The Witten index and the spectral shift function

In [Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99] Atiyah, Patodi Singer introduced spectral flow for elliptic operators on odd-dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an operator a manifold one higher dimension. A general proof this fact was produced by Robbin–Salamon [The Maslov index, Bull. London 27(1) (1995) 1–33, MR 1331677]. [F. Gesztesy, Y. Latushkin, K. Makarov, F. Sukochev Tomilov, The formula shift function relatively trace class perturbations, Adv. 227(1) (2011) 319–420, 2782197], start made extending these ideas to with some essential spectrum as occurs non-compact new ingredient there exploit scattering theory following fundamental paper [A. Pushnitski, flow, function, in Spectral Theory Differential Operators, American Mathematical Society Translations: Series 2, Vol. 225 (American Society, Providence, RI, 2008), pp. 141–155, 2509781]. These results do not apply differential directly, only pseudo-differential manifolds, due restrictive assumption is considered between its perturbation trace-class operator. paper, we extend main earlier papers satisfying pth Schatten condition [Formula: see text], thus allowing manifolds any dimension text]. our result does assume ellipticity or properties at all proves theoretic motivated [M.-T. Benameur, A. Carey, J. Phillips, Rennie, Wojciechowski, An analytic approach von Neumann algebras, Analysis, Geometry Topology Elliptic Operators (World Scientific Publisher, Hackensack, NJ, 2006), 297–352, 2246773; H. Grosse Kaad, On homological 289 (2016) 1106–1156, 3439708]. We illustrate using Dirac type text] arbitrary (see Sec. 8). setting Theorem 6.4 substantially extends 3.5 Grosse, G. Levitina, D. Potapov, Zanin, Trace formulas non-Fredholm operators: review, Rev. Phys. 28(10) 1630002, 3572626], where case d = 1 treated.