By Karl R. Lang

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Using these notations, we arrive at ∞ R2d 0 T0 = B(R0 ) 0 B(R0 ) 0 A T0 t ,Y ε2 T0 B(R0 ) 0 B(R0 ) 0 × χδ (s − t) ζ (t) ds d X = ζ (t) ϕ(H0 (X )) dt d X pδ (s, Y ) aδ,ε (s, Y ) ds dY + oδ (1) T0 = t ,X ε2 (P f ε )δ (t, X ) Aδ δ (X − Y ) ϕ(H0 (X )) dt dY + oδ (1) T0 A (Y ) F(s, H0 (X )) F(s, H0 (X )) B(R0 ) 0 × χδ (s − t) ζ (t) ds d X δ (X − Y ) ϕ(H0 (X )) dt dY + oδ (1), + where the last equality uses the assumed weak convergence of A(t/ε2 , Y ) in L ∞ loc (R × 2d R )-weak- (Hypothesis 3). Hence, we may conclude ∞ R2d 0 (P f ε )δ (t, X ) Aδ ∞ = R2d 0 t ,X ε2 ζ (t) ϕ(H0 (X )) dt d X Fδ (t, X ) A δ (X ) ζ (t) ϕ(H0 (X )) dt d X + oδ (1).

In [3] we noticed that under hypotheses that guaranteed that both the Krein spectral shift function and spectral flow exist then they are essentially the same notion. This led us to the current investigation where we borrow some ideas from spectral flow theory as formulated in a spectral triple to extend the range of situations for which the Krein spectral shift function may be defined. We are then able to obtain new results on the spectral shift function and to prove, by a combination of the methods of [3] and [15, 16] Spectral Shift Function and Spectral Flow 53 that the spectral shift function gives a wide variety of analytic formulae for spectral flow.

The exponent p is as in Hypothesis 3. (iii) There exists a function C(R), bounded for bounded values of R > 0, such that1 sup sup sup A t ,X ε2 ≤ C(R) , γ p+1 sup sup sup B t ,X ε2 ≤ C(R) . γ ε>0 t≥0 X ∈E (R) ε>0 t≥0 X ∈E (R) Here, functions A and B are as in Hypothesis 3. Proof of Proposition 2. The fact that H0 is confining (3) readily implies point (i). Next, point (ii) comes as an immediate consequence of the regularity assumptions we have made on H0 and V . Indeed, we may write, whenever ε > 0, t ≥ 0, and X ∈ E(R), V (t/ε2 − u) , H0 ≤ ∇x V Su 2 L ∞ (R×B(ρ(R))) V (t/ε2 ) , H0 ∇v H0 2 L ∞ (B(ρ(R)) ≤ C(R), where C(R) is a locally bounded function of R.