# Questions tagged [differential-operators]

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

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### 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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### 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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### 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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### 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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### 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 $T_a: D(T_a)\subseteq L^2[-a,a] \to L^2[-a,a]$ ...
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 $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...