Questions tagged [pseudo-differential-operators]

In mathematical analysis, a pseudo-differential operator (or $\Psi$DO for short) is an extension of the concept of differential operator. The role of $\Psi$DOs lie in the fact that there is a number of operations leading outside the class of differential operators but preserving the class of $\Psi$DOs. For example, the resolvent and complex powers of an elliptic differential operator on a compact manifold are classical examples of $\Psi$DOs; they arise when reducing an elliptic boundary value problem to the boundary. They are used extensively in the theory of partial differential equations, quantum field theory, and microlocal analysis.

Definition: Let $~X⊂\mathbb R^n~$ be open. A pseudo-differential operator ($\Psi$DO) is a Fourier integral operator of the form

\begin{align}A&:C^\infty_0(X)→D'(X), \\ Au(x)&=\dfrac 1{(2π)^n}\iint e^{i(x−y)\theta}a(x,y,θ)u(y)dyd\theta. \end{align} The function $a$ is called the symbol of the pseudo-differential operator $A$. An important property of $\Psi$DOs is that many properties of $A$ follow from the so-called symbol calculus applied to $a$.

If $\mathcal F$ denotes the Fourier transform, a short hand notation for this definition is $Au=\mathcal F^{−1}(a\mathcal Fu)$, put in words: Fourier transform $u$, multiply with $a$ and transform back. Another way of saying this is that, at least formally, $A$ is defined by $A(e^{ix\xi)=a(x,\xi) e^{ix\xi}$.

For more details, see following the articles, notes and references therein:

1. Wikipedia article: "Pseudodifferential operator"
2. Encyclopedia of Math article: "Pseudo-differential operator"
3. "Introduction to pseudo-differential operators" by Michael Ruzhansky, TCC notes (2014)
4. "An introduction to pseudo-differential operators" by Jean-Marc Bouclet, notes (2012,2013)
5. "A First Course on Pseudo-Differential Operators" by Nicolas Lerner, notes (2017)