**1**

vote

**1**answer

247 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

**0**answers

96 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

**1**answer

395 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**

votes

**1**answer

423 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} ...

**6**

votes

**1**answer

433 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

**1**answer

303 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**

votes

**1**answer

302 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**

vote

**2**answers

488 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**

votes

**0**answers

318 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**

votes

**2**answers

222 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 ...

**4**

votes

**0**answers

253 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 ...

**5**

votes

**4**answers

887 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 ...

**1**

vote

**0**answers

291 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 ...

**0**

votes

**0**answers

184 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 ...

**1**

vote

**0**answers

156 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 ...

**0**

votes

**2**answers

887 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**

votes

**0**answers

251 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**

votes

**0**answers

355 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**

votes

**1**answer

439 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**

votes

**1**answer

654 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**

votes

**2**answers

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 ...

**1**

vote

**0**answers

301 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**

votes

**1**answer

834 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**

votes

**1**answer

773 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 ...

**3**

votes

**1**answer

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 ...

**11**

votes

**1**answer

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**

votes

**3**answers

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 ...

**1**

vote

**0**answers

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
$c \equiv c_1+i c_2 = (\partial_{xx}-\partial_{yy}+2i \partial_{xy}) a$.
...

**0**

votes

**0**answers

100 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 ...

**5**

votes

**2**answers

486 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**

votes

**3**answers

820 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 ...

**4**

votes

**2**answers

404 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**

votes

**1**answer

711 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 ...

**2**

votes

**5**answers

434 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 ...

**21**

votes

**1**answer

640 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**

votes

**1**answer

346 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 ...

**7**

votes

**2**answers

717 views

### What is the smallest $C^*$-algebra containing the “standard” pseudodifferential operators?

Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying
$
|\partial^\alpha_x \partial^\beta_\xi a(x,\xi)| \leq C_{\alpha,\beta} (1+|\xi|)^{-|\beta|}
$
) a $C^*$-algebra?
In other ...

**1**

vote

**0**answers

249 views

### Differential operator with image of its Frechet derivative in the nullspace of operator adjoint?

Is there an example of a differential operator $A(z)$ with parameter $z \in \mathbb{R}^d$ and Frechet derivative $A_z(z)$ such that $\mathrm{im}(A_z(z)) \subseteq \mathrm{ker}(A^T(z))$. Can this still ...

**1**

vote

**3**answers

637 views

### How to “fill in” 3-dimensional Laplacian kernels

The other day, I asked this question 3x3x3 Laplace Kernel?, regarding what the 3x3x3 kernel was for applying a Laplacian convolution.
On that page, it mentions the kernels were "deduced by using ...

**0**

votes

**1**answer

2k views

### 3x3x3 Laplace Kernel?

Does anyone know what a 3x3x3 Laplacian kernel looks like? I realize that might be an open-ended question, but I need to apply a Laplacian convolution using a 3x3x3 Laplacian kernel, and frankly I ...

**10**

votes

**1**answer

371 views

### Does the image of a differential operator always contain an ideal?

Let $\delta$ denote a non-zero complex algebraic differential operator in a single variable x. That is, it can be written as a sum
$$ \delta = \sum_i f_i\partial_x^i$$
where there $f_i$ are complex ...

**8**

votes

**3**answers

722 views

### Is there a good account of D-affinity and localization theorem for partial flag varieties?

Recall that a topological space is called $A$-affine for a sheaf of algebras $A$ if taking global sections of coherent sheaves of $A$-modules is an equivalence of categories to finitely generated ...

**4**

votes

**0**answers

271 views

### Why should we consider D-module on flag variety of Lie algebra?

Why don't we stay at D-module on base affine space but go to study flag variety of Lie algebra?
I remembered there are nice papers of Bernstein-Gelfand-Gelfand and Gelfand-Kirillov discussing the ...

**3**

votes

**1**answer

196 views

### The space of spectral sections and connections to K-theory

We look at a familiy $D_\alpha$ of Dirac operators over a (compact) base space B. The projection $\Pi^+_\alpha$ onto the positive eigenspaces of $D_\alpha$ is usually not continuous in α. ...

**5**

votes

**1**answer

557 views

### Self-adjoint extension of locally defined differential operators

The following is well known. Given a symmetric differential operator, like $\partial_x^2$, defined on smooth functions of compact support on $\mathbb{R}$, $C_0^\infty(\mathbb{R})$, one can count the ...

**18**

votes

**4**answers

2k views

### Have people successfully worked with the full ring of diferential operators in characteristic p?

This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...

**7**

votes

**0**answers

516 views

### How hard is it to make a differential operator Hermitian?

Let $M$ be a closed finite-dimensional smooth manifold (over $\mathbb R$). Let $C^\infty(M) = C^\infty(M,\mathbb C)$ be the algebra of smooth complex-valued functions on $M$, with the natural complex ...