Spectral Shift Functions and some of their Applications in Mathematical Physics

Transcription

1 Spectral Shift Functions and some of their Applications in Mathematical Physics Fritz Gesztesy (University of Missouri, Columbia, USA) based on several collaborations with A. Carey (Canberra), Y. Latushkin (Columbia MO), K. Makarov (Columbia MO), D. Potapov (Sydney), F. Sukochev (Sydney), Y. Tomilov (Torun) Operator Semigroups meet Complex Analysis, Harmonic Analysis and Mathematical Physics Herrnhut, Germany June 3 7, 2013 Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

2 Outline 1 Krein Lifshitz SSF 2 SSF Applications Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

3 Krein Lifshitz SSF A short course on the spectral shift function ξ Notation: B(H) denotes the Banach space of all bounded operators on H (a complex, separable Hilbert space). B (H) denotes the Banach space of all compact operators on H. B s (H), s 1, denote the usual Schatten von Neumann trace ideals. Given two self-adjoint operators H, H 0 in H, we assume one of the following (we denote V = [H H 0 ] whenever dom(h) = dom(h 0 )): Trace class assumption: V = [H H 0 ] B 1 (H). elative trace class: V (H 0 zi H ) 1 B 1 (H). esolvent comparable: [ (H zi H ) 1 (H 0 zi H ) 1] B 1 (H). Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

4 Krein Lifshitz SSF A short course on the spectral shift function ξ The Krein Lifshitz spectral shift function ξ(λ; H, H 0 ) is a real-valued function on that satisfies the trace formula: For appropriate functions f, tr H (f (H) f (H 0 )) = f (λ) ξ(λ; H, H 0 ) dλ. For example, resolvents, [ tr H (H zih ) 1 (H 0 zi H ) 1] = semigroups (if H, H 0 are bounded from below), etc. ξ(λ; H, H 0 ) dλ (λ z) 2, E.g., for f C 1 () with f (λ) = e iλs dσ(s) and dσ a finite signed measure; f (λ) = (λ z) 1 ; or f L 1 (; (1 + p )dp) (implies f C 1 ()), etc. V. Peller has necessary conditions on f, and also sufficient conditions on f in terms of certain Besov spaces. These spaces are not that far apart... Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

5 Krein Lifshitz SSF A short course on the spectral shift function ξ Generally, ξ(λ; H, H 0 ) = tr H (E H0 (λ) E H (λ)) is not correct (if dim(h) = )! [E H0 (λ) E H (λ)] is NOT necessarily of trace class even for V = H H 0 of rank one! ecall the Trace and the Determinant: Let λ j (T ), j J (J N an index set) denote the eigenvalues of T B 1 (H), counting algebraic multiplicity. Then j J λ j( (T T ) 1/2) < and tr H (T ) = j J λ j (K), det H (I H T ) = j J[1 λ j (T )]. The Perturbation Determinant: H and H 0 self-adjoint in H, H = H 0 + V, D H/H0 (z) = det H ( (H zih )(H 0 zi H ) 1) = det H ( IH + V (H 0 zi H ) 1). Example. If H 0 = (d 2 /dx 2 ), H = (d 2 /dx 2 ) + V ( ) in L 2 (; dx), V L 1 (; (1 + x )dx), real-valued, then D H/H0 (z) = Jost function = Evans function. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

6 Krein Lifshitz SSF A short course on the spectral shift function ξ General Krein s formula for the trace class V = [H H 0 ] B 1 (H): ξ(λ; H, H 0 ) = 1 π lim ε 0 + Im(ln(D H/H 0 (λ + iε))) for a.e. λ, ( ) where D H/H0 (z) = det H ( (H zih )(H 0 zi H ) 1) = det H ( IH + V (H 0 zi H ) 1). Then ξ(λ; H, H 0 ) dλ H H 0 B1(H) If dim(ran(e H0 (a ɛ, b + ɛ))) <, then and ξ(λ; H, H 0 ) dλ = tr(h H 0 ). ξ(b 0; H, H 0 ) ξ(a + 0; H, H 0 ) = dim(ran(e H0 (a, b))) dim(ran(e H (a, b))). ( ) Note. Formulas ( ) and ( ) remain valid in the relative trace class case, where V (H 0 zi H ) 1 B 1 (H). = Eigenvalue counting function. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

7 Krein Lifshitz SSF A short course on the spectral shift function ξ Away from σ ess (H 0 ) = σ ess (H), ξ(λ; H, H 0 ) is piecewise constant: Counting multiplicity: Suppose λ 0 \σ ess (H 0 ). Then, ξ(λ 0 + 0; H, H 0 ) ξ(λ 0 0; H, H 0 ) = m(h 0 ; λ 0 ) m(h; λ 0 ), where m(t, λ) N {0} denotes the multiplicity of λ σ p (T ), T = T. Scattering theory: The Birman Krein formula Let S(H, H 0 ) be the scattering operator for the pair (H, H 0 ). Then S(H, H 0 ) = σ ac (H 0) S(λ; H, H 0 ) dλ in H σ ac (H 0) K dλ, where S(λ; H, H 0 ) is the fixed energy scattering operator in K (assuming uniform spectral multiplicity of σ ac (H 0 ), for simplicity). Then ξ(λ; H, H 0 ) = 1 2πi ln(det K(S(λ; H, H 0 )) for a.e. λ σ ac (H 0 ). Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

8 Applications: Fredholm Indices Computing Fredholm indices: Consider the model operator D A = (d/dt) + A, dom(d A ) = dom(d/dt) dom(a ) in L 2 (; H), where dom(d/dt) = W 1,2 (; H), and A = A ± = A(t) dt, A = A dt in L 2 (; H) H dt, lim A(t) exist in norm resolvent sense and are boundedly t ± invertible, i.e., 0 ρ(a ± ), and where we consider the case of relative trace class perturbations [A(t) A ], A(t) = A + B(t), t, B(t)(A zi H ) 1 B 1 (H), t (plus quite a bit more next page), such that D A becomes a Fredholm operator in L 2 (; H). Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

9 Applications: Fredholm Indices Just for clarity, A = A(t) dt in L 2 (; H) H dt, H a separable, complex H-space, denotes the operator in L 2 (; H) defined by (Af )(t) = A(t)f (t) for a.e. t, { f dom(a) = g L 2 (; H) g(t) dom(a(t)) for a.e. t, } t A(t)g(t) is (weakly) measurable, A(t)g(t) 2 H dt <. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

10 Applications: Fredholm Indices, contd. Main Hypotheses (we re aiming at A(t) = A + B(t)...): A self-adjoint on dom(a ) H, H a complex, separable Hilbert space. B(t), t, closed, symmetric, in H, dom(b(t)) dom(a ). There exists a family B (t), t, closed, symmetric, in H, with dom(b (t)) dom(a ), such that B(t)( A + I H ) 1, t, is weakly locally a.c. and for a.e. t, d ( g, B(t)( A + I H ) 1 h ) dt H = ( g, B (t)( A + I H ) 1 h ), g, h H. H B ( )( A + I H ) 1 B 1 (H), t, B (t)( A + I H ) 1 B1(H) dt <. { ( B(t) 2 + I H ) 1} t and { ( B (t) 2 + I H ) 1} are weakly measurable. t Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

11 Applications: Fredholm Indices, contd. Consequences of these hypotheses: A(t) = A + B(t), dom(a(t)) = dom(a ), t. There exists A + = A(+ ) = A + B(+ ), dom(a + ) = dom(a ), n-lim t ± (A(t) zi H ) 1 = (A ± zi H ) 1, (A + A )(A zi H ) 1 B 1 (H), [ (A(t) zih ) 1 (A ± zi H ) 1] B 1 (H), t, [ (A+ zi H ) 1 (A zi H ) 1] B 1 (H), σ ess (A(t)) = σ ess (A + ) = σ ess (A ), t. Lemma. Under these assumptions, and if 0 ρ(a + ) ρ(a ), Fredholm Property then D A = d dt + A on dom(d A ) = dom(d/dt) dom(a ), is closed and a Fredholm operator in L 2 (; H). Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

12 Applications: Fredholm Indices, contd. The following is proved in F.G., Y. Latushkin, K. Makarov, F. Sukochev, and Y. Tomilov, Adv. Math. 227, (2011): Theorem. Under these assumptions, and if 0 ρ(a + ) ρ(a ), Fredholm Property ind(d A ) = dim(ker(d A )) dim(ker(d A)) Fredholm Index = ξ(0 + ; H 2, H 1 ) H 1 = D AD A, H 2 = D A D A = SpFlow({A(t)} t= ) Spectral Flow = ξ(0; A +, A ) internal SSF A ± = A(± ), 0 ρ(a ± ) = π 1 lim ε 0 Im ( ln ( det H ( (A+ iεi H )(A iεi H ) 1)) Path Independence H 1 = D AD A = d 2 dt + 2 q V 1, V 1 = A 2 A, + q abbreviates the form sum, H 2 = D A D A = d 2 dt + 2 q V 2, V 2 = A 2 + A. T = T (t) dt. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

13 Applications: Fredholm Indices, contd. Two key elements in the proof: Theorem (Trace Identity). ( tr L 2 (;H) (H2 z I) 1 (H 1 z I) 1) = 1 2z tr H(g z (A + ) g z (A )), where g z (x) = x x 2 z, z C\[0, ), x, a smoothed-out sign fct. Theorem (Pushnitski s Formula, an Abel-Type Transform). ξ(λ; H 2, H 1 ) = 1 π λ 1/2 λ 1/2 ξ(ν; A +, A ) dν (λ ν 2 ) 1/2 for a.e. λ > 0. elating the external SSF, ξ( ; H 2, H 1 ) and internal SSF, ξ( ; A +, A ): ecalling, D A = (d/dt) + A, A = A(t) dt in L2 (; H), H 1 = D A D A, H 2 = D A D A, A(t) A ±, etc. t± Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

14 Applications: Fredholm Indices, contd. There s nothing special about resolvent differences of H 2 and H 1 on the l.h.s. of the trace identity! Under appropriate conditions on f one obtains tr L2 (;H) ( f (H2 ) f (H 1 ) ) = tr H ( F (A+ ) F (A ) ), where F is determined by f via respectively, by F (ν) = 1 π F (ν) = 2 1 f (0) sgn(ν) + ν 2π [ν 2, ) f (λ) dλ (λ ν 2 ) 1/2. A Besov space consideration yields that [ F (A+ ) F (A ) ] B 1 (H) if [ν 2, ) f (λ) dλ λ(λ ν 2 ) 1/2, (1 + ν 2 ) 3/4 F L 2 (; dν), (1 + ν 2 ) 9/4 F + 3ν(1 + ν 2 ) 1 F L 2 (; dν), and F L (; dν). Here s the corresponding heat kernel version: f (λ) = e sλ, where erf(x) = 2 π 1/2 x 0 dy e y 2, x. F (ν) = 1 2 erf ( s 1/2 ν ), s (0, ), Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

15 Applications: Fredholm Indices, contd. Note: D A = (d/dt) + A in L 2 (; H) is a model operator: It arises in connection with Dirac-type operators (on compact and noncompact manifolds), the Maslov index, Morse theory (index), Floer homology, winding numbers, Sturm oscillation theory, dynamical systems, etc. The literature on the spectral flow and index theory is endless: M. F. Atiyah, N. Azamov, M.-T. Benameur, B. Booss-Bavnbek, N. V. Borisov, C. Callias, A. Carey, P. Dodds, K. Furutani, P. Kirk, M. Lesch, W. Müller, N. Nicolaescu, J. Phillips, V. Patodi, A. Pushnitski, P. abier, A. ennie, J. obin, D. Salamon,. Schrader, I. Singer, F. Sukochev, C. Wahl, K. P. Wojciechowski, etc. Just scratching the surface... Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

16 Applications: Fredholm Indices, contd. Just a few selections: C. Callias, Axial anomalies and index theorems on open spaces, Commun. Math. Phys. 62, (1978). Started an avalanche in Supersymmetric QM. D. Bolle, F.G., H. Grosse, W. Schweiger, and B. Simon, Witten index, axial anomaly, and Krein s spectral shift function in supersymmetric quantum mechanics, J. Math. Phys. 28, (1987). This treats the scalar case when A(t) is a scalar function and hence dim(h) = 1 (humble beginnings!). The Krein Lifshitz spectral shift function is linked to index theory. See also,. W. Carey and J. D. Pincus, Proc. Symp. Pure Math. 44, (1986), W. Müller, Springer Lecture Notes in Math. Vol (1987). J. obbin and D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27, 1 33 (1995). A very influential paper. A. Pushnitski, The spectral flow, the Fredholm index, and the spectral shift function, in Spectral Theory of Differential Operators: M. Sh. Birman 80th Birthday Collection, AMS, 2008, pp Motivated our recent work. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

17 Applications: The Witten Index Non-Fredholm case: We will not assume 0 ρ(a + ) ρ(a ) and drop the requirement that D A is Fredholm: Definition. Let T be a closed, linear, densely defined operator in H and suppose that for some z C\[0, ), (TT z I H ) 1 (T T z I H ) 1 B 1 (H). Introduce the resolvent regularization (T, z) = z tr H ( (TT z I H ) 1 (T T z I H ) 1), z C\[0, ), and sectors S ϕ := {z C arg(z) < ϕ}. Fix ϕ (0, π/2), then the Witten index W r (T ) of T is defined by whenever this limit exists. W r (T ) = lim z 0, z C\Sϕ (T, z), The subscript r indicates the use of the resolvent regularization. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

18 Applications: The Witten Index Analogously we could use a heat kernel regularization, replacing (T, z) by [ e ttt e tt T ], t > 0, and study the limit t. For brevity, we ll restrict ourselves to the resolvent regularization in this talk. Consistency with the Fredholm index: Theorem. Suppose that T is a Fredholm operator in H. Then the Witten index W r (T ) exists, equals the Fredholm index, ind(t ), of T, and W r (T ) = ind(t ) = ξ(0 + ; TT, T T ). Note. In general (i.e., if T is not Fredholm), W r (T ) is not integer-valued, it can be any real number. In concrete 2d magnetic field systems it can have the meaning of (non-quantized) magnetic flux F. F.G. and B. Simon, Topological invariance of the Witten index, J. Funct. Anal. 79, (1988), showed that W r (T ) has stability properties w.r.t. additive perturbations similar to the Fredholm index, replacing the relative compactness assumption on the perturbation by appropriate relative trace class conditions. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

19 Applications: The Witten Index, contd., dim(h) < Now we return to the model operator D A = (d/dt) + A in L 2 (; H): (1) Special Case: dim(h) <. Assume A is a self-adjoint matrix in H. Suppose there exist families of self-adjoint matrices {B(t)} t such that B( ) is locally absolutely continuous on. Assume that dt B (t) B(H) <. In the special case dim(h) < a complete picture emerges: First, we note Lemma. Under the new set of hypotheses for dim(h) <, D A (equivalently, D A) is Fredholm if and only if 0 / {σ(a + ) σ(a )}. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

20 Applications: The Witten Index, contd., dim(h) < The non-fredholm case if dim(h) < : Theorem. (A. Carey, F.G., D. Potapov, F. Sukochev, Y. Tomilov, in prep.) Assume the new set of hypotheses for dim(h) <. Then ξ( ; H 2, H 1 ) has a continuous representative on the interval (0, ), ξ( ; A +, A ) is piecewise constant a.e. on, the Witten index W r (D A ) exists, and W r (D A ) = ξ(0 + ; H 2, H 1 ) = [ξ(0 + ; A +, A ) + ξ(0 ; A +, A )]/2 = 1 2 [# >(A + ) # > (A )] 1 2 [# <(A + ) # < (A )]. In particular, in the finite-dimensional context, dim(h) <, W r (D A ) is either an integer, or a half-integer (i.e., a Levinson-type theorem in scattering theory). Here # > (A) (resp. # < (A)) denotes the number of strictly positive (resp., strictly negative) eigenvalues of a self-adjoint operator A in H, counting multiplicity. Details rely on scattering theoretic techniques. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

21 Applications: The Witten Index, contd., dim(h) = (2) General Case: dim(h) =. Back to our Main Hypotheses: A self-adjoint on dom(a ) H, H a complex, separable Hilbert space. B(t), t, closed, symmetric, in H, dom(b(t)) dom(a ). There exists a family B (t), t, closed, symmetric, in H, with dom(b (t)) dom(a ), such that B(t)( A + I H ) 1, t, is weakly locally a.c. and for a.e. t, d ( g, B(t)( A + I H ) 1 h ) dt H = ( g, B (t)( A + I H ) 1 h ), g, h H. H B ( )( A + I H ) 1 B 1 (H), t, B (t)( A + I H ) 1 B1(H) dt <. { ( B(t) 2 + I H ) 1} and { ( B (t) 2 + I t H ) 1} are weakly measurable. t Principal objects: A(t) = A + B(t), t, and A = A(t) dt in L2 (; H). Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

22 Applications: The Witten Index, contd., dim(h) = The non-fredholm case if dim(h) = : A first fact: For ϕ (0, π/2) we introduce the sector S ϕ := {z C arg(z) < ϕ}. (3.1) Theorem. (A. Carey, F.G., D. Potapov, F. Sukochev, Y. Tomilov, in prep.) Assume the general hypotheses for dim(h) = and let ϕ (0, π/2) be fixed. If 0 is a right and left Lebesgue point of ξ( ; A +, A ), then ( W r (D A ) = lim z tr L2 (;H) (H2 z I) 1 (H 1 z I) 1) z 0, z C\S ϕ ξ(ν; A +, A ) dν = lim z z 0, z C\S ϕ (ν 2 z) 3/2 = 1 2 [ξ(0 +; A +, A ) + ξ(0 ; A +, A )]. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

23 Applications: The Witten Index, contd., dim(h) = Naturally, the proof relies on a series of careful estimates employing Pushnitski s formula, ξ(λ; H 2, H 1 ) = 1 π and the trace identity, λ 1/2 λ 1/2 ξ(ν; A +, A ) dν (λ ν 2 ) 1/2 for a.e. λ > 0, ( tr L 2 (;H) (H2 z I) 1 (H 1 z I) 1) = 1 2z tr H(g z (A + ) g z (A )), where g z (x) = x x 2 z, z C\[0, ), x, a smoothed-out sign fct. Neither formula depends on the Fredholm property of D A. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

24 Applications: The Witten Index, contd., dim(h) = Lebesgue points: Let f L 1 loc (; dx). Then x 0 is a right Lebesgue point of f if there exists an α + C such that 1 x0+h lim f (x) α + dx = 0. One then defines f (x 0,+ ) = α +. h 0 h x 0 Similarly, x 0 is a left Lebesgue point of f if there exists an α C such that 1 x0 lim f (x) α dx = 0. One then defines f (x 0, ) = α. h 0 h x 0 h Finally, x 0 is a Lebesgue point of f if there exist α C such that 1 lim h 0 2h x0+h x 0 h f (x) α dx = 0. One then defines f (x 0 ) = α. That is, x 0 is a Lebesgue point of f if and only if it is a left and a right Lebesgue point and α + = α = α. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

25 Applications: The Witten Index, contd., dim(h) = A second fact: Theorem. (A. Carey, F.G., D. Potapov, F. Sukochev, Y. Tomilov, in prep.) Assume the general hypotheses for dim(h) =. If 0 is a right and left Lebesgue point of ξ( ; A +, A ), then it is a right Lebesgue point of ξ( ; H 2, H 1 ) and ξ(0 + ; H 2, H 1 ) = 1 2 [ξ(0 +; A +, A ) + ξ(0 ; A +, A )]. The proof employs Pushnitski s formula, ξ(λ; H 2, H 1 ) = 1 π λ 1/2 λ 1/2 ξ(ν; A +, A ) dν (λ ν 2 ) 1/2 for a.e. λ > 0, and combines the right/left Lebesgue point property of ξ( ; A +, A ) at 0 with Fubini s theorem: Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

26 Applications: The Witten Index, contd., dim(h) = Sketch of Proof. Since χ [ λ, λ] (ν) 1 λ ν 2 is even w.r.t. ν, and thus ξ(λ; H 2, H 1 ) 1 2 [ξ(0 +; A +, A ) + ξ(0 ; A +, A )] = 1 π = 1 π λ λ λ 0 ξ(t; A +, A ) dν λ ν [ξ(0 +; A +, A ) + ξ(0 ; A +, A )] [ξ(ν; A +, A ) ξ(0 ; A +, A )] + [ξ( ν, A +, A ) ξ(0 ; A +, A )] dν. λ ν 2 Next, let 0 be a right and Lebesgue point of ξ( ; A +, A ), then abbreviating f ± (ν) := ξ(±ν; A +, A ) ξ(0 ± ; A +, A ), ν, and applying Fubini s theorem yields, Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

27 Applications: The Witten Index, contd., dim(h) = 1 lim h 0+ h h 0 1 = lim h 0+ πh 1 lim h 0+ πh 1 = lim h 0+ πh 2 = lim h 0+ πh 2 = lim h 0+ π h 2 lim h 0+ π h ξ(λ; H 2, H 1 ) 1 2 [ξ(0 +; A +, A ) + ξ(0 ; A +, A )] dλ h λ [f + (ν) + f (ν)] dν 0 0 λ ν 2 dλ h ( λ ) [ f + (ν) + f (ν) ] dν dλ λ ν 2 0 h 0 h 0 h 0 h 0 0 ( h [ f + (ν) + f (ν) ] ν 2 [ f + (ν) + f (ν) ] h ν 2 dν ) dλ dν λ ν 2 [ f + (ν) + f (ν) ] 1 [ν 2 /h] dν [ f + (ν) + f (ν) ] dν = 0 by the right/left L-point hyp. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28

28 Applications: The Witten Index, contd., dim(h) = Combining these results yields: Theorem. (A. Carey, F.G., D. Potapov, F. Sukochev, Y. Tomilov, in prep.) Assume the general hypotheses for dim(h) = and that 0 is a right and left Lebesgue point of ξ( ; A +, A ) (and hence a right Lebesgue point of ξ( ; H 2, H 1 )). Then, for fixed ϕ (0, π/2), W r (D A ) = lim z 0, z C\S ϕ z tr L 2 (;H) = ξ(0 + ; H 2, H 1 ) = lim z 0, z C\S ϕ z ( (H2 z I) 1 (H 1 z I) 1) ξ(ν; A +, A ) dν (ν 2 z) 3/2 = 1 2 [ξ(0 +; A +, A ) + ξ(0 ; A +, A )]. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Herrnhut, June 4, / 28