm operators we know that it has an index in $K^1(B)$. Suppose ...
3
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3answers
878 views

Can the adjoint of the exterior derivative in semi-Riemannian geometry be defined without the Hodge * operator?

The adjoint of the exterior derivarive is defined by $\delta:=(-1)^k\ast^{-1}d\ast$, but I need a way which avoids the Hodge $\ast$ operator. Is there another definition? For example, for ...
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2answers
414 views

Is there a generalization of Floquet theory to elliptic functions?

Hi, Consider a system of linear differential equations $$ {d f \over dz} = A(z) f, $$ where $A(z)$ is a matrix-function. If $z \in \mathbb{R}$ and the function is periodic $A(z) = A(z + T)$, ...
4
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1answer
757 views

How “generalized eigenvalues” combine into producing the spectral measure?

Hi... I am wondering how 'eigenvalues' that don't lie in my Hilbert space combine into producing the spectral measure. I study probability and I am quite ignorant in the field of spectral analysis of ...
3
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5answers
455 views

Analytic hypoellipticity of linear ordinary differential operators

Let $P = a_n(x) D_x^n + a_{n-1}(x) D_x^{n-1} + \ldots + a_0(x)$ be a linear ordinary differential operator with polynomial (or real analytic) coefficients $a_j(x)$. Suppose that $a_n(x)$ doesn't ...
22
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1answer
660 views

Idempotents in Rings of Differential Operators

Differential Operators on General Commutative Rings Let k be an algebraically closed field of characteristic zero, and let R be a commutative k-algebra. Then a (Grothendieck) differential operator ...
3
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1answer
353 views

Are these operators defined on 2D surfaces self-adjoint?

My research group finds/proposes a fundamental operator in quantum mechanics, the Cartesian momentum as I called (I think for mathematician the ref. 2007 is sufficient). However, I do not know whether ...