Topological Interpretation of Levinsons Theorem Johannes Kellendonk - PowerPoint PPT Presentation

Topological Interpretation of Levinson’s Theorem Johannes Kellendonk (mostly j.w. Serge Richard) Institut Camille Jordan, Universit´ e Lyon 1 Wien, September 2014

Motivation ◮ Often Bulk-edge correspondances have their origin in topology. ◮ They can often (and should best) be described by algebraic topology.

Motivation ◮ Often Bulk-edge correspondances have their origin in topology. ◮ They can often (and should best) be described by algebraic topology. ◮ In the context of quantum mechanics this is based on exact sequences (extensions) of operator algebras (Banach algebras): Two algebras J , A which are linked by an extension E : π : E → A surjective algebra morph., J ∼ = ker π . π J ֒ → E → A ◮ Boundary maps, e.g. ind : K 1 ( A ) → K 0 ( J ) , give rise to equations between topologically quantised physical quantities, one related to the system described by J the other to that by A . Example: IQHE [Kellendonk, Richter, Schulz-Baldes]

Motivation ◮ Often Bulk-edge correspondances have their origin in topology. ◮ They can often (and should best) be described by algebraic topology. ◮ In the context of quantum mechanics this is based on exact sequences (extensions) of operator algebras (Banach algebras): Two algebras J , A which are linked by an extension E : π : E → A surjective algebra morph., J ∼ = ker π . π J ֒ → E → A ◮ Boundary maps, e.g. ind : K 1 ( A ) → K 0 ( J ) , give rise to equations between topologically quantised physical quantities, one related to the system described by J the other to that by A . Example: IQHE [Kellendonk, Richter, Schulz-Baldes] ◮ I will show here that Levinson’s theorem is of that type.

Levinson’s theorem Consider H = H 0 + V on L 2 ( R d ) ◮ H 0 is ”nice” free motion (no bound states) (e.g. H 0 = − ∆ , ∂, · · · ) ◮ V (decaying) potential creating finitely many bound states ◮ σ ( H 0 ) = σ ac ( H 0 ) = σ ac ( H ) = I H 0 (interval) Scattering operator S = S ( H 0 ) , S ( E ) the scattering matrix (unitary) Time delay at energy E is iS ∗ ( E ) S ′ ( E ) .

Levinson’s theorem Theorem Integrated time delay = number of bound states + corrections.

Levinson’s theorem Theorem Integrated time delay = number of bound states + corrections. i � ( tr ( S ∗ ( E ) S ′ ( E )) − reg . ) dE = Tr ( P b ) + corr . 2 π σ ( H 0 ) � 1 if ∃ halfbound state corr . = 2 ( d = 3 ) 0 else tr trace on L 2 ( S d − 1 ) , Tr trace on L 2 ( R d ) , P b bound state projection. Halfbound state (0-energy resonance): H Ψ = 0 for Ψ ∈ L 2 loc ( R d ) \ L 2 ( R d ) but in some weighted L 2 Usual proofs involve complex analysis but it is topology!

Topological version of Levinson’s theorem 1 Compare evolution of e − itH Ψ , Ψ ∈ im P ⊥ b with e − itH 0 Ψ ± , Ψ ± ∈ L 2 ( R d ) such that lim t →±∞ � e − itH Ψ − e − itH 0 Ψ ± � = 0. e − itH � ∗ e − itH 0 wave operators. ◮ Ω ± := s − lim t →±∞ � ◮ Ω = Ω − an isometry intertwining dynamics of H 0 with that of H | ac ΩΩ ∗ = 1 − P b Ω ∗ Ω = 1 , � ∗ � S = Ω ∗ e − itH 0 Ω e − itH 0 + Ω − = s − lim t → + ∞

Topological version of Levinson’s theorem 1 Compare evolution of e − itH Ψ , Ψ ∈ im P ⊥ b with e − itH 0 Ψ ± , Ψ ± ∈ L 2 ( R d ) such that lim t →±∞ � e − itH Ψ − e − itH 0 Ψ ± � = 0. e − itH � ∗ e − itH 0 wave operators. ◮ Ω ± := s − lim t →±∞ � ◮ Ω = Ω − an isometry intertwining dynamics of H 0 with that of H | ac ΩΩ ∗ = 1 − P b Ω ∗ Ω = 1 , � ∗ � S = Ω ∗ e − itH 0 Ω e − itH 0 + Ω − = s − lim t → + ∞ Theorem ( [Kellendonk, Richard 2007]) If the wave operator Ω belongs to an extension of C ( S 1 ) by K ( H ) then the number of bound states equals the winding number of π (Ω) .

Topological version of Levinson’s theorem 1 Compare evolution of e − itH Ψ , Ψ ∈ im P ⊥ b with e − itH 0 Ψ ± , Ψ ± ∈ L 2 ( R d ) such that lim t →±∞ � e − itH Ψ − e − itH 0 Ψ ± � = 0. e − itH � ∗ e − itH 0 wave operators. ◮ Ω ± := s − lim t →±∞ � ◮ Ω = Ω − an isometry intertwining dynamics of H 0 with that of H | ac ΩΩ ∗ = 1 − P b Ω ∗ Ω = 1 , � ∗ � S = Ω ∗ e − itH 0 Ω e − itH 0 + Ω − = s − lim t → + ∞ Theorem ( [Kellendonk, Richard 2007]) If the wave operator Ω belongs to an extension of C ( S 1 ) by K ( H ) then the number of bound states equals the winding number of π (Ω) . ◮ May also consider C ( S 1 , K ( H ′ ) + ) in place of C ( S 1 ) . ◮ Part of π (Ω) should be related to the scattering oper. S so that part of the winding number is integrated time delay. ◮ Eigenvalues may be embedded. No gap condition needed! ◮ Conceptual clearness. ◮ Topologically more involved models possible.

New formulae for wave operators The condition on the wave operator is the difficult analytical part! Theorem ( [Kellendonk, Richard (d=1) 2009][Richard, Tiedra (d=3) 2013]) H 0 = − ∆ sur L 2 ( R d ) (d = 1 , 3 ), V ( x ) | 1 + x 2 | ρ d ∈ L 2 ( R d ) . Ω = 1 + R ( A )( S ( H 0 ) − 1 ) + compact x · � ∇ + � A = 1 2 ( � ∇· � x ) (gen. dilation), R ( A ) = ⊕ l ∈ N R l ( A ) (angular mom.) R 0 ( A ) = 1 1 + tanh ( π A ) − i cosh ( π A ) − 1 � � P s − wave 2 R l are smooth functions with R l ( −∞ ) = 0 , R l (+ ∞ ) = 1 .

New formulae for wave operators The condition on the wave operator is the difficult analytical part! Theorem ( [Kellendonk, Richard (d=1) 2009][Richard, Tiedra (d=3) 2013]) H 0 = − ∆ sur L 2 ( R d ) (d = 1 , 3 ), V ( x ) | 1 + x 2 | ρ d ∈ L 2 ( R d ) . Ω = 1 + R ( A )( S ( H 0 ) − 1 ) + compact x · � ∇ + � A = 1 2 ( � ∇· � x ) (gen. dilation), R ( A ) = ⊕ l ∈ N R l ( A ) (angular mom.) R 0 ( A ) = 1 1 + tanh ( π A ) − i cosh ( π A ) − 1 � � P s − wave 2 R l are smooth functions with R l ( −∞ ) = 0 , R l (+ ∞ ) = 1 . ◮ There are results in d = 2 in the absense of half bound states. ◮ Bellissard & Schulz-Baldes have studied H 0 = Laplacian on a lattice.

Some non-commutative topology H inf. dim. sep. Hilbert space, K ( H ) compact operators. W isometry of codim 1. W ∗ W = 1, WW ∗ = 1 − proj. of rank 1 . π K ( H ) ֒ → B ( H ) → B ( H ) / K ( H ) � � � π π ( C ∗ ( W )) ∼ → C ∗ ( W ) = C ( S 1 ) K ( H ) ֒ → C ∗ ( W ) = Toeplitz is C ∗ -algebra gen. by W , W ∗ .

Some non-commutative topology H inf. dim. sep. Hilbert space, K ( H ) compact operators. W isometry of codim 1. W ∗ W = 1, WW ∗ = 1 − proj. of rank 1 . π K ( H ) ֒ → B ( H ) → B ( H ) / K ( H ) � � � π π ( C ∗ ( W )) ∼ → C ∗ ( W ) = C ( S 1 ) K ( H ) ֒ → C ∗ ( W ) = Toeplitz is C ∗ -algebra gen. by W , W ∗ . Theorem (Atkinson) F ∈ B ( H ) is Fredholm whenever π ( F ) is invertible. Theorem (Index theorem; Gochberg, Krein) If F is Fredholm then ind ( F ) = − wind ( π ( F )) . ◮ index and winding number are homotopy invariant and characterise uniquely the homotopy classes.

Heisenberg pair A , B [ A , B ] = ı , σ ( A ) = σ ( B ) = R .

Heisenberg pair A , B [ A , B ] = ı , σ ( A ) = σ ( B ) = R . M = σ ( A ) × σ ( B ) a square ( R = [ −∞ , + ∞ ] ). ∂ M ∼ = S 1 .

Heisenberg pair A , B [ A , B ] = ı , σ ( A ) = σ ( B ) = R . M = σ ( A ) × σ ( B ) a square ( R = [ −∞ , + ∞ ] ). ∂ M ∼ = S 1 . ◮ K ( L 2 ( R )) = C ∗ ( f ( A ) g ( B ) | f , g ∈ C 0 ( R )) ◮ Toeplitz = C ∗ ( f ( A ) g ( B ) | f , g ∈ C ( R )) ◮ π : Toeplitz → C ( ∂ M ) is taking limits A → ±∞ or B → ±∞

Heisenberg pair A , B [ A , B ] = ı , σ ( A ) = σ ( B ) = R . M = σ ( A ) × σ ( B ) a square ( R = [ −∞ , + ∞ ] ). ∂ M ∼ = S 1 . ◮ K ( L 2 ( R )) = C ∗ ( f ( A ) g ( B ) | f , g ∈ C 0 ( R )) ◮ Toeplitz = C ∗ ( f ( A ) g ( B ) | f , g ∈ C ( R )) ◮ π : Toeplitz → C ( ∂ M ) is taking limits A → ±∞ or B → ±∞ ◮ π ( F ) = Γ 1 ◦ Γ 2 ◦ Γ 3 ◦ Γ 4 (concatenation to restrictions on sides) s →−∞ e isA F ( A , B ) e − isA Γ 1 ( A ) = s − lim t → + ∞ e itB F ( A , B ) e − itB Γ 2 ( B ) = s − lim similarily for Γ 3 , Γ 4 .

Heisenberg pair A , B [ A , B ] = ı , σ ( A ) = σ ( B ) = R . M = σ ( A ) × σ ( B ) a square ( R = [ −∞ , + ∞ ] ). ∂ M ∼ = S 1 . ◮ K ( L 2 ( R )) = C ∗ ( f ( A ) g ( B ) | f , g ∈ C 0 ( R )) ◮ Toeplitz = C ∗ ( f ( A ) g ( B ) | f , g ∈ C ( R )) ◮ π : Toeplitz → C ( ∂ M ) is taking limits A → ±∞ or B → ±∞ ◮ π ( F ) = Γ 1 ◦ Γ 2 ◦ Γ 3 ◦ Γ 4 (concatenation to restrictions on sides) s →−∞ e isA F ( A , B ) e − isA Γ 1 ( A ) = s − lim t → + ∞ e itB F ( A , B ) e − itB Γ 2 ( B ) = s − lim similarily for Γ 3 , Γ 4 . wind ( π ( F )) = w 1 + w 2 + w 3 + w 4 , � + ∞ 1 Γ − 1 ( x )Γ ′ w i = ǫ i i ( x ) dx , ǫ 1 = ǫ 2 = 1 , ǫ 3 = ǫ 4 = − 1 i 2 πı −∞ differentiability and integrability assumed.

M as energy-scale space Specify B = 1 2 ln ( − ∆) , A generator of scaling (dilation). π (Ω) = Γ 1 ◦ Γ 2 ◦ Γ 3 ◦ Γ 4 with t → + ∞ e it 1 2 ln H 0 Ω e − it 1 2 ln H 0 = S ( H 0 ) Γ 2 ( H 0 ) = s − lim Γ 4 ( H 0 ) = 1 s → + ∞ e − isA Ω e isA rescale p → e − s p Γ 1 ( A ) = s − lim here 1 + R ( A )( S ( 0 ) − 1 ) = P ⊥ = hb + ( 1 − 2 R 0 ( A )) P hb here Γ 3 ( A ) = 1 + R ( A )( S (+ ∞ ) − 1 ) = 1