Questions tagged [differential-operators]

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

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0answers
56 views

Pseudo-differential operators and differetial operator

Hello I am totally new to Pseudo-differential operators and I’m wondering if a differential operator is a pseudo-differential operator. So, I want to show , using the definition of the symbol given ...
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Is this $1$-form harmonic?

Let $(M^3,g)$ be a compact, connected and oriented Riemannian $3$-manifold with boundary. For a harmonic map $u : M \to \mathbb{S}^1$ satisfying Neumann condition along $\partial M$, let $h = u^*(d \...
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On Grothendieck's abstract definition of differential operators

I have heard that there is the following abstract definition due to Grothendieck of differential operators on a module $M$ over a commutative associative unital algebra $A$ over a field of ...
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1answer
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Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials

There are some known criteria for the Sturm-Liouville Problem \begin{equation} \tag{1} \frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y \...
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Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$

The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by $$ \eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...
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Reference request: Transverse parabolic Schauder estimates

Is there a version of the parabolic Schauder estimates for transversely parabolic linear PDE's on a manifold with a Riemannian foliation for functions that are constant on the leaves of the foliation? ...
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Parameterization of (homogeneous) twisted differential operators

Let $X$ be a smooth algebraic variety over a field $k = \bar{k}$ of characteristic $0$. It's well known that twisted sheaves of differential operators are parameterized by $H^1(X,\mathcal{Z}^1)$, ...
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About generalized binomial theorem and Grünwald-Letnikov fractional derivative

I have run into a problem while computing the fractional derivatives of order $\alpha$ for the Riemann zeta function. My Theorem states Let $s\in\mathbb{C}$, $\mathfrak{Re}(s)>1$, then the ...
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130 views

Differential operators and vector fields [closed]

Let $M$ be a smooth manifold. It is well known that there is a bijective correspondance between vector fields on $M$ and differential operators of order 1. My question is: if we take a differential ...
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Understanding the geometric fibre twisted differential operators

Let $\mathfrak{g}$ be a Complex semisimple Lie algebra with decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}$. For $\lambda \in \mathfrak{h}^*$, we let $\mathcal{D}_{\lambda}$ be ...
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Index of the Fredholm operator

I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $...
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1answer
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Multidifferential operators with vanishing integrals

(Moved from math.stackexchange.) Is the following proposition true? Given a multidifferential operator $D$ on $\mathbb{R}^n$ with constant coefficients, i.e. for all functions $f_1,\dots,f_k \in C^...
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Is there an analog of the Levi–Civita connection for schemes?

Is there an analog of the Levi–Civita connection for schemes? There exists algebraic de Rham theory, $n$-forms on vector bundles (algebraically), and familiar constructions from differential geometry....
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Equivalent definitions of differential operator

This puzzles me from some time and is in parts connected to the questions Symmetrized derivatives version and Symmetrized derivatives version II. For me the linear DO between vector bundles $E$ and $...
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1answer
153 views

Elliptic operator with finite spectrum?

Is it possible for a (non-symmetric) elliptic differential operator to have finite spectrum. If so, is there an explicit example?
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2answers
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Link btw. exponential and derivatives from an algebraic perspective [closed]

I have been attempting to understand my math education (as a bachelor in electrical engineering) from a more algebraic perspective recently. I would like to understand more about the link between ...
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Reference request on the basics of differential operators and symbols

Perhaps it is too much to ask, but I'm searching for some published article or book, to cite some results from it, that treats infinite degree differential opertors (with constant coefficients) and ...
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48 views

Global solutions for an analytic family of differential operators with initial condition

This is related to this other question question of mine. Let $M$ be a $3$-dimensional closed (compact without boundary) strongly pseudoconvex manifold and let $\Delta_t$ be a collection of ...
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48 views

Regularity of eigenfunctions of an ordinary differential operator

Let $I \subseteq \mathbb{R}$ be an open interval and $T:D(T)\subseteq L^2(I) \to L^2(I)$ a differential operator given by $$(Tf)(x):= \sum_{j=0}^n a_j(x)f^{(j)}(x), \quad f\in D(T), \ x \in I,$$ where ...
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Does Peetre's theorem hold in complex analysis?

Let $E, F$ be two smooth vector bundles over a smooth manifold $M$. Peetre's theorem states that any $\mathbb{R}$-linear morphism $D: \mathcal{E} \to \mathcal{F}$ of the sheaves of sections of $E$ and ...
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Generalizing the heat kernel approach

I notice a way of solving equations that goes roughly like this: Want to solve $Tu=v$, i.e. hoping to find "$T^{-1}(v)$". $T^{-1}$ might not exist "on the nose", so instead we consider $\int_0^\infty ...
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References for a proof or interpretation of deficiency indices theorem (von Neumann)

I am looking for a proof or some interpretation around why the domain of the new extension $D(A_U)$ in the Theorem below is given by its specific formula. I have already searched in papers and here ...
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3answers
202 views

Reference request: Long-term behaviour of the heat equation for bounded initial data

Let us consider the heat equation \begin{align*} \frac{\partial}{\partial t}u(t,x) & = \Delta u(t,x), \\ u(0,x) & = f(x) \end{align*} on the whole space $\mathbb{R^d}$. If $f \in L^p := L^...
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Discrete maximum priniciple for parabolic operators

While reading a paper on the topic 'Numerical solutions for generalized Black-Scholes equation', It is given that their numerical scheme can be executed explicitly by solving a linear system $\mathbf ...
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2answers
141 views

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{...
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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 ...
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3answers
289 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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2answers
157 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 ...
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1answer
265 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 ...
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2answers
204 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: ‎$‎‎$‎ ‎\...
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1answer
776 views

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 ...
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2answers
363 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 ...
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1answer
250 views

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 ...
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3answers
372 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, $...
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70 views

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 ...
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0answers
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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) ...
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1answer
99 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 $C^{\infty}(\mathbb{R}^2)$ ...
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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 ...
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57 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 $T_a: D(T_a)\subseteq L^2[-a,a] \to L^2[-a,a]$ ...
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51 views

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 $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...
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60 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_{\...
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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-...
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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 ...
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2answers
671 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 ...
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2answers
181 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}\{\...
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1answer
216 views

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 ...
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3answers
352 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 ...
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2answers
242 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 ...
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201 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 $s\in \Gamma(E)$. ...
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44 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(\...

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