# Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, but I cant find anything in the literature thus far that explains where they could be applied, in a mathematical physics sense.

So basically, I'm wondering if there are any physical, or numerical analysis type applications of pseudodifferential operators.

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There are a few references for your interests:

In particular there is a whole new journal (since 2010) on this:

There are specific uses and applications of such operators, but most are concerned with theoretical aspects of the theory of partial differential equations (like asymptotics or stochastic PDEs). From the point of view of Physics their role is mostly theoretical as well, like their use for constructing relativistic equations (due to their importance for Dirac-type operators, their use in the heat kernel and index theorems, read below), for attempting to solve equations with differential operators with fractional exponents, and for the study of microlocal analysis in quantization procedures. But there has been very active research on more mundane applications such as the analysis of hydrodynamic instabilities and other problems with non self-adjoint operators in applied physics where their pseudo-spectrum plays an important role; also they seem to form a good basis for novel numerical techniques used in the analysis and simulation of physical systems like wave propagation, medical imaging, signal processing, deconvolution of seismic signals and wavefield extrapolations in geophysics. Check this answer in MO for a bit more.

Besides those more scientific/engineering applications, pseudodifferential operators have their most important theoretical application in pure mathematics as the essential objects of the Atiyah-Singer index theorem, one of the most remarkable results of the last century. It gives the Fredholm index of an elliptic differential operator (or the Euler-Poicaré characteristic of a complex sequence of them) acting on smooth sections of vector bundles on a compact manifold in terms of purely topological data in the form of charactersitic classes of the bundles. Basically, you are interested in the number of independent solutions to the homogeneous partial differential equations given by $\operatorname{D}\phi=0$ where $\operatorname{D}:E\rightarrow F$ is an elliptic differential operator taking "global smooth vector fields" $\phi$ of a complex vector bundle $E$ into fields of another vector bundle $F$, over a compact manifold $M$ (this is the one of the most general situations for partial differential equations over an arbitrary space!). In general such important invariant number of solutions is not easy to compute, but for compact manifolds and elliptic operators it is a finite number, and the Fredholm index is defined as $$\operatorname{Ind}\,\operatorname{D}:=\dim \ker \operatorname{D}-\dim \operatorname{coker}\operatorname{D}=\dim \ker \operatorname{D}-\dim \ker \operatorname{D^†}$$ i.e. the number of independent solution of the homogeneous equation $\operatorname{D}\phi=0$ minus the corresponding number of solutions for the complex adjoint equation $\operatorname{D^†}\phi=0$; this is easier to deal with and can be computed by the remarkable Atiyah-Singer index theorem: $$\operatorname{Ind D}=(-1)^{\dim M}\langle\operatorname{ch}(\sigma \,(\operatorname{D}))\smile\operatorname{Td}(TM\otimes\mathbb C),[TM]\rangle\;.$$ The right-hand side is the topological index of the operator (or complex sequence of operators), given in terms of the Todd characteristic class of the complexified tangent bundle of the manifold and the Chern character of the class symbol $\sigma\,(\operatorname{D})$ of the elliptic differential operator, which are all purely topological constructions on $M$. You can compute the characteristic classes in de Rham cohomology so you integrate the highest order part of their product over the tangent bundle manifold, and in most cases, when the Euler class is neither zero nor a zero divisor, the index formula can be simplified by integrating over the fibers to an integral over the base manifold: $$\operatorname{Ind D}=(-1)^{\frac{n(n+1)}{2}}\int_{M}\operatorname{ch}(E-F)\wedge\frac{\operatorname{Td}(TM\otimes\mathbb C)}{e(TX)}\lvert_{\text{vol}}\;.$$ Pseudodifferential operators are fundamental, since the theorem is actually valid and proved for them! Their role appears concerning the symbol class of differential operators in topological K-theory: $\sigma\,(\operatorname{D})\in K_0(TM)$, (although they are as fundamental in the heat kernel version of the proof, I shall discuss the original approach only). The reason is as follows. Since the topological index $B$ (the lhs of the formula) is a purely topological quantity dependent on vector bundles over $TM$, it is easily defined as a homomorphism from the $K$-theory group of the manifold to the integers, i.e. $B:K_0(TM)\rightarrow\mathbb Z$; the ring $K_0(TM)$ is defined in several equivalent ways, e.g. the free abelian group generated by vector bundles with the identification relation $[E]+[F]=[E\oplus F]$ and made into a ring with product the tensor product. Now, $B([a])$ is defined in an axiomatic way, satisfying certain nice properties, and then proved to be the cohomological evaluation given by the lhs of the formulas above (putting $[a]$ instead of $\sigma\,(\operatorname{D})$). Since to any elliptic differential operator an integer can be assigned by the Fredholm index $\operatorname{Ind: D}\rightarrow\mathbb Z$, and also its symbol class $\sigma\,(\operatorname{D})\in K_0(TM)$, it is natural to try to try to compute the first in terms of $B(\sigma\,(\operatorname{D}))$. The symbol class of an elliptic differential operator $\operatorname{D}:E\rightarrow F$ has a deep motivation (which I will not go into) to be defined as $\sigma\,(\operatorname{D}):=[E]-[F]$. Then, the Atiyah-Singer index formula amounts to prove that: $$\operatorname{Ind D}=B(\sigma\,(\operatorname{D})).$$ To establish that formula, one needs to find an inverse to $\sigma:\{\text{Ell. Diff. Operators}\}\rightarrow K_0(TM)$, so that $B=\operatorname{Ind}\circ\,\sigma^{-1}$, thus the question is: given any $[a]\in K_0(TM)$, can one construct an elliptic differential operator $\operatorname D\in\sigma^{-1}([a])$ whose symbol is $[a]$? The answer turns out to be "no" if we restrict ourselves to partial differential operators, thus a larger set of operators on $M$ is required, and as it turns out, the set of elliptic pseudodifferential operators on $M$ is indeed in bijective correspondence with the objects of the K-theory of $TM$: any vector bundle class in $K_0(TM)$ comes from the symbol of a suitable pseudodifferential operator on $M$! Showing this is one of the major steps in the proof of Atiyah-Singer, as one has to build such a suitable operator $\operatorname{D}_{[a]}$ for any $[a]$ and check that any such operators have the same analytical Fredholm index, i.e. $\operatorname{Ind D}=\operatorname{Ind D'}$ if $[a]=\sigma\,(\operatorname{D})=\sigma\,(\operatorname{D'})$. The last step of the proof is showing that $\operatorname{Ind}\circ\,\sigma^{-1}$ satisfies the nice axiomatic properties of the topological index, which characterize it uniquely, so that $\operatorname{Ind}\circ\,\sigma^{-1}$ has to be $B$. This can only be done using pseudodifferential operators, even though our final interest is compute the indices of elliptic differential operators (consequences of Atiyah-Singer are theorems like Chern-Gauß-Bonnet for the differential operators of de Rham's complex, Hirzebruch-Riemann-Roch for the Dolbeault differential complex, Hirzebruch signature for the Hodge complex, Atiyah-Patodi on Dirac operators, Atiyah-Bott fixed point theorem, etc...).

If you want details on pseudodifferential operators and their role on the proof of Atiyah-Singer you may consult the freely available excellent notes by:

Those references stress the importance and uses of pseudodifferential operators in the $K$-theory of $C^\ast$-algebras, thus in noncommutative geometry and topology (but with applications to ordinary geometry!). You can also find a lot of wonderful material on the analysis of pseudodifferential operators, and their applications, on the notes by:

There is a lengthy didactic discussions of pseudodifferential operators on manifolds and their use in index theorems, and how all that has applications to quantum field theories and mathematical physics in general, in the books:

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Wow, lot of information. I especially like the first link. Thanks! – Andrew Jean Bédard Dec 7 '12 at 3:06
Thanks! I know you were asking for more practical applications, but the fundamental role and development of pseudodiff. operators because of index theorems is something every analyst should be aware of, IMHO :D – Javier Álvarez Dec 7 '12 at 9:33
If you liked the first link, I have added many more book and journal references for similar topics (and a link to an answer in MO you should check out). – Javier Álvarez Dec 7 '12 at 9:56

I know I'm a bit late responding to this post, but if you are still interested in some applications of pseudodifferential operators I have a couple of suggestions (though they aren't physical applications). PDOs can be applied to problems in finance (e.g. option pricing). You might check out the book Non-Gaussian Merton-Black-Scholes Theory by Svetlana Boyarchenko and Sergei Levendorskii. The book focuses primarily on option pricing under Regular Levy Processes of Exponential type and uses PDOs to solve the relevant boundary value problems. The same authors also wrote a paper entitled "Perpetual American Options under Levy Processes" which I believe makes some use of PDOs. It was published in the Siam Journal of Control and Optimization, Vol. 40, No. 6, pp. 1663-1696.

Edit: You might also be interested in a set of lecture notes by Mark Joshi. I believe towards the end an application to differential geometry is covered (Hodge Theorem).

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