I am trying to understand the Atiyah–Singer index formula for pseudo-differential operators. As far as I understood, the Fredholm index of the operator $A$ on a manifold can be computed just from the knowledge of its associated principal symbol $\sigma_p(A)$, at least for elliptic operators. However, I am interested in non-local operator of the form
$$ A(f)(x)= \int_{\mathbb R} \tilde A(x-x') f(x')\,\mathrm d x' $$ with $f$ in the Schwartz space.

This kind of operators find application in Phisics as they describe non-local responses of materials.

Suppose $B$ acts on $\mathcal C^{\infty}(\mathbb R,\mathbb R)$ as: $$ B(f)(x)= \int_{\mathbb R} e^{-(x-x')^2/2} f(x')\,\mathrm d x' $$ and has symbol $$ \sigma(B)(x,\xi) = e^{-\xi^2/2},\quad(x,\xi)\in\mathbb R^2 $$ (notice, no dependence on $x$).

$\sigma(B)$ is a Hörmander symbol in the class $\mathcal S^{m}_{1,0}$, $\forall\, m\in \mathbb R$.

Question: Does $\sigma_p(B)$ exist? Is it a generic feature of non-local operator not to have a principal symbol? Can the Atiyah–Singer index formula be applied to such operators?

N.B. For the notion of Hörmander symbol one can look at the Wikipedia page on pseudo-differential operators.

  • $\begingroup$ The Wikipedia page refers to Hörmander, and uses the word 'symbol' in the discussion of linear differential operators with constant coefficients, but seems never explicitly to mention "Hörmander symbol". $\endgroup$ – LSpice Oct 9 '19 at 21:22
  • $\begingroup$ You are right. By "Hörmander symbol" I mean that the symbol is "in the class $S_{1,0}^m$ of Hörmander", using the wording of Wikipedia. I hope there is no big ambiguity in this definition. $\endgroup$ – M. Marciani Oct 10 '19 at 12:04
  • $\begingroup$ To make classical index theory work it isn't strictly necessary that the operators are local, just pseudolocal - meaning they should commute with multiplication by a continuous function modulo compact operators. It doesn't look like this quite holds for the operators that you introduced, which makes me wonder if these operator are even Fredholm - do you know one way or the other? $\endgroup$ – Paul Siegel Oct 10 '19 at 12:38
  • $\begingroup$ If they aren't Fredholm, then one typical way to proceed is to study the algebra generated by the commutators of your preferred class of operators with multiplication operators by continuous functions - your operators will have a Fredholm-like index in the K-theory of this algebra, and often the right way to generalize the Atiyah-Singer index theorem is to calculate this K-theory group. $\endgroup$ – Paul Siegel Oct 10 '19 at 12:43
  • $\begingroup$ I didn't know about the need to be pseudolocal to work out a classical index theory. Do you have any reference (book&chapter or article) about the topic? And also, would you agree with me that these operators (at least the one in the example) don't have any principal symbol? $\endgroup$ – M. Marciani Oct 10 '19 at 13:47

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