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Limit Operators Getting your hands on the essentials. Marko Lindner 14-18 August 2017 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 1 / 44 ...with the help of... This talk is based on joint work

  1. Essential spectrum and norm: a general diagonal matrix For a general (bounded) diagonal matrix   ...   a − 1 , − 1     a 0 , 0   A = ,   a 1 , 1     a 2 , 2     ... it holds that spec ess A = the set of all partial limits of the sequence ( a n , n ) n ∈ Z . In other words: λ ∈ spec ess A ⇐ ⇒ ∃ n 1 , n 2 , · · · → ±∞ : a n k , n k → λ. Moreover, � A � ess = the largest (in modulus) partial limit = lim sup | a n , n | . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 13 / 44

  2. Essential spectrum and norm: a general diagonal matrix For a general (bounded) diagonal matrix   ...   a − 1 , − 1     a 0 , 0   A = ,   a 1 , 1     a 2 , 2     ... it holds that spec ess A = the set of all partial limits of the sequence ( a n , n ) n ∈ Z . In other words: λ ∈ spec ess A ⇐ ⇒ ∃ n 1 , n 2 , · · · → ±∞ : a n k , n k → λ. Moreover, � A � ess = the largest (in modulus) partial limit = lim sup | a n , n | . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 14 / 44

  3. Essential spectrum and norm: a general diagonal matrix For a general (bounded) diagonal matrix   ...   a − 1 , − 1     a 0 , 0   A = ,   a 1 , 1     a 2 , 2     ... it holds that spec ess A = the set of all partial limits of the sequence ( a n , n ) n ∈ Z � A � ess = the largest (in modulus) partial limit = lim sup | a n , n | . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 15 / 44

  4. Essential spectrum and norm: a general diagonal matrix For a general (bounded) diagonal matrix   ...   a − 1 , − 1     a 0 , 0   A = ,   a 1 , 1     a 2 , 2     ... it holds that spec ess A = the set of all partial limits of the sequence ( a n , n ) n ∈ Z � A � ess = the largest (in modulus) partial limit = lim sup | a n , n | . The whole coset A + K ( X ) ∈ L ( X ) / K ( X ) is encoded in the partial limits of ( a n , n ) n ∈ Z . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 15 / 44

  5. Essential spectrum and norm: a general diagonal matrix For a general (bounded) diagonal matrix   ...   a − 1 , − 1     a 0 , 0   A = ,   a 1 , 1     a 2 , 2     ... it holds that spec ess A = the set of all partial limits of the sequence ( a n , n ) n ∈ Z � A � ess = the largest (in modulus) partial limit = lim sup | a n , n | . The whole coset A + K ( X ) ∈ L ( X ) / K ( X ) is encoded in the partial limits of ( a n , n ) n ∈ Z . Restricting consideration to diagonal matrices, the Calkin algebra is L diag ( X ) / K diag ( X ) ∼ = ℓ ∞ / c 0 . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 15 / 44

  6. Table of Contents The essentials 1 Limit operators 2 Stability of approximation methods 3 The Fibonacci Hamiltonian 4 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 16 / 44

  7. From diagonal to band-dominated matrices For A ∈ BDO( X ), the coset A + K ( X ) is still determined by the asymptotics of A = ( a i , j ) at infinity. Again, take a sequence n 1 , n 2 , · · · → ±∞ and follow the entries a n k , n k as k → ∞ . (1) New: Now also the context of the entries (1) is important. TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 17 / 44

  8. From diagonal to band-dominated matrices For A ∈ BDO( X ), the coset A + K ( X ) is still determined by the asymptotics of A = ( a i , j ) at infinity. Again, take a sequence n 1 , n 2 , · · · → ±∞ and follow the entries a n k , n k as k → ∞ . (1) New: Now also the context of the entries (1) is important. Not only the sequence (1) itself shall converge but also its neighbour entries: a n k + i , n k + j → : b i , j ∀ i , j ∈ Z . The existence of such sequences h = ( n k ) is gua- ranteed by the Bolzano-Weierstrass theorem. Definition: limit operator The operator with matrix B = ( b i , j ) i , j ∈ Z is called limit operator of A w.r.t. the sequence h . We write A h for B and σ op ( A ) for the set of all A h . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 17 / 44

  9. Limit operators: Time for examples A periodic matrix: σ op � � � � = all shifts of TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 18 / 44

  10. Limit operators: Time for examples A periodic matrix: σ op � � � � = all shifts of Simple but non-periodic: σ op � � � � = all shifts of , TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 18 / 44

  11. Limit operators: Time for examples Discrete Schr¨ odinger operator in 1D ( Ax ) n = x n − 1 + v ( n ) x n + x n +1 , n ∈ Z with a bounded potential v ∈ ℓ ∞ ( Z ). The matrix looks like this   ... ...    ...    1 v − 2     1 1 v − 1   A =   1 1 v 0     1 v 1 1   ...   1 v 2     ... ... TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 19 / 44

  12. Limit operators: Time for examples Discrete Schr¨ odinger operator in 1D ( Ax ) n = x n − 1 + v ( n ) x n + x n +1 , n ∈ Z with a bounded potential v ∈ ℓ ∞ ( Z ). The matrix looks like this   ... ...    ...    1 v − 2     1 1 v − 1   A =   1 1 v 0     1 v 1 1   ...   1 v 2     ... ... Limit op’s of A : ( Bx ) n = x n − 1 + w ( n ) x n + x n +1 , n ∈ Z with a potential w “locally representing v at infinity”. TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 19 / 44

  13. Example: Discrete Schr¨ odinger operator So it is enough to look at the potential v : Example 1: locally constant potential v = ( · · · , β, β, β, β , α, α, α , β, β , α , β, β , α, α, α , β, β, β, β , · · · ) , α � = β ���� � �� � � �� � ���� ���� � �� � � �� � 1 4 3 2 2 3 4 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 20 / 44

  14. Example: Discrete Schr¨ odinger operator So it is enough to look at the potential v : Example 1: locally constant potential v = ( · · · , β, β, β, β , α, α, α , β, β , α , β, β , α, α, α , β, β, β, β , · · · ) , α � = β ���� � �� � � �� � ���� ���� � �� � � �� � 1 4 3 2 2 3 4 ⇒ 4 limop’s: w = ( · · · , α, α, α, α, α, α, · · · ) = ( · · · , β, β, β, β, β, β, · · · ) w = ( · · · , α, α, α, β, β, β, · · · ) w = ( · · · , β, β, β, α, α, α, · · · ) w ...and shifts of the latter two. TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 20 / 44

  15. Example: Discrete Schr¨ odinger operator So it is enough to look at the potential v : Example 1: locally constant potential v = ( · · · , β, β, β, β , α, α, α , β, β , α , β, β , α, α, α , β, β, β, β , · · · ) , α � = β ���� � �� � � �� � ���� ���� � �� � � �� � 1 4 3 2 2 3 4 ⇒ 4 limop’s: w = ( · · · , α, α, α, α, α, α, · · · ) = ( · · · , β, β, β, β, β, β, · · · ) w = ( · · · , α, α, α, β, β, β, · · · ) w = ( · · · , β, β, β, α, α, α, · · · ) w ...and shifts of the latter two. Example 2: slowly oscillating potential v ( n + 1) − v ( n ) → 0 , n → ∞ � e.g. v ( n ) = cos | n | . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 20 / 44

  16. Example: Discrete Schr¨ odinger operator So it is enough to look at the potential v : Example 1: locally constant potential v = ( · · · , β, β, β, β , α, α, α , β, β , α , β, β , α, α, α , β, β, β, β , · · · ) , α � = β ���� � �� � � �� � ���� ���� � �� � � �� � 1 4 3 2 2 3 4 ⇒ 4 limop’s: w = ( · · · , α, α, α, α, α, α, · · · ) = ( · · · , β, β, β, β, β, β, · · · ) w = ( · · · , α, α, α, β, β, β, · · · ) w = ( · · · , β, β, β, α, α, α, · · · ) w ...and shifts of the latter two. Example 2: slowly oscillating potential v ( n + 1) − v ( n ) → 0 , n → ∞ � e.g. v ( n ) = cos | n | . ⇒ Limop’s (all constant ): w ( n ) ≡ a , a ∈ v ( ∞ ) TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 20 / 44

  17. Example: Discrete Schr¨ odinger operator Example 3: random (actually pseudo-ergodic) potential Take random (i.i.d.) samples v ( n ) from a compact set V ⊂ C . a . s . = ⇒ The infinite “word” ( · · · , v ( − 1) , v (0) , v (1) , · · · ) contains every finite word over V as a subword (up to arbitrary accuracy ε > 0). [ pseudo-ergodic , Davies 2001] TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 21 / 44

  18. Example: Discrete Schr¨ odinger operator Example 3: random (actually pseudo-ergodic) potential Take random (i.i.d.) samples v ( n ) from a compact set V ⊂ C . a . s . = ⇒ The infinite “word” ( · · · , v ( − 1) , v (0) , v (1) , · · · ) contains every finite word over V as a subword (up to arbitrary accuracy ε > 0). [ pseudo-ergodic , Davies 2001] ⇒ lots of limop’s: all functions w : Z → V TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 21 / 44

  19. Example: Discrete Schr¨ odinger operator Example 3: random (actually pseudo-ergodic) potential Take random (i.i.d.) samples v ( n ) from a compact set V ⊂ C . a . s . = ⇒ The infinite “word” ( · · · , v ( − 1) , v (0) , v (1) , · · · ) contains every finite word over V as a subword (up to arbitrary accuracy ε > 0). [ pseudo-ergodic , Davies 2001] ⇒ lots of limop’s: all functions w : Z → V Example 4: (almost-)periodic potential v ( n ) = cos( n α ) , n ∈ Z Case 1: α = p q 2 π ∈ π Q (periodic) ⇒ q limop’s: w k ( n ) = cos(( n + k ) α ) , k = 1 , ..., q TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 21 / 44

  20. Example: Discrete Schr¨ odinger operator Example 3: random (actually pseudo-ergodic) potential Take random (i.i.d.) samples v ( n ) from a compact set V ⊂ C . a . s . = ⇒ The infinite “word” ( · · · , v ( − 1) , v (0) , v (1) , · · · ) contains every finite word over V as a subword (up to arbitrary accuracy ε > 0). [ pseudo-ergodic , Davies 2001] ⇒ lots of limop’s: all functions w : Z → V Example 4: (almost-)periodic potential v ( n ) = cos( n α ) , n ∈ Z Case 1: α = p q 2 π ∈ π Q (periodic) ⇒ q limop’s: w k ( n ) = cos(( n + k ) α ) , k = 1 , ..., q Case 2: α �∈ π Q (almost-periodic, see Almost-Mathieu operator) ⇒ ∞ -many limop’s: w θ ( n ) = cos( n α + θ ) , θ ∈ [0 , 2 π ) TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 21 / 44

  21. Limit operators: The definition revisited For each n ∈ Z , define the n -shift on X = ℓ p ( Z ) via S n : x �→ y with x i = y i + n . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 22 / 44

  22. Limit operators: The definition revisited For each n ∈ Z , define the n -shift on X = ℓ p ( Z ) via S n : x �→ y with x i = y i + n . Then, for h = ( n 1 , n 2 , ... ) with | n k | → ∞ , one has ( S − n k AS n k ) i , j = A i + n k , j + n k , i , j ∈ Z , so that the limit operator A h of A ∈ BDO( X ) equals A h = k →∞ S − n k AS n k , lim the limit taken in the strong topology (pointwise convergence on X ). TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 22 / 44

  23. Limit operators: The definition revisited For each n ∈ Z , define the n -shift on X = ℓ p ( Z ) via S n : x �→ y with x i = y i + n . Then, for h = ( n 1 , n 2 , ... ) with | n k | → ∞ , one has ( S − n k AS n k ) i , j = A i + n k , j + n k , i , j ∈ Z , so that the limit operator A h of A ∈ BDO( X ) equals A h = k →∞ S − n k AS n k , lim the limit taken in the strong topology (pointwise convergence on X ). In this sense, the set σ op ( A ) of all limit operators of A is the set of all partial limits of the operator sequence ( S − n AS n ) n ∈ Z . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 22 / 44

  24. Limit operators: The definition revisited The very same can be done with X = ℓ p ( Z d ) and A ∈ BDO( X ). Again, the set σ op ( A ) of all limit operators of A is the set of all partial limits of the operator “sequence” f A : n ∈ Z d �→ S − n AS n ∈ BDO( X ) . ( S − n AS n ) n ∈ Z d or, likewise, of the function TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 23 / 44

  25. Limit operators: The definition revisited The very same can be done with X = ℓ p ( Z d ) and A ∈ BDO( X ). Again, the set σ op ( A ) of all limit operators of A is the set of all partial limits of the operator “sequence” f A : n ∈ Z d �→ S − n AS n ∈ BDO( X ) . ( S − n AS n ) n ∈ Z d or, likewise, of the function Take a suitable compactification of Z d . Extend the function f A continuously to it. Evaluate f A at the boundary ∂ Z d ⇒ limit operators of A One can enumerate the limit operators of A by the elements of ∂ Z d (rather than by sequences h = ( n 1 , n 2 , . . . ) for which S − n k AS n k converges). TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 23 / 44

  26. Limit operators: The definition revisited The very same can be done with X = ℓ p ( Z d ) and A ∈ BDO( X ). Again, the set σ op ( A ) of all limit operators of A is the set of all partial limits of the operator “sequence” f A : n ∈ Z d �→ S − n AS n ∈ BDO( X ) . ( S − n AS n ) n ∈ Z d or, likewise, of the function Take a suitable compactification of Z d . Extend the function f A continuously to it. Evaluate f A at the boundary ∂ Z d ⇒ limit operators of A One can enumerate the limit operators of A by the elements of ∂ Z d (rather than by sequences h = ( n 1 , n 2 , . . . ) for which S − n k AS n k converges). These ideas can be extended from ℓ p ( Z d ) [Rabinovich, Roch, Silbermann 1998] to ℓ p ( G ) for finitely generated discrete groups G [Roe 2005] ℓ p ( X ) for strongly discrete metric spaces X [Spakula & Willett 2014] L p ( X , µ ) for fairly general metric spaces X and measures µ [Hagger & Seifert 2017 +x ] TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 23 / 44

  27. Limit operators: The definition revisited So we have New enumeration (independent of A ) of the limit operators of A σ op ( A ) = { A h : h = ( n 1 , n 2 , . . . ) in Z d with | n k | → ∞ s.t. lim S − n k AS n k exists } = { A g : g ∈ ∂ Z d } Now one can add or multiply two instances of σ op ( A ) elementwise and get σ op ( A + B ) = σ op ( A ) + σ op ( B ) , σ op ( AB ) = σ op ( A ) σ op ( B ) , σ op ( α A ) = ασ op ( A ) . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 24 / 44

  28. Limit operators: The definition revisited So we have New enumeration (independent of A ) of the limit operators of A σ op ( A ) = { A h : h = ( n 1 , n 2 , . . . ) in Z d with | n k | → ∞ s.t. lim S − n k AS n k exists } = { A g : g ∈ ∂ Z d } Now one can add or multiply two instances of σ op ( A ) elementwise and get σ op ( A + B ) = σ op ( A ) + σ op ( B ) , σ op ( AB ) = σ op ( A ) σ op ( B ) , σ op ( α A ) = ασ op ( A ) . In short: The map ϕ : A �→ σ op ( A ) , BDO( X ) → ℓ ∞ ( ∂ Z d , BDO( X )) is an algebra homomorphism . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 24 / 44

  29. Limit operators: The definition revisited So we have New enumeration (independent of A ) of the limit operators of A σ op ( A ) = { A h : h = ( n 1 , n 2 , . . . ) in Z d with | n k | → ∞ s.t. lim S − n k AS n k exists } = { A g : g ∈ ∂ Z d } Now one can add or multiply two instances of σ op ( A ) elementwise and get σ op ( A + B ) = σ op ( A ) + σ op ( B ) , σ op ( AB ) = σ op ( A ) σ op ( B ) , σ op ( α A ) = ασ op ( A ) . In short: The map ϕ : A �→ σ op ( A ) , BDO( X ) → ℓ ∞ ( ∂ Z d , BDO( X )) is an algebra homomorphism . Key observation The kernel of that homomorphism ϕ : A �→ σ op ( A ) is K ( X ). So A + K ( X ) �→ σ op ( A ) is an isomorphism BDO( X ) / ker ϕ → im ϕ . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 24 / 44

  30. Limit operators and our essentials The result is an identification ∼ σ op ( A ) A + K ( X ) = for all A ∈ BDO( X ) – with the following consequences: σ op ( A ) A + K ( X ) A essential norm � A � ess �A + K ( X ) � L ( X ) / K ( X ) max h � A h � TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 25 / 44

  31. Limit operators and our essentials The result is an identification ∼ σ op ( A ) A + K ( X ) = for all A ∈ BDO( X ) – with the following consequences: σ op ( A ) A + K ( X ) A essential norm � A � ess �A + K ( X ) � L ( X ) / K ( X ) max h � A h � A + K ( X ) invertible in L ( X ) / A is Fredholm all A h are invertible K ( X ) TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 25 / 44

  32. Limit operators and our essentials The result is an identification ∼ σ op ( A ) A + K ( X ) = for all A ∈ BDO( X ) – with the following consequences: σ op ( A ) A + K ( X ) A essential norm � A � ess �A + K ( X ) � L ( X ) / K ( X ) max h � A h � A + K ( X ) invertible in L ( X ) / A is Fredholm all A h are invertible K ( X ) B + K ( X ) = [ A + K ( X )] − 1 B h = A − 1 B is a Φ-regulariser of A h TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 25 / 44

  33. Limit operators and our essentials The result is an identification ∼ σ op ( A ) A + K ( X ) = for all A ∈ BDO( X ) – with the following consequences: σ op ( A ) A + K ( X ) A essential norm � A � ess �A + K ( X ) � L ( X ) / K ( X ) max h � A h � A + K ( X ) invertible in L ( X ) / A is Fredholm all A h are invertible K ( X ) B + K ( X ) = [ A + K ( X )] − 1 B h = A − 1 B is a Φ-regulariser of A h essential spectrum spec ess A spec L ( X ) / K ( X ) ( A + K ( X )) ∪ h spec A h TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 25 / 44

  34. Limit operators and our essentials The result is an identification ∼ σ op ( A ) A + K ( X ) = for all A ∈ BDO( X ) – with the following consequences: σ op ( A ) A + K ( X ) A essential norm � A � ess �A + K ( X ) � L ( X ) / K ( X ) max h � A h � A + K ( X ) invertible in L ( X ) / A is Fredholm all A h are invertible K ( X ) B + K ( X ) = [ A + K ( X )] − 1 B h = A − 1 B is a Φ-regulariser of A h essential spectrum spec ess A spec L ( X ) / K ( X ) ( A + K ( X )) ∪ h spec A h essential pseudospectrum spec ε spec ε ∪ h spec ε A h ess A L ( X ) / K ( X ) ( A + K ( X )) TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 25 / 44

  35. Limit operators and our essentials The result is an identification ∼ σ op ( A ) A + K ( X ) = for all A ∈ BDO( X ) – with the following consequences: σ op ( A ) A + K ( X ) A essential norm � A � ess �A + K ( X ) � L ( X ) / K ( X ) max h � A h � A + K ( X ) invertible in L ( X ) / A is Fredholm all A h are invertible K ( X ) B + K ( X ) = [ A + K ( X )] − 1 B h = A − 1 B is a Φ-regulariser of A h essential spectrum spec ess A spec L ( X ) / K ( X ) ( A + K ( X )) ∪ h spec A h essential pseudospectrum spec ε spec ε ∪ h spec ε A h ess A L ( X ) / K ( X ) ( A + K ( X )) For A ∈ BO( X ) and d = 1: A + K ( X ) invertible in L ( X ) / all A h injective on ℓ ∞ ( Z ) A is Fredholm K ( X ) ∪ h spec ∞ spec ess A spec L ( X ) / K ( X ) ( A + K ( X )) point A h TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 25 / 44

  36. Limit operators and our essentials ∼ σ op ( A ) A + K ( X ) = σ op ( A ) A essential norm � A � ess max h � A h � [Hagger, ML, Seidel 2016] [Lange, Rabinovich 1985+] [Rabinovich, Roch, Silbermann 1998+] A is Fredholm all A h are invertible [ML, Silbermann 2003], [ML 2003+] [Chandler-Wilde, ML 2007] [Seidel, ML 2014] B h = A − 1 B is a Φ-regulariser of A [Seidel 2013] h essential spectrum spec ess A ∪ h spec A h [Seidel, ML 2014] essential pseudospectrum spec ε ∪ h spec ε A h ess A [Hagger, ML, Seidel 2016] For A ∈ BO( X ) and d = 1: all A h injective on ℓ ∞ ( Z ) A is Fredholm [Chandler-Wilde, ML 2008] ∪ h spec ∞ spec ess A point A h TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 26 / 44

  37. Self-similar operators Definition: self-similar operator We say that A ∈ BDO( X ) is self-similar if A ∈ σ op ( A ). Roughly speaking, this means that A contains a copy of itself, at infinity. Each pattern that you see once in A , you will see infinitely often in A . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 27 / 44

  38. Self-similar operators Definition: self-similar operator We say that A ∈ BDO( X ) is self-similar if A ∈ σ op ( A ). Roughly speaking, this means that A contains a copy of itself, at infinity. Each pattern that you see once in A , you will see infinitely often in A . But then, by the above, � A � ess = � A � A is Fredholm ⇐ ⇒ A is invertible spec ess A = spec A spec ε spec ε A = ess A TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 27 / 44

  39. Self-similar operators Definition: self-similar operator We say that A ∈ BDO( X ) is self-similar if A ∈ σ op ( A ). Roughly speaking, this means that A contains a copy of itself, at infinity. Each pattern that you see once in A , you will see infinitely often in A . But then, by the above, � A � ess = � A � A is Fredholm ⇐ ⇒ A is invertible spec ess A = spec A spec ε spec ε A = ess A “essential stuff = real stuff.” TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 27 / 44

  40. Table of Contents The essentials 1 Limit operators 2 Stability of approximation methods 3 The Fibonacci Hamiltonian 4 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 28 / 44

  41. The finite section method Task: Find an approximate solution of the equation Ax = b . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 29 / 44

  42. The finite section method Task: Find an approximate solution of the equation Ax = b . Idea: Approximate A by growing but finite square submatrices A n and, assuming that A is invertible, hope that also the inverses A − 1 exist, at least for sufficiently large n , and n that they converge to the inverse of A , i.e. A − 1 → A − 1 , n TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 29 / 44

  43. The finite section method Task: Find an approximate solution of the equation Ax = b . Idea: Approximate A by growing but finite square submatrices A n and, assuming that A is invertible, hope that also the inverses A − 1 exist, at least for sufficiently large n , and n that they converge to the inverse of A , i.e. A − 1 → A − 1 , n It turns out: This “hope” will come true iff the sequence ( A n ) is stable , meaning that sup n ≥ n 0 � A − 1 all A n with sufficiently large n are invertible and n � < ∞ . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 29 / 44

  44. Stability is just another “essential” The sequence ( A n ) is stable ⇐ ⇒ D := Diag ( A 1 , A 2 , . . . ) is Fredholm . This brings us back to limit operators of D – and hence of A . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 30 / 44

  45. Following the corners as they move out to infinity In the end we have to follow the two “corners” (semi-infinite matrices) � a l n , l n � ... � � . · · · . . and . ... . . · · · a r n , r n of A n as n → ∞ and find (partial) limits of these matrix sequences: TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 31 / 44

  46. Following the corners as they move out to infinity In the end we have to follow the two “corners” (semi-infinite matrices) � a l n , l n � ... � � . · · · . . and . ... . . · · · a r n , r n of A n as n → ∞ and find (partial) limits of these matrix sequences: TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 31 / 44

  47. Following the corners as they move out to infinity In the end we have to follow the two “corners” (semi-infinite matrices) � a l n , l n � ... � � . · · · . . and . ... . . · · · a r n , r n of A n as n → ∞ and find (partial) limits of these matrix sequences: TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 31 / 44

  48. Following the corners as they move out to infinity In the end we have to follow the two “corners” (semi-infinite matrices) � a l n , l n � ... � � . · · · . . and . ... . . · · · a r n , r n of A n as n → ∞ and find (partial) limits of these matrix sequences: TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 31 / 44

  49. Table of Contents The essentials 1 Limit operators 2 Stability of approximation methods 3 The Fibonacci Hamiltonian 4 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 32 / 44

  50. The Fibonacci Hamiltonian The Fibonacci Hamiltonian is a particular discrete Schr¨ odinger operator in 1D: ( Ax ) n = x n − 1 + v n x n + x n +1 , n ∈ Z . So, again, the matrix looks like this   ... ...     ...   v − 2 1     1 v − 1 1   A = .   1 1 v 0     1 1 v 1   ...   1 v 2     ... ... The potential v only assumes the values 0 and 1 – but in a very interesting pattern. 50 letters of the Fibonacci word (“quasiperiodic”) . . . 10110101101101011010110110101101101011010110110101 . . . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 33 / 44

  51. Fibonacci and his rabbit population time population count 1 2 3 4 5 6 7 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 34 / 44

  52. When rabbits become numbers time population count 1 1 1 2 10 2 3 101 3 4 10110 5 5 10110101 8 6 1011010110110 13 7 101101011011010110101 21 8 1011010110110101101011011010110110 34 9 1011010110110101101011011010110110101101011011010110101 55 . . . . . . . . . Three equivalent constructions of the Fibonacci word 2 0 �→ 1 , 1 �→ 10; f k +1 := f k f k − 1 ; v n = χ [1 − α, 1) ( n α mod 1) , α = √ 1+ 5 The last formula is also used to define v n for all n ∈ Z . ( ⇒ bi-infinite Fibonacci word) TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 35 / 44

  53. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  54. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  55. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n 1 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  56. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n 1 0 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  57. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n 1 0 1 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  58. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n 1 0 1 1 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  59. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n 1 0 1 1 0 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  60. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n 1 0 1 1 0 1 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  61. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n 1 0 1 1 0 1 0 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  62. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n 1 0 1 1 0 1 0 1 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  63. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n 1 0 1 1 0 1 0 1 · · · TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  64. The “mod 1” rotation formula and limit operators 2 v n = χ [1 − α, 1) ( n α mod 1 ) , n ∈ Z , α = (“golden mean”) √ 1+ 5 n . . . − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 · · · v n 1 0 1 1 0 1 0 1 · · · TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 36 / 44

  65. Fibonacci word: subword complexity In an infinite random word over the alphabet { 0 , 1 } you can find (almost surely) all 2 n subwords of length n . In contrast: How many can you find in the Fibonacci word v = · · · 1011010110110101101011011010110110101101011011010110101 · · · ? List of subwords of length n length subwords count 1 0 , 1 2 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 37 / 44

  66. Fibonacci word: subword complexity In an infinite random word over the alphabet { 0 , 1 } you can find (almost surely) all 2 n subwords of length n . In contrast: How many can you find in the Fibonacci word v = · · · 1011010110110101101011011010110110101101011011010110101 · · · ? List of subwords of length n length subwords count 1 0 , 1 2 2 01 , 10 , 11 3 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 37 / 44

  67. Fibonacci word: subword complexity In an infinite random word over the alphabet { 0 , 1 } you can find (almost surely) all 2 n subwords of length n . In contrast: How many can you find in the Fibonacci word v = · · · 1011010110110101101011011010110110101101011011010110101 · · · ? List of subwords of length n length subwords count 1 0 , 1 2 2 01 , 10 , 11 3 3 010 , 011 , 101 , 110 4 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 37 / 44

  68. Fibonacci word: subword complexity In an infinite random word over the alphabet { 0 , 1 } you can find (almost surely) all 2 n subwords of length n . In contrast: How many can you find in the Fibonacci word v = · · · 1011010110110101101011011010110110101101011011010110101 · · · ? List of subwords of length n length subwords count 1 0 , 1 2 2 01 , 10 , 11 3 3 010 , 011 , 101 , 110 4 4 0101 , 0110 , 1010 , 1011 , 1101 5 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 37 / 44

  69. Fibonacci word: subword complexity In an infinite random word over the alphabet { 0 , 1 } you can find (almost surely) all 2 n subwords of length n . In contrast: How many can you find in the Fibonacci word v = · · · 1011010110110101101011011010110110101101011011010110101 · · · ? List of subwords of length n length subwords count 1 0 , 1 2 2 01 , 10 , 11 3 3 010 , 011 , 101 , 110 4 4 0101 , 0110 , 1010 , 1011 , 1101 5 . . . . . . · · · n + 1 n Interesting feature: Very moderate (in fact: minimal) growth, compared to 2 n . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 37 / 44

  70. Fibonacci word: subword complexity In an infinite random word over the alphabet { 0 , 1 } you can find (almost surely) all 2 n subwords of length n . In contrast: How many can you find in the Fibonacci word v = · · · 1011010110110101101011011010110110101101011011010110101 · · · ? List of subwords of length n length subwords count 1 0 , 1 2 2 01 , 10 , 11 3 3 010 , 011 , 101 , 110 4 4 0101 , 0110 , 1010 , 1011 , 1101 5 . . . . . . · · · n + 1 n Interesting feature: Very moderate (in fact: minimal) growth, compared to 2 n . One can show: The main diagonal of every limit operator of A has the same list of subwords! TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 37 / 44

  71. Limit operators and their subwords Let v = · · · 10110101101101011010110110101101101011 · · · be the Fibonacci word, A = S − 1 + M v + S 1 be the Fibonacci Hamiltonian, TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 38 / 44

  72. Limit operators and their subwords Let v = · · · 10110101101101011010110110101101101011 · · · be the Fibonacci word, A = S − 1 + M v + S 1 be the Fibonacci Hamiltonian, h = ( n 1 , n 2 , . . . ) be a sequence in Z with n k → ±∞ and limit operator A h = S − 1 + M v h + S 1 with v h = lim k →∞ S − n k v (pointwise). TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 38 / 44

  73. Limit operators and their subwords Let v = · · · 10110101101101011010110110101101101011 · · · be the Fibonacci word, A = S − 1 + M v + S 1 be the Fibonacci Hamiltonian, h = ( n 1 , n 2 , . . . ) be a sequence in Z with n k → ±∞ and limit operator A h = S − 1 + M v h + S 1 with v h = lim k →∞ S − n k v (pointwise). The main diagonal of the limit operator A h has the same subwords as that of A . w ≺ v ⇐ ⇒ w ≺ v h TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 38 / 44

  74. Limit operators and their subwords Let v = · · · 10110101101101011010110110101101101011 · · · be the Fibonacci word, A = S − 1 + M v + S 1 be the Fibonacci Hamiltonian, h = ( n 1 , n 2 , . . . ) be a sequence in Z with n k → ±∞ and limit operator A h = S − 1 + M v h + S 1 with v h = lim k →∞ S − n k v (pointwise). The main diagonal of the limit operator A h has the same subwords as that of A . w ≺ v ⇐ ⇒ w ≺ v h ⇐ w ≺ v h = ⇒ w ≺ S − n k v for large k = ⇒ w ≺ v . TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 38 / 44

  75. Limit operators and their subwords Let v = · · · 10110101101101011010110110101101101011 · · · be the Fibonacci word, A = S − 1 + M v + S 1 be the Fibonacci Hamiltonian, h = ( n 1 , n 2 , . . . ) be a sequence in Z with n k → ±∞ and limit operator A h = S − 1 + M v h + S 1 with v h = lim k →∞ S − n k v (pointwise). The main diagonal of the limit operator A h has the same subwords as that of A . w ≺ v ⇐ ⇒ w ≺ v h ⇐ w ≺ v h = ⇒ w ≺ S − n k v for large k = ⇒ w ≺ v . ⇒ Let w ≺ v , say (w.l.o.g.) w ≺ v + . = ⇒ Every S − n k v contains w in a F n +1 -neighbourhood of zero. = ⇒ Every limit potential v h contains w in a F n +1 -neighbourhood of zero. TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 38 / 44

  76. Limit operators and the “mod 1” rotation formula For the Fibonacci Hamiltonian A = S − 1 + M v + S 1 , one gets � � σ op ( A ) = S − 1 + M v θ + S 1 , S − 1 + M w θ + S 1 : θ ∈ [0 , 1) , where v θ w θ := χ [1 − α, 1) ( θ + n α mod 1) , := χ (1 − α, 1] ( θ + n α mod 1) , n ∈ Z . n n In particular, A ∈ σ op ( A ); so A is self-similar. TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 39 / 44

  77. What do we want from the Fibonacci Hamiltonian? A lot is known of the spectrum of A ; it is a Cantor set on the real line of Lebesgue measure zero, there is no point spectrum (w.r.t. ℓ 2 ) in fact, the spectrum is purely singular continuous... TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 40 / 44

  78. What do we want from the Fibonacci Hamiltonian? A lot is known of the spectrum of A ; it is a Cantor set on the real line of Lebesgue measure zero, there is no point spectrum (w.r.t. ℓ 2 ) in fact, the spectrum is purely singular continuous... Our focus: Applicability of the FSM with arbitrary cut-off points. We show this via invertibility of B , B + and B − for all B ∈ σ op ( A ), including B = A (i.e. 0 is not in the spectrum of any of these operators). TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 40 / 44

  79. Sketch of proof To show that A is invertible on ℓ 2 (hence on any ℓ p ), we show that A is Fredholm ( ⇒ closed range) A is injective on ℓ 2 A ∗ is injective on ℓ 2 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 41 / 44

  80. Sketch of proof To show that A is invertible on ℓ 2 (hence on any ℓ p ), we show that all B ∈ σ op ( A ) are invertible on ℓ 2 A is Fredholm ( ⇒ closed range) ⇐ ⇒ A is injective on ℓ 2 A ∗ is injective on ℓ 2 TUHH Marko Lindner LimOps: Your hands on the essentials 28th IWOTA, Aug 2017, Chemnitz 41 / 44

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