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). B1(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] 2 B1(H). −1 • Relative trace class: V (H0 − zIH) 2 B1(H). −1 −1 • Resolvent comparable: (H − zIH) − (H0 − zIH) 2 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 2 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 2 L1(R; (1 + jpj)dp) (implies f 2 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) = 1)! [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 2 J (J ⊆ N an index set) denote the eigenvalues of T 2 B1(H), counting algebraic multiplicity. Then P ∗ 1=2 j2J λj (T T ) < 1 and X Y trH(T ) = λj (K); detH(IH − T ) = [1 − λj (T )]: j2J j2J 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 2 L1(R; (1 + jxj)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] 2 B1(H): 1 ξ(λ; H; H0)= lim Im(ln(DH=H0 (λ + i"))) for a.e. λ 2 R; (∗) π "!0+ where −1 −1 DH=H0 (z) = detH (H − zIH)(H0 − zIH) = detH IH + V (H0 − zIH) : Then Z Z jξ(λ; H; H0)j dλ 6 kH − H0kB1(H) and ξ(λ; H; H0) dλ = tr(H − H0): R R If dim(ran(EH0 (a − , b + ))) < 1, 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) 2 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 2 Rnσess (H0). Then, ξ(λ0 + 0; H; H0) − ξ(λ0 − 0; H; H0)= m(H0; λ0) − m(H; λ0); ∗ where m(T ; λ) 2 N [ f0g denotes the multiplicity of λ 2 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 2 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 2 ρ(A±), and where we consider the case of relative trace class perturbations [A(t) − A−], A(t)= A− + B(t); t 2 R; −1 B(t)(A− − zIH) 2 B1(H); t 2 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 2 R, 2 f 2 dom(A) = g 2 L (R; H) g(t) 2 dom(A(t)) for a.e. t 2 R; Z 2 t 7! A(t)g(t) is (weakly) measurable, kA(t)g(t)kH dt < 1 : 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 2 R,{ closed, symmetric, in H, dom(B(t)) ⊇ dom(A−). • There exists a family B0(t), t 2 R,{ closed, symmetric, in H, with 0 dom(B (t)) ⊇ dom(A−); such that −1 B(t)(jA−j + IH) ; t 2 R; is weakly locally a.c. and for a.e. t 2 R, d g; B(t)(jA j + I )−1h = g; B0(t)(jA j + I )−1h ; g; h 2 H: dt − H H − H H 0 −1 R 0 −1 • B (·)(jA−j + IH) 2 B1(H), t 2 , B (t)(jA−j + IH) dt < 1. R R B1(H) 2 −1 0 2 −1 • (jB(t)j + IH) and (jB (t)j + IH) are weakly measurable. t2R t2R 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 2 R. There exists A+ = A(+1)= A− + B(+1), dom(A+) = dom(A−), −1 −1 n-limt→±∞(A(t) − zIH) = (A± − zIH) , −1 (A+ − A−)(A− − zIH) 2 B1(H), −1 −1 (A(t) − zIH) − (A± − zIH) 2 B1(H), t 2 R, −1 −1 (A+ − zIH) − (A− − zIH) 2 B1(H), σess(A(t)) = σess(A+) = σess(A−), t 2 R. Lemma. Under these assumptions, and if 0 2 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 2 ρ(A+) \ ρ(A−), ! Fredholm Property ∗ ind(DA) = dim(ker(DA)) − dim(ker(DA)) Fredholm Index ∗ ∗ = ξ(0+; H2; H1) H1 = DADA; H2 = DADA 1 = SpFlow(fA(t)gt=−∞) Spectral Flow = ξ(0; A+; A−) internal SSF A± = A(±∞); 0 2 ρ(A±) −1 −1 = π lim Im ln detH (A+ − i"IH)(A− − i"IH) Path Independence "#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.