Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential operators play a major role in the K-theoretic proof of the index theorem, but it seems to me that they are of no use for any applications of it.
What are examples of applications of the Atiyah-Singer index theorem, which essentially use the computation of the index of a pseudodifferential operator which is not a Dirac-type operator?
Or phrased it another way: what are the benefits of knowing that the Atiyah-Singer index theorem holds for pseudodifferential operators and not only for Dirac-type operators?
Why do I care: for Dirac-type operators we can prove Atiyah-Singer using the heat kernel method, whereas this is in general not possible for pseudodifferential operators (so for them we have to use other proofs). So I was asking myself whether there is any need to have these other proofs that work also for pseudodifferential operators besides the fact that these other proofs further our understanding of the index theorem.
Best Answer
Boundary problems for elliptic differential equations are often studied by reducing to equations on the boundary. These equations are, as a rule, pseudo-differential but not differential. If the boundary problem is elliptic then the pseudo-differential operator is elliptic, thus Fredholm, and its index is of interest for answering the solvability question. See, for example, the chapter on elliptic boundary problems in volume 3 of Hörmanders monograph.
An early paper on this matter is by Fritz Noether (a brother of Emmy Noether) in Math. Ann. 82 (1920), 42-63. Interested in hydrodynamic problems, he considers singular integral operators which turn out to have non-zero index, and he gives a winding-number type formula for the index. The integral operators are pseudo-differential when making some additional smoothness assumptions, and they arise from reduction to the boundary by the use of layer potentials. I believe that the Noether formula is seen as one of the ancestors of the Atiyah-Singer Index Theorem.