The characteristic function for infinite Jacobi matrices, its logarithm, the spectral zeta function, and solvable examples František Štampach 1 , Pavel Št’ovíˇ cek 2 1 Department of Applied Mathematics, Faculty of Information Technology Czech Technical University in Prague, Czech Republic B frantisek.stampach@fit.cvut.cz 2 Department of Mathematics, Faculty of Nuclear Science Czech Technical University in Prague, Czech Republic B stovicek@kmlinux.fjfi.cvut.cz 3rd Najman Conference ON SPECTRAL PROBLEMS FOR OPERATORS AND MATRICES Biograd, Croatia September 16-20, 2013
The function F ( x ) Define F : D → C , ∞ ∞ ∞ ∞ 1 + � ( − 1 ) m � � � F ( x ) = . . . m = 1 k 1 = 1 k 2 = k 1 + 2 k m = k m − 1 + 2 × x k 1 x k 1 + 1 x k 2 x k 2 + 1 . . . x k m x k m + 1 where � ∞ � { x k } ∞ � D = k = 1 ⊂ C ; | x k x k + 1 | < ∞ k = 1 ℓ 2 ( N ) ⊂ D Note that Put F ( x 1 , x 2 , . . . , x n ) = F ( x 1 , x 2 , . . . , x n , 0 , 0 , 0 , . . . ) , F ( ∅ ) = 1 One has � ∞ � ∞ � � � � | F ( x ) | ≤ exp , | F ( x ) − 1 | ≤ exp − 1 | x k x k + 1 | | x k x k + 1 | k = 1 k = 1 František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 2/30
Basic properties of F ( x ) A recurrence rule F ( { x n } ∞ n = 1 ) = F ( { x n + 1 } ∞ n = 1 ) − x 1 x 2 F ( { x n + 2 } ∞ n = 1 ) more generally, for any k ∈ N , F ( { x n } ∞ F ( x 1 , . . . , x k ) F ( { x k + n } ∞ n = 1 ) = n = 1 ) − F ( x 1 , . . . , x k − 1 ) x k x k + 1 F ( { x k + n + 1 } ∞ n = 1 ) For x ∈ D , F ( x ) = lim n →∞ F ( x 1 , x 2 , . . . , x n ) The function F is continuous on ℓ 2 ( N ) František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 3/30
F ( x ) and special functions The Bessel functions of the first kind: for w , ν ∈ C , ν / ∈ − N , � ∞ w ν �� w � J ν ( 2 w ) = Γ( ν + 1 ) F ν + k k = 1 The basic hypergeometric series (q-hypergeometric series): for t , w ∈ C , | t | < 1, ∞ t m ( 2 m − 1 ) w 2 m � ∞ �� t k − 1 w � � 1 + ( − 1 ) m F = ( 1 − t 2 )( 1 − t 4 ) . . . ( 1 − t 2 m ) k = 1 m = 1 0 φ 1 (; 0 ; t 2 , − t w 2 ) = Here k − 1 ∞ q k ( k − 1 ) z k , ( a ; q ) k = � 1 − aq j � � � 0 φ 1 (; b ; q , z ) = ( q ; q ) k ( b ; q ) k k = 0 j = 0 František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 4/30
A class of Jacobi matrices: a convergence condition Consider a symmetric (in general complex) Jacobi (tridiagonal) matrix λ 1 w 1 w 1 λ 2 w 2 J = w 2 λ 3 w 3 ... ... ... where λ = { λ n } ∞ and w = { w n } ∞ n = 1 ⊂ C \ { 0 } n = 1 ⊂ C Denote der ( λ ) := the set of all finite accumulation points of λ C λ 0 := C \ { λ n ; n ∈ N } The convergence condition ∞ w 2 � � � n for some and hence any z ∈ C λ � � � < ∞ 0 � � ( λ n − z )( λ n + 1 − z ) � n = 1 w 2 Then � ∞ ( λ n − z )( λ n + 1 − z ) converges locally uniformly on C λ n n = 1 0 František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 5/30
The characteristic polynomial of a finite Jacobi matrix One has F ( x 1 , x 2 , . . . , x n ) = det X n where 1 x 1 1 x 2 x 2 ... ... ... X n = ... ... ... 1 x n − 1 x n − 1 1 x n γ 2 k − 1 := � k − 1 w 2 j � k − 1 w 2 j + 1 Put w 2 j , k = 1 , 2 , 3 , . . . w 2 j − 1 , γ 2 k := w 1 j = 1 j = 1 Then γ k γ k + 1 = w k Let J n be the n × n truncation of the Jacobi matrix. Then � n � � γ 2 γ 2 � γ 2 � 1 2 det ( J n − zI n ) = n ( λ k − z ) F λ 1 − z , λ 2 − z , . . . , λ n − z k = 1 František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 6/30
The characteristic function of J Put �� � ∞ � γ 2 n F J ( z ) := F λ n − z n = 1 and for z / ∈ C \ der ( λ ) , ∞ � r ( z ) := δ z ,λ k k = 1 (the number of occurrences of z in λ ; r ( z ) = 0 for z / ∈ { λ n ; n ∈ N } ) Lemma Suppose J fulfills the convergence condition. Then F J ( z ) is a well defined analytic function on C \ { λ n ; n ∈ N } , meromorphic on C \ der ( λ ) with poles at the points λ n , n ∈ N , (not belonging to der ( λ ) , however) of order at most r ( λ n ) . František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 7/30
The zero set of the characteristic function of J Define the zero set � � z ∈ C \ der ( λ ); lim u → z ( u − z ) r ( z ) F J ( u ) = 0 Z ( J ) := and the functions ξ k ( z ) , k = 0 , 1 , 2 , ..., on C \ der ( λ ) ( w 0 = 1), γ 2 � ∞ k � w j − 1 j � ξ k ( z ) := lim u → z ( u − z ) r ( z ) F u − λ j λ j − u j = 1 j = k + 1 � � = F − 1 Remarks . (i) Notice that ( 0 ) Z ( J ) ∩ C \ { λ n ; n ∈ N } J (ii) z ∈ Z ( J ) iff ξ 0 ( z ) = 0 (iii) For k sufficiently large, − 1 γ 2 � ∞ k � k � � j � � ξ k ( z ) = w j − 1 ( z − λ j ) F λ j − z j = 1 j = 1 j = k + 1 λ j � = z František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 8/30
The spectrum of J in C \ der ( λ ) Theorem Suppose the convergence condition on J is fulfilled and F J ( z ) does not vanish identically on C \ { λ n ; n ∈ N } . Then J determines unambiguously a closed operator in ℓ 2 ( N ) (denoted again by J), spec ( J ) \ der ( λ ) = spec p ( J ) \ der ( λ ) = Z ( J ) The point spectrum of J is simple. If z ∈ Z ( J ) , i.e. ξ 0 ( z ) = 0 , then ξ ( z ) = ( ξ 1 ( z ) , ξ 2 ( z ) , ξ 3 ( z ) , . . . ) is a corresponding eigenvector. František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 9/30
The case J is real Corollary Suppose J is is real and fulfills the convergence condition. Then J determines unambiguously a self-adjoint operator in ℓ 2 ( N ) and spec ( J ) ∩ ( C \ der ( λ )) = Z ( J ) consists of simple real eigenvalues which have no accumulation points in R \ der ( λ ) . František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 10/30
Example 1 J. Gard, E. Zakrajšek: J. Inst. Math. Appl. 11 (1973) Y. Ikebe, Y. Kikuchi, I. Fujishiro: J. Comput. Appl. Math. 38 (1991) λ n = n and w n = w > 0 for all n ∈ N , 1 w 2 w w J = 3 w w ... ... ... One has der ( λ ) = ∅ , spec ( J ) = Z ( J ) , and for r ∈ Z + , � γ 2 � ∞ � � = w − r + z Γ( 1 + r − z ) J r − z ( 2 w ) n F n − z n = r + 1 Hence spec ( J ) = { z ∈ C ; J − z ( 2 w ) = 0 } Corresponding eigenvectors v ( z ) can be chosen as v k ( z ) = ( − 1 ) k J k − z ( 2 w ) , k ∈ N František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 11/30
Example 2 λ n = 1 / n , w n = β/ � n ( n + 1 ) , for β > 0 and all n ∈ N , √ 1 2 β/ √ √ 2 1 / 2 6 β/ β/ √ √ J = 6 1 / 3 12 β/ β/ ... ... ... Then, for r ∈ Z + , r − 1 / z �� γ 2 � ∞ � r + 1 − 1 � 2 β � z � � � � n F = Γ J r − 1 / z λ n − z β z z n = r + 1 and � 2 β � � � z ∈ R \ { 0 } ; J − 1 / z = 0 ∪ { 0 } spec ( J ) = z Corresponding eigenvectors v ( z ) can be chose as √ � 2 β � v k ( z ) = k J k − 1 / z , k ∈ N z František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 12/30
Example 3 λ n = q n − 1 w n = β q ( n − 1 ) / 2 , with and 0 < q < 1, β > 0, 1 β β √ q β q β √ q J = q 2 β q ... ... ... Then, for r ∈ Z + , �� � ∞ � γ 2 z ; q , − q r β 2 ; q r � � n = 0 φ 1 F z 2 λ n − z n = r + 1 and so z ; q , − β 2 � 1 ; 1 � � � � � z ∈ R \ { 0 } ; = 0 ∪ { 0 } spec ( J ) = z ; q ∞ 0 φ 1 z 2 A corresponding eigenvector v ( z ) can be written in the form k − 1 � q k z ; q , − q k β 2 ; q k � β � � � � v k ( z ) = q ( k − 1 )( k − 2 ) / 4 z ; q ∞ 0 φ 1 , k ∈ N z 2 z František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 13/30
Example 4 A modification of Example 3 → an unbounded Jacobi operator λ n = q − n + 1 and w n = β q − ( n − 1 ) / 2 where again 0 < q < 1, β > 0 1 β q − 1 β q − 1 / 2 β J = β q − 1 / 2 q − 2 β q − 1 ... ... ... Then, for r ∈ Z + , � ∞ �� γ 2 � = 0 φ 1 (; q r z ; q , − q r + 1 β 2 ) n F λ n − z n = r + 1 and so � � z ∈ R ; ( z ; q ) ∞ 0 φ 1 (; z ; q , − q β 2 ) = 0 spec ( J ) = The k th entry of a corresponding eigenvector v ( z ) v k ( z ) = q k ( k + 1 ) / 4 ( − β ) k − 1 ( q k z ; q ) ∞ 0 φ 1 (; q k z ; q , − q k + 1 β 2 ) , k ∈ N František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 14/30
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