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Spectral Shift Functions and Some of Their Applications in Mathematical Physics

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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). 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.
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