Spectral Shift Functions and Some of Their Applications in Mathematical Physics

Spectral Shift Functions and some of their Applications in Mathematical Physics

Fritz Gesztesy (University of Missouri, Columbia, USA)

based on collaborations with

A. Carey (Canberra), Y. Latushkin (Columbia–MO), K. Makarov (Columbia–MO), D. Potapov (Sydney), F. Sukochev (Sydney), Y. Tomilov (Torun)

Quantum Spectra and Transport The Hebrew University of Jerusalem, Israel June 30–July 4, 2013

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 1 / 29 Outline

1 Krein–Lifshitz SSF

2 SSF Applications

3 Final Thoughts

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 2 / 29 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.

Bs (H), s ≥ 1, denote the usual Schatten–von Neumann trace ideals.

Given two self-adjoint operators H, H0 in H, we assume one of the following (we denote V = [H − H0] whenever dom(H) = dom(H0)):

• Trace class assumption: V = [H − H0] ∈ B1(H). −1 • Relative trace class: V (H0 − zIH) ∈ B1(H). −1 −1 • Resolvent comparable: (H − zIH) − (H0 − zIH) ∈ B1(H).

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 3 / 29 Krein–Lifshitz SSF A short course on the spectral shift function ξ

The Krein–Lifshitz spectral shift function ξ(λ; H, H0) is a real-valued function on R that satisﬁes the trace formula: For appropriate functions f ,

Z 0 trH(f (H) − f (H0)) = f (λ) ξ(λ; H, H0) dλ. R

For example, resolvents,

Z ξ(λ; H, H ) dλ tr (H − zI )−1 − (H − zI )−1 = − 0 , H H 0 H (λ − z)2 R

semigroups (if H, H0 are bounded from below), etc. E.g., for f ∈ C 1( ) with f 0(λ) = R e−iλs dσ(s) and dσ a ﬁnite signed measure; R R f (λ) = (λ − z)−1; or bf ∈ L1(R; (1 + |p|)dp) (implies f ∈ C 1(R)), 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 Jerusalem, June 30, 2013 4 / 29 Krein–Lifshitz SSF A short course on the spectral shift function ξ

Generally, ξ(λ; H, H0)=tr H(EH0 (λ) − EH (λ)) is not correct (if dim(H) = ∞)!

[EH0 (λ) − EH (λ)] is NOT necessarily of trace class even for V = H − H0 of rank one!

Recall the Trace and the Determinant: Let λj (T ), j ∈ J (J ⊆ N an index set) denote the eigenvalues of T ∈ B1(H), counting algebraic multiplicity. Then P ∗ 1/2 j∈J λj (T T ) < ∞ and X Y trH(T ) = λj (K), detH(IH − T ) = [1 − λj (T )]. j∈J j∈J

The Perturbation Determinant: H and H0 self-adjoint in H, H = H0 + V , −1 −1 DH/H0 (z) = detH (H − zIH)(H0 − zIH) = detH IH + V (H0 − zIH) .

2 2 2 2 2 Example. If H0 = −(d /dx ), H = −(d /dx ) + V (·) in L (R; dx), V ∈ L1(R; (1 + |x|)dx), real-valued, then

DH/H0 (z)= Jost function = Evans function.

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 5 / 29 Krein–Lifshitz SSF A short course on the spectral shift function ξ

General Krein’s formula for the trace class V = [H − H0] ∈ B1(H): 1 ξ(λ; H, H0)= lim Im(ln(DH/H0 (λ + iε))) for a.e. λ ∈ R, (∗) π ε→0+ where

−1 −1 DH/H0 (z) = detH (H − zIH)(H0 − zIH) = detH IH + V (H0 − zIH) .

Then Z Z

|ξ(λ; H, H0)| dλ 6 kH − H0kB1(H) and ξ(λ; H, H0) dλ = tr(H − H0). R R

If dim(ran(EH0 (a − , b + ))) < ∞, then

ξ(b − 0; H, H0)−ξ(a + 0; H, H0) = dim(ran(EH0 (a, b)))−dim(ran(EH (a, b))). (∗∗)

Note. Formulas (∗) and (∗∗) remain valid in the relative trace class case, −1 where V (H0 − zIH) ∈ B1(H).= ⇒ Eigenvalue counting function.

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 6 / 29 Krein–Lifshitz SSF A short course on the spectral shift function ξ

Away from σess (H0) = σess (H), ξ(λ; H, H0) is piecewise constant:

Counting multiplicity: Suppose λ0 ∈ R\σess (H0). Then,

ξ(λ0 + 0; H, H0) − ξ(λ0 − 0; H, H0)= m(H0; λ0) − m(H; λ0),

∗ where m(T , λ) ∈ N ∪ {0} denotes the multiplicity of λ ∈ σp(T ), T = T .

Scattering theory: The Birman–Krein formula

Let S(H, H0) be the scattering operator for the pair (H, H0). Then

Z ⊕ Z ⊕ S(H, H0)= S(λ; H, H0) dλ in H' K dλ, σac (H0) σac (H0)

where S(λ; H, H0) is the ﬁxed energy scattering operator in K (assuming uniform spectral multiplicity of σac (H0), for simplicity). Then 1 ξ(λ; H, H )= − ln(det (S(λ; H, H )) for a.e. λ ∈ σ (H ). 0 2πi K 0 ac 0

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 7 / 29 SSF Applications Applications: Fredholm Indices

Computing Fredholm indices: Consider the model operator

2 DA = (d/dt) + A, dom(DA) = dom(d/dt) ∩ dom(A−) in L (R; H),

where dom(d/dt) = W 1,2(R; H), and Z ⊕ Z ⊕ Z ⊕ 2 A = A(t) dt, A− = A− dt in L (R; H) ' H dt, R R R

A± = 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 ∈ R,

−1 B(t)(A− − zIH) ∈ B1(H), t ∈ R (plus quite a bit more −→ next page),

2 such that DA becomes a Fredholm operator in L (R; H).

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 8 / 29 SSF Applications Applications: Fredholm Indices

Just for clarity,

Z ⊕ Z ⊕ 2 A = A(t) dt in L (R; H) ' H dt, H a separable, complex H-space, R R

denotes the operator in L2(R; H) deﬁned by

(Af )(t) = A(t)f (t) for a.e. t ∈ R, 2 f ∈ dom(A) = g ∈ L (R; H) g(t) ∈ dom(A(t)) for a.e. t ∈ R,

Z 2 t 7→ A(t)g(t) is (weakly) measurable, kA(t)g(t)kH dt < ∞ . R

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 9 / 29 SSF Applications 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 ∈ R,– closed, symmetric, in H, dom(B(t)) ⊇ dom(A−).

• There exists a family B0(t), t ∈ R,– closed, symmetric, in H, with

0 dom(B (t)) ⊇ dom(A−), such that −1 B(t)(|A−| + IH) , t ∈ R, is weakly locally a.c. and for a.e. t ∈ R, d g, B(t)(|A | + I )−1h = g, B0(t)(|A | + I )−1h , g, h ∈ H. dt − H H − H H

0 −1 R 0 −1 • B (·)(|A−| + IH) ∈ B1(H), t ∈ , B (t)(|A−| + IH) dt < ∞. R R B1(H)

2 −1 0 2 −1 • (|B(t)| + IH) and (|B (t)| + IH) are weakly measurable. t∈R t∈R

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 10 / 29 SSF Applications Applications: Fredholm Indices, contd.

Consequences of these hypotheses:

A(t)= A− + B(t), dom(A(t)) = dom(A−), t ∈ R.

There exists A+ = A(+∞)= A− + B(+∞), dom(A+) = dom(A−),

−1 −1 n-limt→±∞(A(t) − zIH) = (A± − zIH) ,

−1 (A+ − A−)(A− − zIH) ∈ B1(H),

−1 −1 (A(t) − zIH) − (A± − zIH) ∈ B1(H), t ∈ R,

−1 −1 (A+ − zIH) − (A− − zIH) ∈ B1(H),

σess(A(t)) = σess(A+) = σess(A−), t ∈ R. Lemma.

Under these assumptions, and if 0 ∈ ρ(A+) ∩ ρ(A−), ←→ Fredholm Property d then DA = dt + A on dom(DA) = dom(d/dt) ∩ dom(A−), is closed and a Fredholm operator in L2(R; H).

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 11 / 29 SSF Applications Applications: Fredholm Indices, contd.

The following is proved in F.G., Y. Latushkin, K. Makarov, F. Sukochev, and Y. Tomilov, Adv. Math. 227, 319–420 (2011):

Theorem.

Under these assumptions, and if 0 ∈ ρ(A+) ∩ ρ(A−), ←→ Fredholm Property

∗ ind(DA) = dim(ker(DA)) − dim(ker(DA)) Fredholm Index ∗ ∗ = ξ(0+; H2, H1) H1 = DADA, H2 = DADA ∞ = SpFlow({A(t)}t=−∞) Spectral Flow

= ξ(0; A+, A−) internal SSF A± = A(±∞), 0 ∈ ρ(A±) −1 −1 = π lim Im ln detH (A+ − iεIH)(A− − iεIH) Path Independence ε↓0

∗ d 2 2 0 H1 = DADA = − dt2 +q V1, V1 = A − A , “+q” abbreviates the form sum, ∗ d 2 2 0 R ⊕ H2 = D D = − 2 +q V2, V2 = A + A . T = T (t) dt. A A dt R Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 12 / 29 SSF Applications Applications: Fredholm Indices, contd.

Two key elements in the proof:

Theorem (Trace Identity).

−1 −1 1 tr 2 (H − z I) − (H − z I) = tr (g (A ) − g (A )), L (R;H) 2 1 2z H z + z − where g (x) = √ x , z ∈ \[0, ∞), x ∈ , a smoothed-out sign fct. z x2−z C R

Theorem (Pushnitski’s Formula, an Abel-Type Transform).

1/2 1 Z λ ξ(ν; A , A ) dν ξ(λ; H , H )= + − for a.e. λ > 0. 2 1 2 1/2 π −λ1/2 (λ − ν )

Relating the external SSF, ξ( · ; H2, H1) and internal SSF, ξ( · ; A+, A−):

Recalling, D = (d/dt) + A, A = R ⊕ A(t) dt in L2( ; H), A R R ∗ ∗ H1 = D D , H2 = D D , A(t) −→ A±, etc. A A A A t±∞

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 13 / 29 SSF Applications Applications: Fredholm Indices, contd.

There’s nothing special about resolvent diﬀerences of H2 and H1 on the l.h.s. of the trace identity! Under appropriate conditions on f one obtains 2 trL (R;H) f (H2) − f (H1) = trH F (A+) − F (A−) , where F is determined by f via

−1 ν R f (λ) dλ F (ν) = −2 f (0) sgn(ν) + 2π [ν2,∞) λ(λ−ν2)1/2 ,

0 0 1 R f (λ) dλ respectively, by F (ν) = π [ν2,∞) (λ−ν2)1/2 . A Besov space consideration yields that F (A+) − F (A−) ∈ B1(H) if

(1 + ν2)3/4F 0 ∈ L2(R; dν), (1 + ν2)9/4 F 00 + 3ν(1 + ν2)−1F 0 ∈ L2(R; dν), and F 0 ∈ L∞(R; dν). Here’s the corresponding heat kernel version: −sλ 1 1/2 f (λ) = e , F (ν) = − 2 erf s ν , s ∈ (0, ∞), 2 R x −y 2 where erf(x) = π1/2 0 dy e , x ∈ R. Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 14 / 29 SSF Applications Applications: Fredholm Indices, contd.

2 Note:D A = (d/dt) + A in L (R; 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. M¨uller,N. Nicolaescu, J. Phillips, V. Patodi, A. Pushnitski, P. Rabier, A. Rennie, J. Robin, D. Salamon, R. 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 Jerusalem, June 30, 2013 15 / 29 SSF Applications Applications: Fredholm Indices, contd.

Just a few selections: C. Callias, Axial anomalies and index theorems on open spaces, Commun. Math. Phys. 62, 213–234 (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, 1512–1525 (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, R. W. Carey and J. D. Pincus, Proc. Symp. Pure Math. 44, 149–161 (1986), W. Müller, Springer Lecture Notes in Math. Vol. 1244 (1987). J. Robbin 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. 141–155. Motivated our recent work.

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 16 / 29 SSF Applications Applications: The Witten Index

Non-Fredholm case: We will not assume 0 ∈ ρ(A+) ∩ ρ(A−) and drop the

requirement that DA is Fredholm: Deﬁnition. Let T be a closed, linear, densely deﬁned operator in H and suppose that for some z ∈ C\[0, ∞), ∗ −1 ∗ −1 (TT − z IH) − (T T − z IH) ∈ B1(H). Introduce the resolvent regularization ∗ −1 ∗ −1 ∆(T , z) = z trH (TT − z IH) − (T T − z IH) , z ∈ C\[0, ∞),

and sectors Sϕ := {z ∈ C | | arg(z)| < ϕ}. Fix ϕ ∈ (0, π/2), then the Witten index Wr (T ) of T is deﬁned by W (T ) = lim ∆(T , z), r z→0, z∈C\Sϕ whenever this limit exists.

The subscript “r” indicates the use of the resolvent regularization.

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 17 / 29 SSF Applications 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 Wr (T ) exists, equals the Fredholm index, ind(T ), of T , and ∗ ∗ Wr (T ) = ind(T ) = ξ(0+; TT , T T ).

Note. In general (i.e., if T is not Fredholm), Wr (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 ∈ R. F.G. and B. Simon, Topological invariance of the Witten index, J. Funct. Anal. 79, 91–102 (1988), showed that Wr (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 Jerusalem, June 30, 2013 18 / 29 SSF Applications Applications: The Witten Index, contd., dim(H) < ∞

2 Now we return to the model operatorD A = (d/dt) + A in L (R; 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∈R such that B(·) is locally absolutely continuous on R. R 0 • Assume that dt B (t) < ∞. R 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) < ∞, DA (equivalently, DA) is Fredholm if and only if 0 ∈/ {σ(A+) ∪ σ(A−)}.

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 19 / 29 SSF Applications 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( · ; H2, H1) has a continuous representative on the interval (0, ∞), ξ( · ; A+, A−) is piecewise constant a.e. on R, the Witten index Wr (DA) exists, and

Wr (DA)= ξ(0+; H2, H1) = [ξ(0+; A+, A−)+ ξ(0−; A+, A−)]/2 1 1 = 2 [#>(A+) − #>(A−)] − 2 [#<(A+) − #<(A−)].

In particular, in the ﬁnite-dimensional context, dim(H) < ∞, Wr (DA) 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 Jerusalem, June 30, 2013 20 / 29 SSF Applications 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 ∈ R,– closed, symmetric, in H, dom(B(t)) ⊇ dom(A−). • There exists a family B0(t), t ∈ R,– closed, symmetric, in H, with

0 −1 R 0 −1 • B (·)(|A−| + IH) ∈ B1(H), t ∈ , B (t)(|A−| + IH) dt < ∞. R R B1(H) 2 −1 0 2 −1 • (|B(t)| + IH) and (|B (t)| + IH) are weakly measurable. t∈R t∈R

R ⊕ 2 Principal objects: A(t)= A− + B(t), t ∈ , and A = A(t) dt in L ( ; H). R R R

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 21 / 29 SSF Applications 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

Sϕ := {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

−1 −1 2 Wr (DA) = lim z trL (R;H) (H2 − z I) − (H1 − z I) z→0, z∈C\Sϕ Z ξ(ν; A , A ) dν = − lim z + − 2 3/2 z→0, z∈ \Sϕ (ν − z) C R 1 = [ξ(0 ; A , A ) + ξ(0 ; A , A )]. 2 + + − − + −

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 22 / 29 SSF Applications Applications: The Witten Index, contd., dim(H) = ∞

Naturally, the proof relies on a series of careful estimates employing Pushnitski’s formula,

1/2 1 Z λ ξ(ν; A , A ) dν ξ(λ; H , H )= + − for a.e. λ > 0, 2 1 2 1/2 π −λ1/2 (λ − ν ) and the trace identity,

−1 −1 1 tr 2 (H − z I) − (H − z I) = tr (g (A ) − g (A )), L (R;H) 2 1 2z H z + z −

where g (x) = √ x , z ∈ \[0, ∞), x ∈ , a smoothed-out sign fct. z x2−z C R

Neither formula depends on the Fredholm property of DA.

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 23 / 29 SSF Applications Applications: The Witten Index, contd., dim(H) = ∞

1 Lebesgue points: Let f ∈ Lloc(R; dx).

Then x0 ∈ R is a right Lebesgue point of f if there exists an α+ ∈ C such that

1 Z x0+h lim |f (x) − α+| dx = 0. One then deﬁnes f (x0,+) = α+. h↓0 h x0

Similarly, x0 ∈ R is a left Lebesgue point of f if there exists an α− ∈ C such that 1 Z x0 lim |f (x) − α−| dx = 0. One then deﬁnes f (x0,−) = α−. h↓0 h x0−h

Finally, x0 ∈ R is a Lebesgue point of f if there exist α ∈ C such that

1 Z x0+h lim |f (x) − α| dx = 0. One then deﬁnes f (x0) = α. h↓0 2h x0−h

That is, x0 ∈ R 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 Jerusalem, June 30, 2013 24 / 29 SSF Applications 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( · ; H2, H1) and 1 ξ(0 ; H , H )= [ξ(0 ; A , A ) + ξ(0 ; A , A )]. + 2 1 2 + + − − + −

The proof employs Pushnitski’s formula,

1/2 1 Z λ ξ(ν; A , A ) dν ξ(λ; H , H )= + − for a.e. λ > 0, 2 1 2 1/2 π −λ1/2 (λ − ν )

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 Jerusalem, June 30, 2013 25 / 29 SSF Applications Applications: The Witten Index, contd., dim(H) = ∞

Sketch of Proof.

√ √ 1 Since χ[− λ, λ](ν) λ−ν2 is even w.r.t. ν ∈ R, and thus 1 ξ(λ; H , H ) − [ξ(0 ; A , A )+ ξ(0 ; A , A )] 2 1 2 + + − − + − √ Z λ 1 ξ(t; A+, A−) dν 1 = √ − [ξ(0+; A+, A−)+ ξ(0−; A+, A−)] √ 2 π − λ λ − ν 2 √ 1 Z λ [ξ(ν; A , A ) − ξ(0 ; A , A )] + [ξ(−ν, A , A ) − ξ(0 ; A , A )] = + − − + −√ + − − + − dν. 2 π 0 λ − ν

Next, let 0 be a right and a left Lebesgue point of ξ( · ; A+, A−), then abbreviating

f±(ν) := ξ(±ν; A+, A−) − ξ(0±; A+, A−), ν ∈ R, and applying Fubini’s theorem yields,

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 26 / 29 SSF Applications Applications: The Witten Index, contd., dim(H) = ∞

Z h 1 1 lim ξ(λ; H2, H1) − [ξ(0+; A+, A−)+ ξ(0−; A+, A−)] dλ h↓0+ h 0 2 √ 1 Z h Z λ [f (ν)+ f (ν)] dν = lim + √ − dλ 2 h↓0+ πh 0 0 λ − ν √ 1 Z h Z λ [|f (ν)| + |f (ν)|] dν ≤ lim + √ − dλ 2 h↓0+ πh 0 0 λ − ν √ 1 Z h Z h dλ = lim [|f+(ν)| + |f−(ν)|] √ dν 2 h↓0+ πh 0 ν2 λ − ν √ Z h 2 p 2 = lim [|f+(ν)| + |f−(ν)|] h − ν dν h↓0+ πh 0 √ Z h 2 p 2 = lim √ [|f+(ν)| + |f−(ν)|] 1 − [ν /h] dν h↓0+ π h 0 √ 2 Z h ≤ lim √ [|f+(ν)| + |f−(ν)|] dν = 0 by the right/left L-point hyp. h↓0+ π h 0 Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 27 / 29 SSF Applications 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( · ; H2, H1)). Then, for ﬁxed ϕ ∈ (0, π/2),

−1 −1 2 Wr (DA) = lim z trL (R;H) (H2 − z I) − (H1 − z I) z→0, z∈C\Sϕ

= ξ(0+; H2, H1) Z ξ(ν; A , A ) dν = − lim z + − 2 3/2 z→0, z∈ \Sϕ (ν − z) C R 1 = [ξ(0 ; A , A ) + ξ(0 ; A , A )]. 2 + + − − + −

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 28 / 29 Final Thoughts Final Thoughts:

A VERY HAPPY (belated) BIRTHDAY, YOSI!

Fritz Gesztesy (Univ. of Missouri, Columbia) The Spectral Shift Function Jerusalem, June 30, 2013 29 / 29