Spectral analysis of two doubly infinite Jacobi operators

Transcription

1 Spectral analysis of two doubly infinite Jacobi operators František Štampach jointly with Mourad E. H. Ismail Stockholm University Spectral Theory and Applications conference in memory of Boris Pavlov March 13, 2016 František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

2 Contents 1 Introduction 2 Spectral resolution of A 3 Spectral resolution of B František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

3 Two doubly-infinite Jacobi matrices We analyze the spectral properties of Jacobi operators A and B acting on vectors of the standard basis of l 2 Z) as: and Ae n = q n+1 e n 1 + q n e n+1, n Z, where q 0, 1) and α R. Be n = e n 1 + αq n e n + e n+1, n Z, František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

4 Two doubly-infinite Jacobi matrices We analyze the spectral properties of Jacobi operators A and B acting on vectors of the standard basis of l 2 Z) as: and Ae n = q n+1 e n 1 + q n e n+1, n Z, where q 0, 1) and α R. Be n = e n 1 + αq n e n + e n+1, n Z, The semi-infinite parts of A and B has been studied before mainly within the theory of Orthogonal Polynomials: František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

5 Two doubly-infinite Jacobi matrices We analyze the spectral properties of Jacobi operators A and B acting on vectors of the standard basis of l 2 Z) as: and Ae n = q n+1 e n 1 + q n e n+1, n Z, where q 0, 1) and α R. Be n = e n 1 + αq n e n + e n+1, n Z, The semi-infinite parts of A and B has been studied before mainly within the theory of Orthogonal Polynomials: A with n < 0 [Al-Salam, Ismail 83]: Rogers Ramanujan s functions and continued fractions. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

6 Two doubly-infinite Jacobi matrices We analyze the spectral properties of Jacobi operators A and B acting on vectors of the standard basis of l 2 Z) as: and Ae n = q n+1 e n 1 + q n e n+1, n Z, where q 0, 1) and α R. Be n = e n 1 + αq n e n + e n+1, n Z, The semi-infinite parts of A and B has been studied before mainly within the theory of Orthogonal Polynomials: A with n < 0 [Al-Salam, Ismail 83]: Rogers Ramanujan s functions and continued fractions. A with n > 0 [Chen, Ismail 98, FS 16]: Indeterminate moment problem, Nevanlinna functions. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

7 Two doubly-infinite Jacobi matrices We analyze the spectral properties of Jacobi operators A and B acting on vectors of the standard basis of l 2 Z) as: and Ae n = q n+1 e n 1 + q n e n+1, n Z, where q 0, 1) and α R. Be n = e n 1 + αq n e n + e n+1, n Z, The semi-infinite parts of A and B has been studied before mainly within the theory of Orthogonal Polynomials: A with n < 0 [Al-Salam, Ismail 83]: Rogers Ramanujan s functions and continued fractions. A with n > 0 [Chen, Ismail 98, FS 16]: Indeterminate moment problem, Nevanlinna functions. B with n 0 [Ismail, Mulla 87]: q-chebyshev polynomials. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

8 Two doubly-infinite Jacobi matrices We analyze the spectral properties of Jacobi operators A and B acting on vectors of the standard basis of l 2 Z) as: and Ae n = q n+1 e n 1 + q n e n+1, n Z, where q 0, 1) and α R. Be n = e n 1 + αq n e n + e n+1, n Z, The semi-infinite parts of A and B has been studied before mainly within the theory of Orthogonal Polynomials: A with n < 0 [Al-Salam, Ismail 83]: Rogers Ramanujan s functions and continued fractions. A with n > 0 [Chen, Ismail 98, FS 16]: Indeterminate moment problem, Nevanlinna functions. B with n 0 [Ismail, Mulla 87]: q-chebyshev polynomials. The spectrum of any associated semi-infinite Jacobi operator is never known explicitly α 0) but is expressible in terms of zeros of certain special functions.functions František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

9 Special functions 1/2 - basic hypergeometric series Let 0 < q < 1, r, s Z +. Recall the basic hypergeometric function [ ] a1, a r φ 2,... a r s ; q, z b 1, b 2,... b s is defined by the power series a 1 ; q) na 2 ; q) n... a r ; q) n 1) s r+1)n q s r+1)nn 1)/2 z n b n=0 1 ; q) nb 2 ; q) n... b s; q) n q; q) n where z, a 1, a 2,..., a r C, b 1, b 2,..., b s C \ q Z and a; q) 0 = 1, a; q) n = 1 a)1 aq)... 1 aq n 1 ) is the q-pochhammer symbol. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

10 Special functions 1/2 - basic hypergeometric series Let 0 < q < 1, r, s Z +. Recall the basic hypergeometric function [ ] a1, a r φ 2,... a r s ; q, z b 1, b 2,... b s is defined by the power series a 1 ; q) na 2 ; q) n... a r ; q) n 1) s r+1)n q s r+1)nn 1)/2 z n b n=0 1 ; q) nb 2 ; q) n... b s; q) n q; q) n where z, a 1, a 2,..., a r C, b 1, b 2,..., b s C \ q Z and a; q) 0 = 1, a; q) n = 1 a)1 aq)... 1 aq n 1 ) is the q-pochhammer symbol. Here we will need only 0 φ 1 and 1 φ 1. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

11 Special functions 2/2 - theta functions The theta function: 1 θ qz) := z; q) q/z; q) = q nn 1)/2 z) n q; q) n= František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

12 Special functions 2/2 - theta functions The theta function: 1 θ qz) := z; q) q/z; q) = q nn 1)/2 z) n q; q) n= Jacobi s theta functions: ϑ 1 z q) = iq 1/4 e iz q 2 ; q 2 ) θ q 2 e 2iz) ϑ 2 z q) = q 1/4 e iz q 2 ; q 2 ) θ q 2 e 2iz) ϑ 3 z q) = q 2 ; q 2 ) θ q 2 qe 2iz) ϑ 4 z q) = q 2 ; q 2 ) θ q 2 qe 2iz) František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

13 Contents 1 Introduction 2 Spectral resolution of A 3 Spectral resolution of B František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

14 Operators associated with the Jacobi matrix A Operator A with Dom A = spane n n Z} acting as has deficiency indices 1, 1). Ae n = q n+1 e n 1 + q n e n+1 František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

15 Operators associated with the Jacobi matrix A Operator A with Dom A = spane n n Z} acting as Ae n = q n+1 e n 1 + q n e n+1 has deficiency indices 1, 1). Thus, there is a one-parameter family of self-adjoint extensions A t, t R }. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

16 Operators associated with the Jacobi matrix A Operator A with Dom A = spane n n Z} acting as Ae n = q n+1 e n 1 + q n e n+1 has deficiency indices 1, 1). Thus, there is a one-parameter family of self-adjoint extensions A t, t R }. Let D := ψ l 2 Aψ l 2 }. By using the theory of self-adjoint extensions and simple structure of matrix A one gets: František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

17 Operators associated with the Jacobi matrix A Operator A with Dom A = spane n n Z} acting as Ae n = q n+1 e n 1 + q n e n+1 has deficiency indices 1, 1). Thus, there is a one-parameter family of self-adjoint extensions A t, t R }. Let D := ψ l 2 Aψ l 2 }. By using the theory of self-adjoint extensions and simple structure of matrix A one gets: Proposition self-adjoint extensions) For t R }, operators A t, acting as A t ψ = Aψ, with domains Dom A t = ψ D if t R, or are all self-adjoint extensions of A. lim n q n ψ 2n+1 + tψ 2n ) = 0 Dom A = ψ D } lim n q n qψ 2n 1 tψ 2n ) = 0, } lim n q n ψ 2n = 0, František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

18 Operators associated with the Jacobi matrix A Operator A with Dom A = spane n n Z} acting as Ae n = q n+1 e n 1 + q n e n+1 has deficiency indices 1, 1). Thus, there is a one-parameter family of self-adjoint extensions A t, t R }. Let D := ψ l 2 Aψ l 2 }. By using the theory of self-adjoint extensions and simple structure of matrix A one gets: Proposition self-adjoint extensions) For t R }, operators A t, acting as A t ψ = Aψ, with domains Dom A t = ψ D if t R, or are all self-adjoint extensions of A. In addition, lim n q n ψ 2n+1 + tψ 2n ) = 0 Dom A = ψ D } lim n q n qψ 2n 1 tψ 2n ) = 0, } lim n q n ψ 2n = 0, σ ca t ) = σessa t ) = 0}, t R }. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

19 Solutions of the eigenvalue equation The q-exponential function: E qz) = n=0 q n2 /4 ) z n = 1 φ 1 0; q 1/2 ; q 1/2, q 1/4 z q; q) n František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

20 Solutions of the eigenvalue equation The q-exponential function: Sequences ψ ±, where E qz) = n=0 q n2 /4 ) z n = 1 φ 1 0; q 1/2 ; q 1/2, q 1/4 z q; q) n ψ n ± := ±i) n q n/2 E q 2 ±ixq n ), are two linearly independent solutions of the difference equation Aψ = xψ. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

21 Solutions of the eigenvalue equation The q-exponential function: Sequences ψ ±, where E qz) = n=0 q n2 /4 ) z n = 1 φ 1 0; q 1/2 ; q 1/2, q 1/4 z q; q) n ψ n ± := ±i) n q n/2 E q 2 ±ixq n ), are two linearly independent solutions of the difference equation Aψ = xψ. By inspection of the asymptotic behavior of ψ ± n, as n ±, one gets: ψ ± l 2 + ), however, ψ ± / l 2 ). František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

22 Solutions of the eigenvalue equation The q-exponential function: Sequences ψ ±, where E qz) = n=0 q n2 /4 ) z n = 1 φ 1 0; q 1/2 ; q 1/2, q 1/4 z q; q) n ψ n ± := ±i) n q n/2 E q 2 ±ixq n ), are two linearly independent solutions of the difference equation Aψ = xψ. By inspection of the asymptotic behavior of ψ ± n, as n ±, one gets: ψ ± l 2 + ), however, ψ ± / l 2 ). Hence, one expects there are non-trivial coefficients a = ax) and b = bx) such that aψ + + bψ l 2 ). František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

23 The l 2 -solution Proposition For all x C \ 0}, the sequence ) ) ϕx) := θ q iq 1/2 x ψ ) x) + θ q iq 1/2 x ψ +) x), is the non-trivial solution of Aφ = xφ which belongs to l 2 Z). In addition, within the space l 2 Z), this solution is given uniquely up to a multiplicative constant. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

24 The l 2 -solution Proposition For all x C \ 0}, the sequence ) ) ϕx) := θ q iq 1/2 x ψ ) x) + θ q iq 1/2 x ψ +) x), is the non-trivial solution of Aφ = xφ which belongs to l 2 Z). In addition, within the space l 2 Z), this solution is given uniquely up to a multiplicative constant. Moreover, ϕ nx) = 1; q) x n q nn 1)/2 0φ 1 ; 0; q 2, q 2n+4 x 2) and František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

25 The l 2 -solution Proposition For all x C \ 0}, the sequence ) ) ϕx) := θ q iq 1/2 x ψ ) x) + θ q iq 1/2 x ψ +) x), is the non-trivial solution of Aφ = xφ which belongs to l 2 Z). In addition, within the space l 2 Z), this solution is given uniquely up to a multiplicative constant. Moreover, and ϕ nx) = 1; q) x n q nn 1)/2 0φ 1 ; 0; q 2, q 2n+4 x 2) q 2 ϕx) 2 ; q 2) 2 l 2 = 4 ) θ q; q 2 2 q 2 z 2) František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

26 The secular equation Theorem secular equation) For t R }, one has spec c A t ) = 0} and spec p A t ) coincides with the set of roots of the secular equation: xθ q 4 q 2 x 2) + tθ q 4 x 2) = 0, for t R, and θ q 4 x 2) = 0, for t =. In addition, all eigenvalues of A t are simple. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

27 The secular equation Theorem secular equation) For t R }, one has spec c A t ) = 0} and spec p A t ) coincides with the set of roots of the secular equation: xθ q 4 q 2 x 2) + tθ q 4 x 2) = 0, for t R, and θ q 4 x 2) = 0, for t =. In addition, all eigenvalues of A t are simple. Corollary: } spec p A 0 ) = ±q 2n+1 n Z } and spec p A ) = ±q 2n n Z. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

28 The secular equation Theorem secular equation) For t R }, one has spec c A t ) = 0} and spec p A t ) coincides with the set of roots of the secular equation: xθ q 4 q 2 x 2) + tθ q 4 x 2) = 0, for t R, and θ q 4 x 2) = 0, for t =. In addition, all eigenvalues of A t are simple. Corollary: } spec p A 0 ) = ±q 2n+1 n Z } and spec p A ) = ±q 2n n Z. How to solve the secular equation in general? František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

29 The secular equation Theorem secular equation) For t R }, one has spec c A t ) = 0} and spec p A t ) coincides with the set of roots of the secular equation: xθ q 4 q 2 x 2) + tθ q 4 x 2) = 0, for t R, and θ q 4 x 2) = 0, for t =. In addition, all eigenvalues of A t are simple. Corollary: } spec p A 0 ) = ±q 2n+1 n Z } and spec p A ) = ±q 2n n Z. How to solve the secular equation in general? Reparametrize t = Φs) and use nice properties of Jacobi s theta functions... František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

30 Reparametrization t = Φs) The function Φs) := iq 1/2 ϑ 4 is q 2 ) ϑ ) 1 is q 2 is real-valued, strictly decreasing on 0, 2 ln q), and maps [0, 2 ln q) onto R }. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

31 Reparametrization t = Φs) The function Φs) := iq 1/2 ϑ 4 is q 2 ) ϑ ) 1 is q 2 is real-valued, strictly decreasing on 0, 2 ln q), and maps [0, 2 ln q) onto R }. For the inverse function, one has Φ 1 dx t) = Cq) Dq) ) t + x 2 q + x 2) where and Cq) = q 1/2 ϑ 2 0 q 2 ) ϑ 3 0 q 2 ) 0 q 2 ) Dq) = qϑ2 3 ϑ 2 ). 2 0 q 2 František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

32 Spectrum fully explicitly Using the reparametrization t = Φs), the secular equation simplifies to θ q 2 e s x ) θ q 2 e s x ) = 0. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

33 Spectrum fully explicitly Using the reparametrization t = Φs), the secular equation simplifies to θ q 2 e s x ) θ q 2 e s x ) = 0. Theorem Let t R }, then where spec p A t ) = e s q 2Z e s q 2Z dx s = Cq) Dq) ) t + x 2 q + x 2). In addition, the family of corresponding eigenvectors ϕ±e s q 2N ) N Z}, where forms an orthogonal basis of l 2 Z). ϕ nx) = 1; q) x n q nn 1)/2 0φ 1 ; 0; q 2, q 2n+4 x 2), František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

34 Contents 1 Introduction 2 Spectral resolution of A 3 Spectral resolution of B František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

35 The discrete Schrödinger operator B The second Jacobi matrix B determines the unique operator B = U + U + αv where U is the forward shift operator and V is the self-adjoint diagonal operator: Ue n = e n+1 and Ve n = q n e n, n Z. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

36 The discrete Schrödinger operator B The second Jacobi matrix B determines the unique operator B = U + U + αv where U is the forward shift operator and V is the self-adjoint diagonal operator: Ue n = e n+1 and Ve n = q n e n, n Z. Proposition essential spectrum) The operator B is self-adjoint and one has σessb) = [ 2, 2]. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

37 Solutions of the eigenvalue equation Using properties of the Hahn-Exton q-bessel functions one verifies: f nz) := 1) n α n q 2 1 nn+1) z 1 α 1 q n+1 ; q ) 1 φ 1 0; z 1 α 1 q n+1 ; q, zα 1 q n+1) and g nz) := z n zαq 1 n ; q ) 1 φ 1 0; zαq 1 n ; q, qz 2). are two solutions of the equation for all α, z 0. Bψ = z + z 1 )ψ. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

38 Solutions of the eigenvalue equation Using properties of the Hahn-Exton q-bessel functions one verifies: f nz) := 1) n α n q 2 1 nn+1) z 1 α 1 q n+1 ; q ) 1 φ 1 0; z 1 α 1 q n+1 ; q, zα 1 q n+1) and g nz) := z n zαq 1 n ; q ) 1 φ 1 0; zαq 1 n ; q, qz 2). are two solutions of the equation for all α, z 0. Note that z z + z 1 : Bψ = z + z 1 )ψ. z 0 < z < 1} C \ [ 2, 2], outside σessa) ) e iθ θ [0, π]} [ 2, 2], inside σessa) ) František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

39 Solutions of the eigenvalue equation Using properties of the Hahn-Exton q-bessel functions one verifies: f nz) := 1) n α n q 2 1 nn+1) z 1 α 1 q n+1 ; q ) 1 φ 1 0; z 1 α 1 q n+1 ; q, zα 1 q n+1) and g nz) := z n zαq 1 n ; q ) 1 φ 1 0; zαq 1 n ; q, qz 2). are two solutions of the equation for all α, z 0. Note that z z + z 1 : Bψ = z + z 1 )ψ. z 0 < z < 1} C \ [ 2, 2], outside σessa) ) e iθ θ [0, π]} [ 2, 2], inside σessa) ) The solutions f z) and gz) are linearly independent iff z / α 1 q Z 0} since W f, g) = z 1 θ q αz). František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

40 The point spectrum Detailed asymptotic analysis of solutions f and g yields: If 0 < z < 1 and z / α 1 q Z 0}, then l 2 + ) / l 2 + ) f z) / l 2 gz) ) l 2 ). František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

41 The point spectrum Detailed asymptotic analysis of solutions f and g yields: If 0 < z < 1 and z / α 1 q Z 0}, then l 2 + ) / l 2 + ) f z) / l 2 gz) ) l 2 ) If z = 1 the asymptotic behavior of solutions is very different and, in the end, it implies that: For α R and x [ 2, 2], there is no non-trivial solution of Bψ = xψ belonging to l 2 Z).. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

42 The point spectrum Detailed asymptotic analysis of solutions f and g yields: If 0 < z < 1 and z / α 1 q Z 0}, then l 2 + ) / l 2 + ) f z) / l 2 gz) ) l 2 ) If z = 1 the asymptotic behavior of solutions is very different and, in the end, it implies that: For α R and x [ 2, 2], there is no non-trivial solution of Bψ = xψ belonging to l 2 Z). Theorem point spectrum) If α 0, then } σb) \ [ 2, 2] = σ pb) = α 1 q m + αq m m > log q α and all points from this set are simple eigenvalues of B.. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

43 The point spectrum Detailed asymptotic analysis of solutions f and g yields: If 0 < z < 1 and z / α 1 q Z 0}, then l 2 + ) / l 2 + ) f z) / l 2 gz) ) l 2 ) If z = 1 the asymptotic behavior of solutions is very different and, in the end, it implies that: For α R and x [ 2, 2], there is no non-trivial solution of Bψ = xψ belonging to l 2 Z). Theorem point spectrum) If α 0, then } σb) \ [ 2, 2] = σ pb) = α 1 q m + αq m m > log q α and all points from this set are simple eigenvalues of B. Further, eigenvectors v m corresponding to eigenvalues α 1 q m + αq m, can be chosen as v m = v m,j } j=, with v m,j = f j α 1 q m) = 1) j α j q 1 2 jj+1) q m+j+1 ; q) 1 φ 1 0; q m+j+1 ; q, α 2 q m+j+1).. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

44 The point spectrum Detailed asymptotic analysis of solutions f and g yields: If 0 < z < 1 and z / α 1 q Z 0}, then l 2 + ) / l 2 + ) f z) / l 2 gz) ) l 2 ) If z = 1 the asymptotic behavior of solutions is very different and, in the end, it implies that: For α R and x [ 2, 2], there is no non-trivial solution of Bψ = xψ belonging to l 2 Z). Theorem point spectrum) If α 0, then } σb) \ [ 2, 2] = σ pb) = α 1 q m + αq m m > log q α and all points from this set are simple eigenvalues of B. Further, eigenvectors v m corresponding to eigenvalues α 1 q m + αq m, can be chosen as v m = v m,j } j=, with v m,j = f j α 1 q m) = 1) j α j q 1 2 jj+1) q m+j+1 ; q) 1 φ 1 0; q m+j+1 ; q, α 2 q m+j+1).. In addition, v m l 2 Z) = α m q mm+1)/2 1 α 2 q 2m q; q), m > log q α. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

45 The absolutely continuous part of the spectral measure Let us denote E k,l ) := e k, E B )e l, k, l Z, where E B stands for the spectral measure of B. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

46 The absolutely continuous part of the spectral measure Let us denote E k,l ) := e k, E B )e l, k, l Z, where E B stands for the spectral measure of B. To determine the spectral measure in the essential spectrum we use the formula E k,l a, b)) = lim lim 1 b δ Gk,l x + iɛ) G k,l x iɛ) ) dx, δ 0+ ɛ 0+ 2πi a+δ where G k,l z) := e k, B z) 1 e l = 1 g k z)f l z), k l, W f, g) g l z)f k z), k l, František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

47 The absolutely continuous part of the spectral measure Let us denote E k,l ) := e k, E B )e l, k, l Z, where E B stands for the spectral measure of B. To determine the spectral measure in the essential spectrum we use the formula E k,l a, b)) = lim lim 1 b δ Gk,l x + iɛ) G k,l x iɛ) ) dx, δ 0+ ɛ 0+ 2πi a+δ where G k,l z) := e k, B z) 1 e l = 1 g k z)f l z), k l, W f, g) g l z)f k z), k l, Proposition Let α 0 and 2 a < b 2. Then for any k, l Z, it holds E k,l [a, b]) = 1 2π φa f l φ b e iφ) f k e iφ) e 2iφ ; q ) 2 αe iφ, qα 1 e iφ ; q ) dφ where φ a = arccos a/2) and φ b = arccos b/2). Consequently, σ acb) = [ 2, 2]. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

48 Summary Theorem If α 0, then and σ essb) = σ acb) = [ 2, 2], } σ pb) = σ d B) = α 1 q m + αq m m > log q α. František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

49 Summary Theorem If α 0, then and σ essb) = σ acb) = [ 2, 2], } σ pb) = σ d B) = α 1 q m + αq m m > log q α. In addition, for M R a Borel set, we have E k,l M) = 1 f l e iφ) f k e iφ) e 2iφ ; q ) 2 2π 2 cos φ [ 2,2] A αe iφ, qα 1 e iφ ; q ) dφ α 2 q 2m) α 2m q mm+1) f l α 1 q m) f k α 1 q m). q; q) 2 m> log α α 1 q m +αq m M František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

50 One consequence for special functions Recall the Hanhn-Exton or third Jackson s) q-bessel function is defined as J νz; q) = z ν qν+1 ; q) q; q) 1φ 1 0; q ν+1 ; q, qz 2). František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

51 One consequence for special functions Recall the Hanhn-Exton or third Jackson s) q-bessel function is defined as J νz; q) = z ν qν+1 ; q) q; q) 1φ 1 0; q ν+1 ; q, qz 2). Elements of the eigenvectors v m are expressible in terms of Hanhn-Exton q-bessel function and the formula for the norm of the eigenvectors yields n Z Jn 2 z; q) = 1, z < 1. 1 z2 František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

52 One consequence for special functions Recall the Hanhn-Exton or third Jackson s) q-bessel function is defined as J νz; q) = z ν qν+1 ; q) q; q) 1φ 1 0; q ν+1 ; q, qz 2). Elements of the eigenvectors v m are expressible in terms of Hanhn-Exton q-bessel function and the formula for the norm of the eigenvectors yields n Z Jn 2 z; q) = 1, z < 1. 1 z2 This formula seems to be new Really?) and it generalizes the well-known summation formula for the Bessel functions of the first kind: Jn 2 z) = 1, z < 1. n Z František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20

53 The end Thank you! František Štampach Stockholm University) Jacobi operators Spectr. Theor. Appl.) March 13, / 20