Questions tagged [differential-operators]

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

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Example of differential equation with periodic boundary conditions that has at least two simple eigenvalues

I need an example of a periodic function $q:\mathbb{R} \to \mathbb{R}$ with period $\pi$ such that if we consider the differential equation \begin{equation}\tag{1} y''(x)+(\lambda -q(x))y(x)=0 \end{...
Mark Bans's user avatar
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5 votes
1 answer
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Fredholm theory of non elliptic operators

In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in ...
3 votes
3 answers
372 views

Density of a functional space

Let $D$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial D$. Is the following subspace dense in space $L^2(D)\times L^2(\partial D)$: $$\{(f,f\rvert_{\partial D}) : f\in C^\infty(\...
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1 vote
2 answers
461 views

Non-closed range space of Laplace operators?

Set $ -\Delta: H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3) $. Then $ \mathcal{R}(-\Delta) $ is non-closed? Sorry if this question is trivial. I am not familiar with theory of ...
Yidong Luo's user avatar
6 votes
2 answers
2k views

The spectrum of the Hodge Laplacian on a Riemannian manifold

The Hodge Laplacian operator on differential forms on a (compact?) Riemannian manifold carries useful information about the topology of the manifold. In particular, the multiplicity of the zero ...
Asvin's user avatar
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4 votes
3 answers
608 views

Question about the regularity of fractional Heat equation

Let $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$, $0<s<1$ and $(-\Delta)^s$ denotes the restricted fractional Laplacian. Let consider the following fractional Heat equation: ‎$‎‎$‎ ‎\...
Hheepp's user avatar
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14 votes
1 answer
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Computing spectra without solving eigenvalue problems

There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda ...
Victor Galitski's user avatar
2 votes
2 answers
586 views

Explanation of definition of George Wilson's adelic Grassmannian

How is George Wilson's adelic Grassmannian from e.g. the paper https://link.springer.com/article/10.1007%2Fs002220050237 related to the adeles or (especially) the affine Grassmannian (a.k.a. the loop ...
Yellow Pig's user avatar
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3 votes
1 answer
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Differentiability of operator norm [closed]

Is there any known results about differentiability properties of the function $\mathbb f:\mathbb R \to\mathbb R,$ $f(t):=\|A+tB\|_{op}$ where $\|.\|_{op}$ denotes the usual operator norm of the ...
A beginner mathmatician's user avatar
3 votes
3 answers
612 views

Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation

The entry OEIS A139605 (also related OEIS A145271) has a matrix computation for the partition polynomials that represent the expansions of iterated derivatives, or vectors in differential geometry, $...
Tom Copeland's user avatar
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1 vote
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Spectrum of a differential operator on $L^2(0, \infty)$

Let $A:H^n(0, \infty) \subset L^2(0, \infty) \to L^2(0, \infty)$ be the differential operator defined by $$Af:= \sum_{j=0}^na_j f^{(j)}$$ for all $f \in H^n(0, \infty),$ where $a_j \in \mathbb{C}$ and ...
Mark Bans's user avatar
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1 vote
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How to solve or analyse the smallest eigenvalue of 2 coupled 1st-order linear ODEs?

I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities \begin{align} -\mathrm{i} u'(x) +f^*(x) ...
xiaohuamao's user avatar
2 votes
1 answer
119 views

Keeping track of limit cycles via certain second order differential operator

Inspired by the two posts which are linked bellow we ask the following question: Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ ...
Ali Taghavi's user avatar
58 votes
22 answers
12k views

Which high-degree derivatives play an essential role?

Q. Which high-degree derivatives play an essential role in applications, or in theorems? Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration), and the ...
2 votes
0 answers
64 views

Inclusion of the spectrum of two differential operators defined on $L^2[-a,a]$ and $L^2[0, \infty)$

Let $T$ be the formal operator defined by $$Tu:= \sum_{j=0}^{2n} a_j\frac{d^ju}{dx^j}$$ where $a_j \in \mathbb{C}$. Consider the differential operators $T_a: D(T_a)\subseteq L^2[-a,a] \to L^2[-a,a]$ ...
Mark Bans's user avatar
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0 answers
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A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact parallelizable manifold

Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...
Ali Taghavi's user avatar
0 votes
0 answers
75 views

A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow of a vector field

Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\...
Ali Taghavi's user avatar
9 votes
0 answers
336 views

Geometric motivation behind the Fredholm module definition

If $A$ is an involutive algebra over the complex numbers $\mathbb{C}$, then a Fredholm module over $A$ consists of an involutive representation of $A$ on a Hilbert space $H$, together with a self-...
Max Schattman's user avatar
4 votes
0 answers
204 views

Exterior derivative on loop space

Notations: Let $X$ be a manifold, and denote by $LX := C^\infty(S^1,X)$ its loop space. For a loop $\gamma \in LX$ we can think at the tangent space of $LX$ at the point $\gamma$ as the space of ...
Mattia Coloma's user avatar
18 votes
2 answers
2k views

Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold

Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta_g$ be its Laplace-Beltrami operator. A "well-known fact" is that the eigenvalues of $\Delta_g$ have finite multiplicity and tend to ...
Max Schattman's user avatar
5 votes
2 answers
313 views

For which tempered distributions is the fractional derivative well-defined?

Let $\gamma \geq 0$ and consider the fractional derivative operator defined in Fourier domain by $$\mathcal{F} \{\mathrm{D}^{\gamma} \varphi \} (\omega) = (\mathrm{i} \omega)^{\gamma} \mathcal{F}\{\...
Goulifet's user avatar
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1 answer
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An alternative representation of the principal symbol of the Laplace operator

Assume that $(M,g)$ is a $n$ dimensional Riemannian manifold. We denote by $\Delta$, the Laplacian associated to this Riemannian structure. Are the following two conditions equivalent? First condition ...
Ali Taghavi's user avatar
2 votes
3 answers
694 views

Generating all rotationally invariant linear differential operators

It is relatively easy to show that the Laplacian $$ \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} $$ Is the unique second order linear differential operator that is ...
henrikr's user avatar
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0 votes
2 answers
281 views

Derivations of $\chi^{\infty}(M)$ which are elliptic operator

What is an example of a manifold $M$ with $\dim(M)>1$ whose Lie algebra $\chi^{\infty}(M)$ of smooth vector fields admit an elliptic operator $D:\chi^{\infty}(M)\to \chi^{\infty}(M)$ such ...
Ali Taghavi's user avatar
5 votes
0 answers
215 views

A differential operator analogy of certain fact in real analysis of smooth functions

Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$. Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$. ...
Ali Taghavi's user avatar
3 votes
0 answers
53 views

Controlling a Schwartz kernel near the diagonal

Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...
geometricK's user avatar
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1 vote
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Showing a differential operator is positive semidefinite

Let $R>\lambda>\chi$ be positive real constants and $\alpha$ be a real number. The following differential operator \begin{multline} \mathcal{L}g = -\frac{d}{d\xi}\left[(1-\xi^2)\frac{dg}{d\xi}\...
eyeballfrog's user avatar
1 vote
0 answers
75 views

PDE on an open ball with prescribed value on some open subsets

Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it ...
Overflowian's user avatar
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2 votes
1 answer
139 views

Perturbation of the adiabatic limit

Let $(M,g_M)$ be a closed oriented Riemannian manifold that has a fibration structure $$ Y \rightarrow M \overset{\pi}{\rightarrow} B $$ where $(Y,g_Y)$ and $(B,g_B)$ are closed Riemannian manifolds ...
Totoro's user avatar
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2 votes
0 answers
90 views

A quantity associated to a compact Riemannian manifold with boundary(The pair of Laplacian)

Let $M$ be a Riemannian manifold with boundary $\partial M$. Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of ...
Ali Taghavi's user avatar
2 votes
0 answers
97 views

1D Schrödinger Equation with Measure-Valued Coefficients

I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following: $$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}...
S. Thornton's user avatar
5 votes
1 answer
136 views

Construction of non-split extension of simple modules of Lie algebras using linear differential operators

Consider the natural action of $W_1=k\left\langle x,\frac{d}{dx}\right\rangle$ on $X=\mathbb C[x]$. Then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentially a $\mathfrak{sl}_2$-tuple ($\left[...
sawdada's user avatar
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2 votes
2 answers
362 views

Why we use Caputo fractional derivative in application?

I'm working on some papers which use Caputo fractional evolution equation (see on Wikipedia) as application for thier main result: For example: $$\left\{\begin{matrix} ^CD^{\sigma}_tx(t)+Ax(t)=&...
Motaka's user avatar
  • 291
3 votes
1 answer
420 views

Hochschild Homology and Formal Geometry

My question concerns the degeneration of the spectral sequence computing Hochschild homology of differential operators on a smooth affine variety $X$. The spectral sequence arises from the ...
user avatar
2 votes
0 answers
92 views

Vector bundle endomorphism diffeomorphism invariant?

Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \...
phydev's user avatar
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5 votes
0 answers
262 views

Is there any geometrical/homological intuition behind symmetrized gradient?

The gradient/differential/exterior differential/divergence/curl are all strictly related first order differential operators. As far as I understood, they are the base of (co)homological theories in ...
Romeo's user avatar
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5 votes
0 answers
139 views

Family of Hodge decomposition

It is known that a metric $g$ gives a Hodge decomposition: $$ \Omega^*(M)=\mathcal H^*(M)\oplus d\Omega^*(M) \oplus \delta_g \Omega^*(M) $$ Note that the usual differential restricts to an isomorphism ...
Hang's user avatar
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1 vote
0 answers
53 views

How to choose Eigenvaues for system extending in perpendicular direction?

I have two linear third order ODEs with a separation constant (eigenvalue parameter) on a rectangular domain where $x \in (0,1)$ and $y \in (0,1)$as follows: $$ \lambda_h F''' - 2 \lambda_h \beta_h ...
Avrana's user avatar
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3 votes
0 answers
80 views

Conformal manifolds produce Fredholm modules-pseudodifferential operator

This question is a continuation of the discussion which can be found here. From the exterior derivative one constructs an operator $S$ with the property that the graph of $S$ is the (closure of) the ...
truebaran's user avatar
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0 votes
2 answers
429 views

Spectrum equals eigenvalues for unbounded operator

Let $D$ be an unbounded densely defined operator on a separable Hilbert space $H$. If $D$ is diagonalisable with all eigenvalues having finite multiplicity and growing towards infinity, does it follow ...
Bas Winkelman's user avatar
14 votes
1 answer
549 views

Projective-invariant differential operator

This question was originally asked on Math StackExchange. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = ...
user76284's user avatar
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0 votes
0 answers
64 views

Addition property of Laplace-Beltrami eigenfunctions in symmetric spaces

[Originally posted on math stackexchage but have not received feedback in over a month, I'm hoping someone here could point me in the right direction]. Consider the eigenvalue equation for the ...
GreaterThanZero's user avatar
4 votes
0 answers
148 views

Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”

I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...
MrMatzetoni's user avatar
3 votes
0 answers
81 views

Some questions on defining the analytic index

The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the ...
Ho Man-Ho's user avatar
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8 votes
1 answer
359 views

Spectrum of a first-order elliptic differential operator

Suppose that I have a first-order elliptic differential operator $A: \mathrm{dom}(A) \subset L^2(E) \to L^2(E)$, where $(E,h^E) \to M$ is a hermitian vector bundle and $M$ is a compact manifold. I ...
Lashi Bandara's user avatar
2 votes
0 answers
278 views

Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum

I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...
Hugo Chapdelaine's user avatar
2 votes
0 answers
194 views

Existence of a certain kind of compact spin manifold with boundary

For a compact spin Riemannian manifold $(M^n,g)$ without boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has real, ...
M. L. Nguyen's user avatar
4 votes
1 answer
325 views

Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$

Consider the following PDE: \begin{equation} p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$} \end{equation} where $g$ is a flat function at the point (...
Ilia's user avatar
  • 307
4 votes
2 answers
293 views

Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability

We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...
Ali Taghavi's user avatar
8 votes
3 answers
404 views

Surjectivity of differential operators with constant coefficients

I would like a proof or a reference (or a counter-example...) for the following fact. Let $P\in \mathbb{C}[x_1,\ldots ,x_n]$ and $D\in \mathbb{C}[\frac{\partial }{\partial x_1} ,\ldots ,\frac{\...
abx's user avatar
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