Sharp spectral estimates for periodic matrix-valued Jacobi operators
aa r X i v : . [ m a t h . F A ] N ov SHARP SPECTRAL ESTIMATES FOR PERIODIC MATRIX-VALUEDJACOBI OPERATORS
ANTON KUTSENKO
Abstract.
For the periodic matrix-valued Jacobi operator J we obtain the estimate of theLebesgue measure of the spectrum | σ ( J ) | n Tr( a n a ∗ n ) , where a n are off-diagonalelements of J . Moreover estimates of width of spectral bands are obtained. Introduction
We consider a self-adjoint matrix-valued Jacobi operator J : ℓ ( Z ) m → ℓ ( Z ) m given by( J y ) n = a n y n +1 + b n y n + a ∗ n − y n − , n ∈ Z , y n ∈ C m , y = ( y n ) n ∈ Z ∈ ℓ ( Z ) m , (1.1)where a n and b n = b ∗ n are p -periodic sequences of the complex m × m matrices. It is well known(see e.g. [CG], [CGR], [KKu], [KKu1]) that the spectrum of this operator σ ( J ) = σ ac ( J ) ∪ σ p ( J ), where absolutely continuous part σ ac ( J ) is a union of finite number of intervals and σ p ( J ) consists of finite number of eigenvalues of infinite multiplicity. Note that if det a n = 0for all n = 1 , ..., p , then σ ( J ) = σ ac ( J ) and σ p ( J ) = ∅ always. The main goal of this paperis to obtain estimate of length of spectrum σ ( J ). We don’t know such estimates for thematrix-valued Jacobi operators. Theorem 1.1.
The Lebesgue measure of spectrum of J satisfy the following estimate mes( σ ( J )) n Tr( a n a ∗ n ) . (1.2) Remark 1.
Note that this estimate does not depend on period p and coefficients b n , i.e.changing p and b n we can’t sufficiently increase the length of spectrum of J . If a n = 0 forsome n ∈ Z , then this estimate is sharp.
2. ( m = 1 ) For the scalar case m = 1 we have the estimate (see e.g. [DS], [Ku], [KKr])mes( σ ( J )) | a a ...a p | p , (1.3)We reach equality for the case of discrete Shr¨odinger operator J with a n = 1, b n = 0.Estimate (1.2) is better than (1.3), since min | a n | | a a ...a p | p . The result similar to (1.3) was obtained in [PR] for general non periodic scalar case ( m = 1). From the Proof of Theorem 1.1 we obtain estimates of spectral bands, see (2.9).
Example (sharpness).
We construct the Jacobi matrix J whose spectrum satisfy mes( σ ( J )) =4 min n (Tr a n a ∗ n ) . Let J be Jacobi matrix with elements a n = I m ( m × m identical matrix)and b n = diag(4 k ) mk =1 for any n . Since all a n and b n are diagonal matrix, then J is unitarily Date : November 21, 2018.1991
Mathematics Subject Classification.
Key words and phrases. matrix-valued Jacobi operator, Jacobi matrix, spectral estimates, measure ofspectrum. equivalent to the direct sum of scalar Jacobi operators. In our case this is the direct sum ofshifted discrete Shr¨odinger operators ⊕ mk =1 ( J + 4 kI ) ( I is identical operator). Then σ ( J ) = m [ k =1 σ ( J + 4 kI ) = m [ k =1 [ − k, k ] = [2 , m ] , which gives us mes( σ ( J )) = 4 m = 4(Tr a n a ∗ n ) .2. Proof of Theorem 1.1
Without lost of generality we may assume that min n Tr( a n a ∗ n ) = Tr( a a ∗ ) and p > J is unitarily equivalent to the operator J = R ⊕ [0 , π ) K ( x ) dx acting in R ⊕ [0 , π ) H dx , where H = C pm and pm × pm matrix K ( x ) is givenby K ( x ) = b a ... e − ix a ∗ a ∗ b a ... a ∗ b ... ... ... ... ... ...e ix a ... b p = K + K ( x ) , (2.4)where K = b a ... a ∗ b a ... a ∗ b ... ... ... ... ... ... ... b p , K ( x ) = ... e − ix a ∗ ...
00 0 0 ... ... ... ... ... ...e ix a ... . (2.5)The spectrum σ ( J ) is σ ( J ) = [ x ∈ [0 , π ] σ ( K ( x )) . (2.6)From (2.4)-(2.5) we obtain K − | K | K ( x ) K + | K | , x ∈ [0 , π ] (2.7)where | K | = ( K K ∗ ) = ( a ∗ a ) ...
00 0 0 ...
00 0 0 ... ... ... ... ... ... ... ( a a ∗ ) (2.8)does not depend on x . Let λ ( x ) ... λ N ( x ) be eigenvalues of K ( x ) and let λ ± ... λ ± N be eigenvalues of K ± | K | . Using (2.7) we obtain λ − n λ n ( x ) λ + n , x ∈ [0 , π ] (2.9)which with (2.6) gives us σ ( J ) = [ x ∈ [0 , π ] { λ n ( x ) } Nn =1 ⊂ N [ n =1 [ λ − n , λ + n ] . (2.10) STIMATES OF LENGTH OF SPECTRUM 3
Then mes( σ ( J )) N X n =1 ( λ + n − λ − n ) = 2 Tr | K | = 4 Tr( a n a ∗ n ) (2.11) Acknowledgements.
I would to express thanks to prof. E. Korotyaev for useful discussionsand remarks. Also I want to thank prof. B. Simon for useful comments and refferences to thepaper [PR].Many thanks to Michael J. Gruber, who told me that instead of k a n k rank a n (which was inthe first version of this paper) is better to use Tr( a n a ∗ n ) in (1.2). References [CG] Clark, S.; Gesztesy, F. On Weyl-Titchmarsh theory for singular finite difference Hamiltonian systems,J. Comput. Appl. Math., 171 (2004) 151184.[CGR] Clark, S.; Gesztesy, F.; Renger, W. Trace formulas and Borg-type theorems for matrix-valued Jacobiand Dirac finite difference operators. J. Diff. Eq. 219 (2005), 144–182.[DS] P. Deift, B. Simon. Almost periodic Schr¨odinger operators III. The absolutely continuous spectrum inone dimension. Commun. Math. Phys., 90, 389411 (1983).[K] Kato T. Perturbation Theory for Linear Operators. Springer (February 15, 1995).[Ku] Kutsenko A. Estimates of Parameters for Conformal Mappings Related to a Periodic Jacobi Matrix.Journal of Mathematical Sciences, Volume 134, Number 4 / April 2006, Pages 2295-2304.[KKr] Korotyaev, E.; Krasovsky, I. Spectral estimates for periodic Jacobi matrices, Commun. Math. Phys.234(2003), 517-532.[KKu] Korotyaev, E., Kutsenko, A. Lyapunov functions for periodic matrix-valued Jacobi operators, AMStranslations Series 2, 225 (2008), 117-131.[KKu1] Korotyaev, E., Kutsenko, A. Borg type uniqueness Theorems for periodic Jacobi operators with matrixvalued coefficients. Proc. of the AMS, Volume 137, Number 6, June 2009, Pages 1989-1996.[L] Y. Last. On the measure of gaps and spectra for discrete 1D Schr¨odinger operators. Commun. Math.Phys., 149, 347-360 (1992).[PR] A. Poltoratski, C. Remling. Reflectionless Herglotz Functions and Jacobi Matrices, Commun. Math.Phys. Volume 288 Number 3(2009), 1007–1021.[RS] M. Reed ; B. Simon. Methods of modern mathematical physics. IV. Analysis of operators. AcademicPress, New York-London, 1978.[S] B. Simon, Orthogonal polynomials on the unit circle, Part 1 and Part 2, AMS, Providence, RI, 2005.[S1] B. Simon, Trace Ideals and Their Applications: Second Edition. Mathematical Surveys and Monographsvol. 120, 2005.[Te] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys andMonographs, vol. 72, AMS, Rhode Island, 2000.[vM] P. van Moerbeke. The spectrum of Jacobi matrices. Invent. Math. 37 (1976), no. 1, 45–81.[CG] Clark, S.; Gesztesy, F. On Weyl-Titchmarsh theory for singular finite difference Hamiltonian systems,J. Comput. Appl. Math., 171 (2004) 151184.[CGR] Clark, S.; Gesztesy, F.; Renger, W. Trace formulas and Borg-type theorems for matrix-valued Jacobiand Dirac finite difference operators. J. Diff. Eq. 219 (2005), 144–182.[DS] P. Deift, B. Simon. Almost periodic Schr¨odinger operators III. The absolutely continuous spectrum inone dimension. Commun. Math. Phys., 90, 389411 (1983).[K] Kato T. Perturbation Theory for Linear Operators. Springer (February 15, 1995).[Ku] Kutsenko A. Estimates of Parameters for Conformal Mappings Related to a Periodic Jacobi Matrix.Journal of Mathematical Sciences, Volume 134, Number 4 / April 2006, Pages 2295-2304.[KKr] Korotyaev, E.; Krasovsky, I. Spectral estimates for periodic Jacobi matrices, Commun. Math. Phys.234(2003), 517-532.[KKu] Korotyaev, E., Kutsenko, A. Lyapunov functions for periodic matrix-valued Jacobi operators, AMStranslations Series 2, 225 (2008), 117-131.[KKu1] Korotyaev, E., Kutsenko, A. Borg type uniqueness Theorems for periodic Jacobi operators with matrixvalued coefficients. Proc. of the AMS, Volume 137, Number 6, June 2009, Pages 1989-1996.[L] Y. Last. On the measure of gaps and spectra for discrete 1D Schr¨odinger operators. Commun. Math.Phys., 149, 347-360 (1992).[PR] A. Poltoratski, C. Remling. Reflectionless Herglotz Functions and Jacobi Matrices, Commun. Math.Phys. Volume 288 Number 3(2009), 1007–1021.[RS] M. Reed ; B. Simon. Methods of modern mathematical physics. IV. Analysis of operators. AcademicPress, New York-London, 1978.[S] B. Simon, Orthogonal polynomials on the unit circle, Part 1 and Part 2, AMS, Providence, RI, 2005.[S1] B. Simon, Trace Ideals and Their Applications: Second Edition. Mathematical Surveys and Monographsvol. 120, 2005.[Te] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys andMonographs, vol. 72, AMS, Rhode Island, 2000.[vM] P. van Moerbeke. The spectrum of Jacobi matrices. Invent. Math. 37 (1976), no. 1, 45–81.