On some tridiagonal k-toeplitz matrices: Algebraic and analytical aspects. Applications

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1 Journal of Computational and Applied Mathematics 184 (005) wwwelseviercom/locate/cam On some tridiagonal k-toeplitz matrices: Algebraic and analytical aspects Applications R Álvarez-Nodarse a,b,, J Petronilho c, NR Quintero b,d a Dpto Análisis Matemático, Universidad de Sevilla, Apdo 1160, Sevilla, Spain b Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E Granada, Spain c Dpto de Matemática, FCTUC, Universidade de Coimbra, Apdo Coimbra, Portugal d Dpto Física Aplicada I, EUP, Universidad de Sevilla, Virgen de África 7, Sevilla, Spain Received July 004; received in revised form 7 December 004 Abstract In this paper, we use the analytic theory for and 3-Toeplitz matrices to obtain the explicit expressions for the eigenvalues, eigenvectors and the spectral measure associated to the corresponding infinite matrices As an application we consider two solvable models related with the so-called chain model Some numerical experiments are also included 005 Elsevier BV All rights reserved Keywords: k-toeplitz matrices; Jacobi matrices; Orthogonal polynomials; Chain models 1 Introduction k-toeplitz matrices are tridiagonal matrices of the form A =[a i,j ] n i,j=1 (with n k) such that a i+k,j+k = a i,j (i, j = 1,,,n k), so that they are k-periodic along the diagonals parallel to the main diagonal [8] When k = 1 it reduces to a tridiagonal Toeplitz matrix Corresponding address Departamento de Análisis Matemático, Universidad de Sevilla, Apdo 1160, Sevilla, Spain address: ran@uses (R Álvarez-Nodarse) /$ - see front matter 005 Elsevier BV All rights reserved doi:101016/jcam

2 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) The interest of the study of k-toeplitz matrices appears to be very important not only from a theoretical point of view(in linear algebra or numerical analysis, eg), but also in applications For instance, it is useful in the study of sound propagation problems (see [,9]) In this paper, we present a complete study of the eigenproblems for tridiagonal and 3-Toeplitz matrices, including the spectral measure associated to the corresponding infinite Jacobi matrices, and then we apply the results to the study of the so called chain models in Quantum Physics Since we hope that our discussion could be of interest both to readers working on Applied Mathematics and Orthogonal Polynomials as well as physicists, we will further develop the works [5,6,14 17] including several results on tridiagonal k-toeplitz matrices that followfrom the results in those papers but not explicitly included there Of particular interest is the symmetric case because of its interest in the study of quantum chain models In fact, one of the main problems in Quantum Physics is to find the solutions of the stationary Schrödringer equation H Φ =ε Φ, (11) where H is the Hamiltonian of the system and ε is the energy corresponding to the state Φ A usual method for solving Eq (11) is to expand the unknown wave functions Φ in the discrete basis (not necessarily orthogonal) { Φ k } k=1, ie, N Φ = C Nk Φ k (1) k=1 Substituting Eq (1) in (11) and multiplying by Φ m and taking into account the orthogonality of the functions Φ k we obtain the following linear system of equations N C Nk Φ m H Φ k =εc Nm (13) k=1 If we denote the matrix elements Φ m H Φ k by h mk then we can rewrite (13) in the matrix form h 11 h 1 h 1N 1 h 1N C N1 C N1 h 1 h h N 1 h N C N C N h 31 h 3 h 3N 1 h 3N C N3 C = ε N3 (14) h N 1 h N 1 h N 1N 1 h N 1N C N 1N C N 1N h N1 h N h NN 1 h NN C NN C NN In general, the computation of the eigenvalues (and also the eigenvectors) for an arbitrary matrix, as the previous one in (14), is very difficult since it is equivalent to the problem of finding the roots of a polynomial of degree N Moreover, the numerical algorithms (Newton s Method, etc) are, in general, unstable for large N For this reason simpler models (which are easier to solve and numerically more stable and economic) have been introduced One of these models is the so called chain model which has been successfully used in Solid State Physics [1], Nuclear Physics [], and Quantum Mechanics [11], etc In fact, any quantum model can be transformed into the corresponding chain model (see [10] and references therein)

3 50 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) In our previous paper [1] we have briefly considered the -periodic chain model the constant chain model, ie, when the sequences {a n } and {b n } are constants equal to a and b, respectively, has been considered in [10] ie, when the sequences {a n } and {b n } are periodic sequences with period, ie, {a n }={a,b,a,b,} and {b n }={c,d,c,d,} For such models it is possible to obtain analytic formulae for the values of the energy (eigenvalues) of H and its corresponding wave functions Here we will conclude the study started in [1] for the two chain model and will present the complete study of the 3-periodic chain model The structure of the paper is as follows In Section, we give the needed mathematical background Section 3 is devoted to some applications of the theory of tridiagonal k-toeplitz matrices in quantum physics: concretely to the so-called chain model Mathematical background We start with some basic results from the general theory of orthogonal polynomials (see eg [3]) It is known that any orthogonal polynomial sequence (OPS) {P n } n 0 it is characterised by a three-term recurrence relation xp n (x) = α n P n+1 (x) + β n P n (x) + γ n P n 1 (x), n = 0, 1,, (1) with initial conditions P 1 =0 and P 0 =1 where {α n } n 0, {β n } n 0 and {γ n } n 1 are sequences of complex numbers such that α n γ n+1 = 0 for all n 0, or in matrix form xp n (x) = J n+1 P n (x) + α n P n+1 (x)e n, () where P n (x) =[P 0 (x),,p n (x)] T, e n =[0, 0,,0, 1] T R n+1 and J n+1 is the tridiagonal matrix of order n + 1 β 0 α γ 1 β 1 α γ β α 0 0 J n+1 = 0 0 γ 3 β β n 1 α n γ n β n If {x nj } 1 j n are the zeros of the polynomial P n, then it follows from () that each x nj is an eigenvalue of the corresponding tridiagonal matrix J n and P n 1 (x nj ) := [P 0 (x nj ),,P n 1 (x nj )] T is a corresponding eigenvector

4 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) When α n =1 and γ n > 0 for all n=1,,the (monic) polynomials {P n } n 0 defined by the recurrence relation (1) arise as denominators of the approximants of the continued fraction 1 γ x β 0 1 γ x β 1 x β x β n 1 γ n 1 γ n x β n Under these conditions, by Favard s theorem [3], {P n } n 0 constitutes an orthogonal polynomial sequence with respect to a positive definite moment functional, and if the moment problem associated with the continued fraction is determined, then this linear functional can be characterised by a unique distribution function, ie, a function σ : R R which is nondecreasing, it has infinitely many points of increase and all the moments + xn dσ(x), n = 0, 1,,, are finite The numerators of the continued fraction, denoted by {P n (1) } n 0, can be given by the shifted recurrence relation xp n (1) (1) (x) = P n+1 (x) + β n+1p n (1) (x) + γ n+1p (1) n 1 (x), n 0 with initial conditions P (1) (1) 1 = 0 and P 0 = 1 This continued fraction converges to a function F(z; σ) and the general theory of the moment problem ensures that F is analytic in the complex plane with a cut along the support of σ (ie, the set of points of increase of σ) This fact can be summarised by Markov Stieltjes s theorem lim n + μ 0 P (1) n 1 (z) + = F(z; σ) := P n (z) dσ(x), z C\supp(σ), (3) x z where μ 0 = + dσ(x) is the first moment of the distribution σ(x) and F is its Stieltjes function Now, the function σ(x) can be recovered from (3) by applying the Stieltjes inversion formula x 1 σ(x) σ(y) = lim IF(t + iε, σ) dt, ε 0 + π y where it is assumed that σ is normalised in the following way σ(x + 0) + σ(x 0) σ( ) = 0, σ(x) = and Iz denotes the imaginary part of z An important family of orthogonal polynomials are the orthonormal Chebyshev polynomials of second kind {U n (x)} n 0 defined in terms of trigonometric polynomials in cos θ as sin(n + 1)θ U n (x) =, x = cos θ sin θ For these polynomials (1) takes the form U 1 = 0, U 0 = 1, xu n (x) = U n+1 (x) + U n 1 (x), n = 0, 1,,

5 5 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) They are orthonormal with respect to the distribution function dσ U (x) = 1 x dx, supp(σ U ) =[ 1, 1], π ie, 1 U n (x)u m (x) dσ U (x) = δ n,m, 1 where δ n,m is the Kronecker symbol: δ k,m = 1 for k = m, elsewhere δ k,m = 0 The corresponding Stieltjes function is ( ) F U (z) = z + z 1 = z z 1, z C\[ 1, 1], (4) where the complex square root is such that z + z 1 > 1 whenever z/ [ 1, 1] The Chebychev polynomials U n are closely related with the tridiagonal and 3 Toeplitz matrices as it is shown in the next two sections 1 Remarks on tridiagonal -Toeplitz matrices Let B N be the irreducible tridiagonal -Toeplitz matrix a 1 b c 1 a b c B N = a 1 b c 1 a b R (N,N), N N, (5) c a 1 where b 1,b,c 1 and c are positive numbers Define the polynomials π (x) = (x a 1 )(x a ) and ( ) x P n (x) = (b 1 b c 1 c ) n/ b1 c 1 b c U n, n= 0, 1,,, b 1 b c 1 c where U n is the Chebyshev polynomial of second kind The following theorem holds Theorem 1 (Gover [7] and Marcellán and Petronilho [14]) Let B N, N =1,, 3,,be the irreducible tridiagonal -Toeplitz matrix given by (5), where b 1, b, c 1, and c are positive numbers The sequence {S n } n 0 of orthogonal polynomials associated with the matrices B N is S k (x) = (b 1 b ) k {P k (π (x)) + b c P k 1 (π (x))}, S k+1 (x) = (b 1 1 (b 1b ) k (x a 1 )P k (π (x)), k = 0, 1,

6 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) Then the eigenvalues λ N,m of B N are the zeros of S N, and the corresponding eigenvectors v N,m are given by S 0 (λ N,m ) S v N,m = 1 (λ N,m ), m= 1,,,N S N 1 (λ N,m ) In particular, 1 the eigenvalues λ n+1,m of B n+1 (m=1,,,n+1) are λ n+1,1 =a 1 and the solutions of the quadratic equations [ π (λ) b 1 c 1 + b c + ] kπ b 1 b c 1 c cos = 0, n + 1 k = 1,,n (6) Notice that since the sequence {S k } k is an orthogonal polynomial sequence corresponding to a positive definite case, then the zeros are simple and interlace, ie, if {x k,j } k j=1 denotes the zeros of the polynomial S k, then x k,j <x k 1,j <x k,j+1, j = 1,,,k 1 Therefore using the values (6) we can obtain bounds for the eigenvalues of the corresponding matrices for the even case Moreover (cf [0]; see also [15]), the Stieltjes function associated to the sequence {S n } n 0 reads as F S (z) = 1 a 1 z 1 b c 1 z a 1 ( π (z) β (π (z) β) α ), (7) where α = b 1 b c 1 c and β = b 1 c 1 + b c Furthermore, {S n } n 0 is orthogonal with respect to the distribution function dσ S (x) = Mδ(x a 1 ) dx α πb c x a 1 (π (x) β) dx, (8) where M = 1 min{b 1 c 1,b c }/(b c ) and which support is the union of two intervals if M = 0 and the union of two intervals with a singular set if M>0, ie, { ΣS if b supp(σ S ) = 1 c 1 b c, Σ S {a 1 } if b 1 c 1 >b c, 1 For the case when b 1 c 1 >b c the eigenvalues λ n,m of B n (m=1,,,n) are the solutions of the quadratic equations π (λ) [b 1 c 1 + b c + b 1 b c 1 c cos θ nk ]=0, k = 1,,n, where θ nk s are the nonzero solutions of the trigonometric equation b1 c 1 sin[(n + 1)θ]+ b c sin(nθ) = 0, (0 < θ < π)

7 54 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) where Σ S =[ a 1+a s, a 1+a r] [ a 1+a + r, a 1+a + s] and r = b 1 c 1 b c + a 1 a, s = b 1 c 1 + b c + a 1 a Remarks on tridiagonal 3-Toeplitz matrices Let us nowconsider the irreducible tridiagonal 3-Toeplitz matrix Define a 1 b c 1 a b c a 3 b c B N = 3 a 1 b c 1 a b 0 R (N,N), N N (9) c a 3 b c 3 a 1 P n (x) = (b 1 b b 3 c 1 c c 3 ) n/ U n ( x b1 c 1 b c b 3 c 3 b 1 b b 3 c 1 c c 3 Let ξ 1 and ξ be the zeros of the quadratic polynomial ), n= 0, 1,, (x a 1 )(x a ) b 1 c 1 (10) and define the polynomial π 3 (x) := (x a 1 )(x a )(x a 3 ) (b 1 c 1 + b c + b 3 c 3 )(x a 3 ) + b c (a 1 a 3 ) + b 3 c 3 (a a 3 ) + b 1 c 1 + b c + b 3 c 3 (11) In this case we have the following Theorem (Marcellán and Petronilho [16]) Let B N,N=1,, 3,be the irreducible tridiagonal 3- Toeplitz matrix given by (9), where b 1,b,b 3,c 1,c and c 3 are positive numbers The sequence {S n } n 0 of orthogonal polynomials associated with the matrices B N is S 3k (x) = (b 1 b b 3 ) k {P k (π 3 (x)) + b 3 c 3 (x a )P k 1 (π 3 (x))}, S 3k+1 (x) = b 1 1 (b 1b b 3 ) k {(x a 1 )P k (π 3 (x)) + b 1 c 1 b 3 c 3 P k 1 (π 3 (x))}, S 3k+ (x) = (b 1 b ) 1 (b 1 b b 3 ) k (x ξ 1 )(x ξ )P k (π 3 (x)), k = 0, 1,,

8 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) where ξ 1 and ξ are the roots of the polynomial (10) and π 3 is the polynomial given by (11) Then the eigenvalues λ N,m of B N are the zeros of S N, and the corresponding eigenvectors v N,m are given by S 0 (λ N,m ) S v N,m = 1 (λ N,m ), m= 1,,,N S N 1 (λ N,m ) In particular, when N = 3n +, the eigenvalues λ 3n+,m of B 3n+ (m = 1,,,3n + ) are λ 3n+,1 = ξ 1, λ 3n+, = ξ and the solutions of the cubic equations [ π 3 (λ) b 1 c 1 + b c + b 3 c 3 + ] kπ b 1 b b 3 c 1 c c 3 cos = 0, k = 1,,n (1) n + 1 As in the previous case, we can use the values (1) to give bounds for the eigenvalues of the corresponding matrices in the N = 3n and N = 3n + 1 cases In this case [17] the Stieltjes function associated to the sequence {S n } n 0 is F S (z) = ξ a 1 1 ξ ξ 1 ξ 1 z + a 1 ξ 1 1 ξ ξ 1 ξ z ( 1 (a 1 ξ 1 )(a 1 ξ ) b 1 c 1 b 3 c 3 (z ξ 1 )(z ξ ) π 3 (z) β (π 3 (z) β) α ), (13) where α = b 1 b b 3 c 1 c c 3 and β = b 1 c 1 + b c + b 3 c 3 In (13) the square root is such that z β + (z β) α > α whenever z/ [β α, β + α] Moreover, {S n } n 0 is orthogonal with respect to the distribution function dσ S (x) = M 1 δ(x ξ 1 ) dx + M δ(x ξ ) dx 1 (a 1 ξ 1 )(a 1 ξ ) α πb 1 c 1 b 3 c 3 (x ξ 1 )(x ξ ) (π 3 (x) β) dx, which support is contained in the union of the three intervals Σ S = π 1 3 ([β α, β + α]) (see Fig 1) with two possible mass points at ξ 1 and ξ, ie, Σ S if M 1 = 0,M = 0, Σ supp(σ S ) = S {ξ } if M 1 = 0,M > 0, Σ S {ξ 1 } if M 1 > 0,M = 0, Σ S {ξ 1, ξ } if M 1 > 0,M > 0, where and M 1 = a 1 ξ ξ 1 ξ M = ξ [ 1 a 1 ξ 1 ξ [ 1 b ( c (ξ 1 a 1 ) π3 (ξ 1 ) β F U α α 1 b ( c (ξ a 1 ) π3 (ξ ) β F U α α )] )]

9 56 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) α+β β α -1 π 3 ([β α,β+α]) Fig 1 The polynomial mapping [β α, β + α] π 1 3 ([β α, β + α]) If we now take into account the identity 4[(π 3 (ξ 1 ) β) α ]=[(b 3 c 3 b c )Δ + (a 1 a )(b 3 c 3 + b c )], where Δ= (a 1 a ) + 4b 1 c 1 as well as the right choice of the branch of the square root in the definition of F U (4) we find M 1 = max{0,(b 3c 3 b c )Δ + (a 1 a )(b 3 c 3 + b c )} 0, b 3 c 3 Δ M = max{0,(b 3c 3 b c )Δ (a 1 a )(b 3 c 3 + b c )} 0 (14) b 3 c 3 Δ A simple inspection of the values of M 1 and M leads to the following four cases: 1 b 3 c 3 b c and a 1 a In this case M 1 = 0iffa 1 = a and b 3 c 3 = b c, M = 0iffb 3 c 3 b c + a 1 a (b 3 c 3 + b c )Δ 1 b 3 c 3 b c and a 1 a Then M 1 = 0iffb 3 c 3 b c + a 1 a (b 3 c 3 + b c )Δ 1, M = 0iffa 1 = a and b 3 c 3 = b c 3 b 3 c 3 b c and a 1 a Then M 1 = 0iffb 3 c 3 b c a 1 a (b 3 c 3 + b c )Δ 1, In this case always M = 0

10 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) b 3 c 3 b c and a 1 a In this case always M 1 = 0, M = 0iffb 3 c 3 b c a 1 a (b 3 c 3 + b c )Δ 1 3 Some remarks on a matrix theoretic approach Here we want to emphasize another approach to the problem concerning the study of the spectral properties (eigenvalues, eigenvectors and asymptotic (limit) spectral measure) of the sequences of matrices defined by (5) and (9), based on recent results by Serra Capizzano, Fasino, Kuijlaars and Tilli (cf [4,13,18,19]) To simplify we will consider the case when the order N of the matrix B N in (5) is even Then B N is the block Toeplitz matrix A 0 A 1 A 1 B N = A 1 A 1 A 0 generated by the matrix valued polynomial f (x) := A 0 + A 1 e ix + A 1 e ix with [ a1 b A 0 := 1 c 1 a ], A 1 := [ ] 0 0, A b 0 1 := [ ] 0 c 0 0 Since, in general, f (x) is not hermitian then not very much can be said on the eigenvalues However, according to Theorem 1 the conditions b 1 c 1 > 0 and b c > 0 hold, and so it is well-known that, under such conditions, B N is similar (via diagonal transformations) to the block Toeplitz matrix ˆB N generated by the matrix valued polynomial ˆf (x) :=  0 +  1 e ix +  1 e ix with  0 := [ ] a1 b1 b1 c 1,  c 1 a 1 := [ 0 0 b c 0 ],  1 := [ ] 0 b c 0 0 Similar considerations remains true for the generalized case of a tridiagonal k-toeplitz matrix (see Eq (1) in [4]) Now, the limit distribution is described in Theorems 1 and in [4] In our specific case the spectra of the matrix B N distributes as the eigenvalues of ˆf (x), which are λ ± (x) := a 1 + a (a1 ) a ± + b 1 c 1 + b c + b 1 c 1 b c cos x

11 58 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) More precisely, it follows from Theorem in [4] that, with possible exception of at most a denumerable set of point masses, the support of the measure of orthogonality for the orthogonal polynomials corresponding to B N is contained in the set S := [λ 1, λ+ 1 ] [λ, λ+ ] (and the zeros of the orthogonal polynomials are dense in this set), where λ 1 := min{λ (0), λ (π)}, λ := min{λ +(0), λ + (π)}, Therefore, since λ + 1 := max{λ (0), λ (π)}, λ + := max{λ +(0), λ + (π)} and λ ± (0) = a 1 + a ± (a1 a ) ( + b1 c 1 + ) b c λ ± (π) = a 1 + a ± (a1 a ) ( + b1 c 1 ), b c we see that S is the same union of two intervals given in the end of Section 1 Also, the limit spectral measure follows from asymptotic spectral theory of Toeplitz matrices We remark that the spectral distribution holds for odd N as well since constant rank corrections do not modify the asymptotic spectral distribution Further, the results in [4] are true for every k-toeplitz matrix sequences (and so, in particular, for k = and k = 3), as well as for variable recurrence coefficients (see also [13]) and in the multidimensional case (cf also [18,19]) On the other hand, our results in Theorems 1 and gives more precise information on the localization of the zeros As a final remark, we would like to point out that in the present paper the theory of orthogonal polynomials is used for giving spectral information, while in [4,13] the idea is exactly the opposite since matrix theoretic tools are used for deducing information on the zeros of orthogonal polynomials 3 Applications: the chain model Here we will resume some important properties of the chain model For a more detailed study we refer to the nice paper by Haydock [10] Definition 31 (Haydocle [10]) The Chain Model is a quantum model determined by a sequence of orthonormal orbits (states) {u 0, u 1,} and two sets of real parameters {a 1,a,} and {b 1,b,}, which describe the action of the Hamiltonian H on the orbitals by a symmetric three-term recurrence relation of the form Hu n = b n+1 u n+1 + a n u n + b n u n 1 (31) The sequence {u 0, u 1,} may be finite or infinite In the first case we need to take the orbitals u 1 and u N+1 equal to zero Moreover, in [10] it has been shown that this model is equivalent to expressing

12 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) the matrix H by using an appropriate basis as a Jacobi (tridiagonal symmetric) matrix a 0 b b 1 a 1 b 0 H = 0 b a b b 3 a 3 (3) In the following we will suppose that the solution u of the Schrödinger equation (11) can be written as a linear combination of the states u 0, u 1, u,,ie, u = C k u k (33) k=0 For this model it is possible to obtain analytic formulae for the so-called general diagonal Green function G 0 (ε) In [10,1] it is shown that G 0 (ε) for the chain model (31) is related to the continued fraction G 0 (ε) = ε a 0 ε a 1 1 b 1 b ε a b n ε a n, ε C (34) For the finite chain model, this continued fraction reduces to the ratio of two polynomials which conforms the well known Padé approximants of order n of the infinite continued fraction This and some other results concerning the calculation of the Green function will be considered in detail in the next section Of particular interest are the function G 0 (ε) the real part of G 0 (ε) describes the response of the system 1 to be driven at a given energy and n 0 (ε) = lim ε 0 π I(G 0(ε + iε)) is the local density of the initial state (see [11] for more details) In the case of the infinite chain, it is possible to obtain an analytic expression for the Green function G 0 (ε) (34) In this case, using Rational Approximation Theory [1], we obtain that the continued fraction (34) converges to the Stieltjes function associated with the measure of orthogonality of the polynomial sequence {S n } n 0, ie, G 0 (ε) = F S (ε) Moreover, we can obtain the local density n 0 (ε) which coincides with the corresponding measure of orthogonality dσ S (ε) 31 The -periodic chain model We will suppose that the sequences of coupling constants {a n } and {b n } are periodic sequences with period, ie, {a n }={a,b,a,b,} and {b n }={c,d,c,d,} Then the matrix (3) becomes a c H = c b d d a c 0

13 530 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) The eigenvalues and eigenstates of the -periodic chain In order to obtain the eigenvalues and eigenstates of the -periodic chain we use (33) Then, (11) can be rewritten in the form a c C 1 C 1 c b d 0 0 C 0 d a c 0 C 3 = ε C C 3, (35) where C k are the coefficients in the linear combination (33), and ε is the corresponding eigenvalue of the matrix Hamiltonian H Here, it is important to remark that we need to consider finite or infinite chains The explicit solution of this eigenvalue problem for the finite chain with N = n + 1 states is given by Theorem 1 In this case, we have a 1 = a,a = b, b 1 = c 1 = c and b = c = d and then the eigenvalues of (35) are the following ε 0 = a, ε ±k = a + b (a b) ( ) kπ ± + c 4 + d + cd cos, (36) n + 1 where k = 1,,,n Moreover, the corresponding eigenvectors are S 0 (ε l ) S v l = 1 (ε l ), l= 0, ±1, ±,,±n, (37) where S n (ε l ) S k (ε) = (cd) k {P k ((ε a)(ε b)) + d P k 1 ((ε a)(ε b))}, S k+1 (ε) = c 1 (cd) k (x a)p k ((ε a)(ε b)), k = 0, 1, ( ) and P n (x) = (cd) n U x c d n cd For the particular case when a = b and c = d Eq (36) gives ε 0 = a, ε ±k = a ± c cos k = 1,,,n, which is in agreement with [11] 31 The Green function and the local density n 0 (ε) Using (7) the following expression for the Green function follows ( ) 1 G 0 (ε) = d + φ (a ε) (ε) φ (ε) (cd), ( ) kπ n+ for where φ (ε) = (ε a)(ε b) c d To obtain the local density n 0 (ε), we use the distribution function (8) ( n 0 (ε) = 1 min(c,d ) ) d δ(ε a) πd (cd) φ (ε), (38) ε a

14 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) Fig The eigenvalues of the -chain model for a =,b= 1,c= 4 and d = 3 that is located in the union of the following two intervals [ a + b τ +, a + b ] [ a + b τ + τ, a + b ] + τ + with a possible mass point at ε = a, where τ ± = 1 (b a) + 4(c ± d) 313 Some numerical experiments In this section, we will show some numerical results corresponding to the case of -Toeplitz matrix To check the validity of the analytic formulas we have computed numerically the eigenvalues of the matrix (35) using MATLAB and compare them with the analytic values given, for the case N = n + 1 by (36) The corresponding analytic expressions for the eigenvectors can be obtained from (37) For the case N = n we can use the bounds ε n+1,j < ε n,j < ε n+1,j+1,j = 1,,,n In Fig we show the eigenvalues of the -Toeplitz symmetric N N matrix, N = n + 1 for a =,b= 1,c= 4 and d = 3 We showthe numerical eigenvalues (stars) and the analytical ones (open circles) for n = 30 (left panel) and n = 500 (right panel) With this choice of parameters the density function n 0 of the initial state, represented in Fig 4 (left panel), has not any mass point at ε=1 (see (38)), ie, it is an absolute continuous function supported on two disjoint intervals In Fig 3 we show the eigenvalues of the -Toeplitz symmetric N N matrix, N = n + 1 for a =,b= 1,c= 3 and d = 4 We showthe numerical eigenvalues (stars) and the analytical ones (open circles) for n = 30 (left panel) and n = 500 (right panel) With this choice of the parameters the density function n 0 of the initial state has a mass point M = 7/16 at ε = 1, ie, it has an absolute continuous part supported on two disjoint intervals, represented in Fig 4 (right panel), plus a delta Dirac mass at x = 1 Here we have shown only the case of N =n+1 matrices for which always one has an isolate eigenvalue ε 1 = a For the case of matrices of order N = n we have not this isolated eigenvalue Also notice that the spectrum of H has two branches 3 The 3-periodic chain model Let nowsuppose that the sequences of coupling constants {a n } and {b n } are periodic sequences with period 3, ie, {a n }={a,b,c,a,b,c,} and {b n }={d,e,f,d,e,f,} Then the matrix (3) becomes

15 53 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) Fig 3 The eigenvalues of the -chain model for a = 1,b=,c= 3 and d = 4 Fig 4 The density function n 0 (ε) of the -chain model for a =,b= 1,c= 4 and d = 3 (left) and a = 1,b=,c= 4 and d = 3 (right) a d d b e e c f f a d 0 0 H = d b e e c f f a 31 The eigenvalues and eigenstates of the 3-periodic chain Again we will suppose that the solution u of the Schrödinger equation (11) can be written as (33), thus (11) takes the form a d C 1 C 1 d b e 0 0 C C 0 e c f 0 C f a d C 4 = ε C 3 C 4, (39)

16 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) where C k are the coefficients in the linear combination (33), and ε is the corresponding eigenvalue of the matrix Hamiltonian H In this case we use Theorem which gives an explicit expression for the eigenvalue problem in the case of N = 3n + In this case, we have a 1 = a,a = b, a 3 = c, b 1 = c 1 = d,b = c = e, and b 3 = c 3 = f and then the eigenvalues of (39) are the solutions ε i,k,i= 1,, 3, k = 1,,,n, of the polynomial equations and x 3 (a + b + c)x + (ab + ac + bc d e f )x ( ) + cd + ae + bf kπ abc + def cos = 0 (310) n + 1 ε 3n+1 = a + b (a b) + 4d, ε 3n+ = a + b + (a b) + 4d The corresponding eigenvectors are S 0 (ε l ) S v = 1 (ε l ), l {(i, k), 3n + 1, 3n + i = 1,, 3, k= 1,,n}, (311) S 3n+1 (ε l ) where S 3k (x) = (def ) k {P k (π 3 (x)) + f (x b)p k 1 (π 3 (x))}, S 3k+1 (x) = d 1 (def ) k {(x a)p k (π 3 (x)) + d f P k 1 (π 3 (x))}, S 3k+ (x) = (de) 1 (def ) k (x ξ 1 )(x ξ )P k (π 3 (x)), k = 0, 1,, being ( x d P n (x) = (def ) n e f ) U n def and π 3 (x) = d + e + f + (a c)e + (b c)f (d + e + f )(x c) + (x a)(x b)(x c) (31) ( ) The particular case a = b = c and d = e = f gives ε 1,k = a + d cos kπ 3n+3, ( ) ( ) (n + 1 k)π (n k)π ε,k = a d cos, ε 3,k = a + d cos, 3n + 3 3n + 3 for k = 1,,,n, ε 3n+1 = a d, ε 3n+ = a + d, that is in agreement with [10] 3 The Green function and the local density n 0 (ε) Using (13) the following expression for the Green function follows ( 1 G 0 (ε) = (ε a)(ε b) d b ε + 1 ( )) f φ 3 (ε) φ 3 (ε) (def),

17 534 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) where φ 3 (ε) = π 3 (ε) d e f To obtain the local density n 0 (ε), we use the distribution function (313) (def) φ 3 n 0 (ε) = M 1 δ(ε ξ 1 ) + M δ(ε ξ ) + (ε) f π (ε a)(ε b) d, (313) where ξ 1 = a + b + (a b) + 4d, ξ = a + b (a b) + 4d, (314) which support is contained in the union of the three intervals defined by π 1 3 ([d + e + f def,d + e + f + def]), where π 3 is the polynomial defined in (31), and two possible mass points M 1 and M (see (14)) M 1 = max{0,(f e ) (a b) + 4d + (a b)(f + e )} (a b) + 4d f, M = max{0,(f e ) (a b) + 4d (a b)(f + e )}, (a b) + 4d f located at ε=ξ 1 and ε=ξ, respectively Moreover, the following four situations are possible (see Section ): 1 f e and a b In this case M 1 = 0iffa = b and f = e, and M = 0ifff e + a b (f +e ) (a b) +4d f e and a b Then M 1 = 0ifff e + a b (f +e ) (a b) +4d, and M = 0iffa = b and f = e 3 f e and a b Then M 1 = 0ifff e a b (f +e ) (a b) +4d In this case always M = 0 4 f e and a b With this choice always M 1 = 0, and M = 0ifff e a b (f +e ) (a b) +4d 33 Some numerical experiments In this section, we will present some numerical experiments related to the three periodic chain model As in the previous case we represent with stars the values obtained by using the analytic expression (310) and with circles the values obtained numerically The eigenvectors can be easily obtained using (311) In Fig 5 we show the eigenvalues of the three 3-Toeplitz symmetric N N matrix, N = 3n + with a =,b= 1,c= 3,d= 4,e= and f = 3 for n = 0 (left panel) and for n = 300 (right panel) In Fig 7 (left panel) we represent the absolute continuous part of the density function n 0 of the initial state This case corresponds to the situation 1 discussed above for which we have two mass points M 1 = , M =

18 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) Fig 5 The 3-chain model with a =,b= 1,c= 3,d = 4,e= and f = 3 Fig 6 The 3-chain model for a = 3,b=,c= 1,d =,e= 3 and f = Fig 7 The function n 0 (ε) of the 3-periodic chain model with a =,b = 1,c = 3,d = 4,e = and f = 3 (left) and a =,b= 1,c= 3,d = 4,e= and f = 3 (right) at ξ 1 = and ξ = 3 65, respectively (see (314)) In this case n 0 is supported in three disjoint intervals plus two isolated points at ξ 1 and ξ In Fig 6 we show the eigenvalues of the 3-Toeplitz symmetric N N matrix, N = 3n + with a = 3,b=,c= 1,d=,e= 3 and f = for n = 0 (left panel) and for n = 300 (right panel) In Fig 7 (right panel) we represent the absolute continuous part of the density function n 0 of the initial state This

19 536 R Álvarez-Nodarse et al / Journal of Computational and Applied Mathematics 184 (005) case corresponds to the situation 3 discussed above for which M 1 = 0 and M = 0, ie, there is no mass points, so the support of n 0 are three disjoint intervals Programs: For the numerical simulations presented here we have used the commercial program MATLAB The used source code can be obtained by request via to niurka@euleruses or ran@uses Acknowledgements We thank Profs JS Dehesa and F Marcellán for stimulating discussions This research has been supported by the Programa de Acciones Integradas Hispano-Lusas HP & E-6/03 The authors were also partially supported by the DGES Grant BFM C03-01 and PAI Grant FQM-06 (RAN); the DGES Grant BFM C0 and PAI Grant FQM-007 (NRQ); and CMUC (JP) We are very grateful to the anonymous referee for pointing out the alternative pure matrix theoretic approach to the problem treated in the paper (see Section 3) References [1] R Álvarez-Nodarse, F Marcellán, J Petronilho, On some polynomial mappings for measures Applications, in: M Alfaro, et al (Eds), Proceedings of the International Workshop on Orthogonal Polynomials in Mathematical Physics, Servicio de Publicaciones de la Universidad Carlos III de Madrid, Leganés, 1997, pp 1 1 (free available at renato/iwop/proceedingshtml) [] SN Chandler-Wilde, MJC Gover, On the application of a generalization of Toeplitz matrices to the numerical solution of integral equations with weakly singular convolution kernels, IMA J Numer Anal 9 (1989) [3] TS Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, NewYork, 1978 [4] D Fasino, S Serra Capizzano, From Toeplitz matrix sequence to zero distribution of orthogonal polynomials, Contemp Math 33 (003) [5] CM Fonseca, J Petronilho, Explicit inverses of some tridiagonal matrices, Linear Algebra Appl 35 (001) 7 1 [6] CM Fonseca, J Petronilho, Explicit inverse of a tridiagonal k-toeplitz matrix, Numer Math, to appear [7] MJC Gover, The eigenproblem of a tridiagonal -Toeplitz matrix, Linear Algebra Appl (1994) [8] MJC Gover, S Barnett, Inversion of Toeplitz matrices which are not strongly singular, IMA J Numer Anal 5 (1985) [9] MJC Gover, S Barnett, DC Hothersall, Sound propagation over inhomogeneous boundaries, in: Internoise 86, Cambridge, MA, 1986, pp [10] R Haydock, Recursive solution of Schrödinger s equation, Solid State Phys 35 (1980) [11] R Haydock, The recursion method and the Schrödinger s equation, in: P Nevai (Ed), Orthogonal Polynomials: Theory and Practice, NATO ASI Series C, vol 94, Kluwer Acad Publ, Dordrecht, 1990, pp 17 8 [1] R Haydock, V Heine, MJ Kelly, Electronic structure based on the local atomic environment for tight-binding bands: II, J Phys C 8 (1975) [13] A Kuijlaars, S Serra Capizzano, Asymptotic zero distribution of orthogonal polynomials with discontinuously varying recurrence coefficients, J Approx Theory 113 (001) [14] F Marcellán, J Petronilho, Eigenproblems for tridiagonal -Toeplitz matrices and quadratic polynomial mappings, Linear Algebra Appl 60 (1997) [15] F Marcellán, J Petronilho, Orthogonal polynomials and quadratic transformations, Portugal Math 56 (1) (1999) [16] F Marcellán, J Petronilho, Orthogonal polynomials and cubic polynomial mappings I, Comm Anal Theory Contin Fractions 8 (000) [17] F Marcellán, J Petronilho, Orthogonal polynomials and cubic polynomial mappings II, Comm Anal Theory Contin Fractions 9 (001) 11 0 [18] S Serra Capizzano, Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations, Linear Algebra Appl 366 (003)

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