J-theory in application to the spectral theory of periodic GMP matrices Benjamin Eichinger Institute of Analysis Johannes Kepler University Linz OTIND , 17st-20th December 1 / 19
Jacobi matrices Let d σ be a real compactly supported measure and P n = P n ( d σ ) the corresponding orthonormal polynomials. It is well known, that they satisfy a three-term recurrence relation. xP n ( x ) = a n P n − 1 ( x ) + b n P n ( x ) + a n + 1 P n + 1 , a n > 0 . That is, multiplication by x in the basis { P n } n ≥ 0 has the one-sided Jacobi matrix 0 b 0 a 1 a 1 b 1 a 2 J + = ... ... ... . 0 ... ... 2 / 19
Jacobi matrices Let d σ be a real compactly supported measure and P n = P n ( d σ ) the corresponding orthonormal polynomials. It is well known, that they satisfy a three-term recurrence relation. xP n ( x ) = a n P n − 1 ( x ) + b n P n ( x ) + a n + 1 P n + 1 , a n > 0 . That is, multiplication by x in the basis { P n } n ≥ 0 has the one-sided Jacobi matrix 0 b 0 a 1 a 1 b 1 a 2 J + = ... ... ... . 0 ... ... 2 / 19
Shift on Jacobi matrices Let us define the resolvent function r + ( z ) = � ( J + − z ) − 1 e 0 , e 0 � , � � e 0 = 1 0 0 . . . . Let r ( 1 ) be the resolvent function of J ( 1 ) + , + b 0 a 1 · · · 0 · · · . a 1 J + = J ( 1 ) 0 + Then − 1 r + ( z ) = . 1 r ( 1 ) z − b 0 − a 2 + ( z ) That is, � 0 � � � r ( 1 ) � r + ( z ) � − 1 / a 1 + ( z ) ∼ , 1 a 1 ( z − b 0 ) / a 1 1 where � u � � x � � u � � x � ∼ ⇐ ⇒ ∃ c = c v y v y 3 / 19
Shift on Jacobi matrices Let us define the resolvent function r + ( z ) = � ( J + − z ) − 1 e 0 , e 0 � , � � e 0 = 1 0 0 . . . . Let r ( 1 ) be the resolvent function of J ( 1 ) + , + b 0 a 1 · · · 0 · · · . a 1 J + = J ( 1 ) 0 + Then − 1 r + ( z ) = . 1 r ( 1 ) z − b 0 − a 2 + ( z ) That is, � 0 � � � r ( 1 ) � r + ( z ) � − 1 / a 1 + ( z ) ∼ , 1 a 1 ( z − b 0 ) / a 1 1 where � u � � x � � u � � x � ∼ ⇐ ⇒ ∃ c = c v y v y 3 / 19
Shift on Jacobi matrices Let us define the resolvent function r + ( z ) = � ( J + − z ) − 1 e 0 , e 0 � , � � e 0 = 1 0 0 . . . . Let r ( 1 ) be the resolvent function of J ( 1 ) + , + b 0 a 1 · · · 0 · · · . a 1 J + = J ( 1 ) 0 + Then − 1 r + ( z ) = . 1 r ( 1 ) z − b 0 − a 2 + ( z ) That is, � 0 � � � r ( 1 ) � r + ( z ) � − 1 / a 1 + ( z ) ∼ , 1 a 1 ( z − b 0 ) / a 1 1 where � u � � x � � u � � x � ∼ ⇐ ⇒ ∃ c = c v y v y 3 / 19
Shift on Jacobi matrices Let us define the resolvent function r + ( z ) = � ( J + − z ) − 1 e 0 , e 0 � , � � e 0 = 1 0 0 . . . . Let r ( 1 ) be the resolvent function of J ( 1 ) + , + b 0 a 1 · · · 0 · · · . a 1 J + = J ( 1 ) 0 + Then − 1 r + ( z ) = . 1 r ( 1 ) z − b 0 − a 2 + ( z ) That is, � 0 � � � r ( 1 ) � r + ( z ) � − 1 / a 1 + ( z ) ∼ , 1 a 1 ( z − b 0 ) / a 1 1 where � u � � x � � u � � x � ∼ ⇐ ⇒ ∃ c = c v y v y 3 / 19
periodic Jacobi matrices For periodic Jacobi matrices, i.e., ∃ p ∀ k a k + p = a k , b k + p = b k , it is convenient to extend the sequences periodically to Z and consider the two-sided Jacobi matrix ... ... ... 0 a p − 1 J − a p − 1 b p − 1 a 0 a 0 = a 0 b 0 a 1 a 0 ... ... 0 J + a 1 ... We define r − ( z ) = � ( J − − z ) − 1 e − 1 , e − 1 � . 4 / 19
periodic Jacobi matrices For periodic Jacobi matrices, i.e., ∃ p ∀ k a k + p = a k , b k + p = b k , it is convenient to extend the sequences periodically to Z and consider the two-sided Jacobi matrix ... ... ... 0 a p − 1 J − a p − 1 b p − 1 a 0 a 0 = a 0 b 0 a 1 a 0 ... ... 0 J + a 1 ... We define r − ( z ) = � ( J − − z ) − 1 e − 1 , e − 1 � . 4 / 19
We see that � 0 � 0 � � � r ( p ) � r + ( z ) � � − 1 / a 1 − 1 / a p + ( z ) ∼ ... 1 a 1 ( z − b 0 ) / a 1 a p ( z − b p − 1 ) / a p 1 � r + ( z ) � = T p ( z ) . 1 Theorem Let us define the polynomial of degree p, ∆( z ) = tr T p ( z ) . The spectrum of a periodic two-sided Jacobi matrix is purely absolutely continuous and it is given by E = ∆ − 1 ([ − 2 , 2 ]) . Moreover the resolvent functions are reflectionless on E , i.e., 1 r + ( x + i 0 ) = a 2 0 r − ( x + i 0 ) . Note that for this reason spectra of periodic Jacobi matrices are very special finite union of intervals. 5 / 19
We see that � 0 � 0 � � � r ( p ) � r + ( z ) � � − 1 / a 1 − 1 / a p + ( z ) ∼ ... 1 a 1 ( z − b 0 ) / a 1 a p ( z − b p − 1 ) / a p 1 � r + ( z ) � = T p ( z ) . 1 Theorem Let us define the polynomial of degree p, ∆( z ) = tr T p ( z ) . The spectrum of a periodic two-sided Jacobi matrix is purely absolutely continuous and it is given by E = ∆ − 1 ([ − 2 , 2 ]) . Moreover the resolvent functions are reflectionless on E , i.e., 1 r + ( x + i 0 ) = a 2 0 r − ( x + i 0 ) . Note that for this reason spectra of periodic Jacobi matrices are very special finite union of intervals. 5 / 19
We see that � 0 � 0 � � � r ( p ) � r + ( z ) � � − 1 / a 1 − 1 / a p + ( z ) ∼ ... 1 a 1 ( z − b 0 ) / a 1 a p ( z − b p − 1 ) / a p 1 � r + ( z ) � = T p ( z ) . 1 Theorem Let us define the polynomial of degree p, ∆( z ) = tr T p ( z ) . The spectrum of a periodic two-sided Jacobi matrix is purely absolutely continuous and it is given by E = ∆ − 1 ([ − 2 , 2 ]) . Moreover the resolvent functions are reflectionless on E , i.e., 1 r + ( x + i 0 ) = a 2 0 r − ( x + i 0 ) . Note that for this reason spectra of periodic Jacobi matrices are very special finite union of intervals. 5 / 19
Functional model Let E be a finite union of intervals E = [ a 0 , b 0 ] \ � g j = 1 ( a j , b j ) and Ω = C \ E . Let g Ω ( z , z 0 ) denote the Green’s function of Ω and define the complex Green’s function by B z 0 ( z ) = e − g Ω ( z , z 0 ) − i � g Ω ( z , z 0 ) . Note that we have the properties: B z 0 ( z 0 ) = 0, | B z 0 | = 1 on E , γ j = e 2 π i ω Ω ( E j , z 0 ) B z 0 . | B z 0 | < 1 in Ω , B z 0 ◦ ˜ 6 / 19
Functional model Let E be a finite union of intervals E = [ a 0 , b 0 ] \ � g j = 1 ( a j , b j ) and Ω = C \ E . Let g Ω ( z , z 0 ) denote the Green’s function of Ω and define the complex Green’s function by B z 0 ( z ) = e − g Ω ( z , z 0 ) − i � g Ω ( z , z 0 ) . Note that we have the properties: B z 0 ( z 0 ) = 0, | B z 0 | = 1 on E , γ j = e 2 π i ω Ω ( E j , z 0 ) B z 0 . | B z 0 | < 1 in Ω , B z 0 ◦ ˜ 6 / 19
Functional model Let E be a finite union of intervals E = [ a 0 , b 0 ] \ � g j = 1 ( a j , b j ) and Ω = C \ E . Let g Ω ( z , z 0 ) denote the Green’s function of Ω and define the complex Green’s function by B z 0 ( z ) = e − g Ω ( z , z 0 ) − i � g Ω ( z , z 0 ) . Note that we have the properties: B z 0 ( z 0 ) = 0, | B z 0 | = 1 on E , γ j = e 2 π i ω Ω ( E j , z 0 ) B z 0 . | B z 0 | < 1 in Ω , B z 0 ◦ ˜ 6 / 19
Functional model Let E be a finite union of intervals E = [ a 0 , b 0 ] \ � g j = 1 ( a j , b j ) and Ω = C \ E . Let g Ω ( z , z 0 ) denote the Green’s function of Ω and define the complex Green’s function by B z 0 ( z ) = e − g Ω ( z , z 0 ) − i � g Ω ( z , z 0 ) . Note that we have the properties: B z 0 ( z 0 ) = 0, | B z 0 | = 1 on E , γ j = e 2 π i ω Ω ( E j , z 0 ) B z 0 . | B z 0 | < 1 in Ω , B z 0 ◦ ˜ 6 / 19
Functional model Let E be a finite union of intervals E = [ a 0 , b 0 ] \ � g j = 1 ( a j , b j ) and Ω = C \ E . Let g Ω ( z , z 0 ) denote the Green’s function of Ω and define the complex Green’s function by B z 0 ( z ) = e − g Ω ( z , z 0 ) − i � g Ω ( z , z 0 ) . Note that we have the properties: B z 0 ( z 0 ) = 0, | B z 0 | = 1 on E , γ j = e 2 π i ω Ω ( E j , z 0 ) B z 0 . | B z 0 | < 1 in Ω , B z 0 ◦ ˜ 6 / 19
Functional model Let E be a finite union of intervals E = [ a 0 , b 0 ] \ � g j = 1 ( a j , b j ) and Ω = C \ E . Let g Ω ( z , z 0 ) denote the Green’s function of Ω and define the complex Green’s function by B z 0 ( z ) = e − g Ω ( z , z 0 ) − i � g Ω ( z , z 0 ) . Note that we have the properties: B z 0 ( z 0 ) = 0, | B z 0 | = 1 on E , γ j = e 2 π i ω Ω ( E j , z 0 ) B z 0 . | B z 0 | < 1 in Ω , B z 0 ◦ ˜ 6 / 19
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