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Our main result shows that these equations have a natural symplectic structure, that is, every symmetric three-term recurrence equation is a special discrete symplectic system.. In addit

Volume 2010, Article ID 626942, 17 pages

doi:10.1155/2010/626942

Research Article

Symmetric Three-Term Recurrence Equations and Their Symplectic Structure

Roman ˇSimon Hilscher1 and Vera Zeidan2

1 Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotl´aˇrsk´a 2,

61137 Brno, Czech Republic

2 Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA

Correspondence should be addressed to Roman ˇSimon Hilscher,hilscher@math.muni.cz

Received 11 March 2010; Accepted 1 May 2010

Academic Editor: Martin Bohner

Copyrightq 2010 R ˇSimon Hilscher and V Zeidan This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We revive the study of the symmetric three-term recurrence equations Our main result shows that these equations have a natural symplectic structure, that is, every symmetric three-term recurrence equation is a special discrete symplectic system The assumptions on the coefficients in this paper are weaker and more natural than those in the current literature In addition, our result implies that symmetric three-term recurrence equations are completely equivalent with Jacobi difference equations arising in the discrete calculus of variations Presented applications of this study include the Riccati equation and inequality, detailed Sturmian separation and comparison theorems, and the eigenvalue theory for these three-term recurrence and Jacobi equations

1 Introduction

In this paper, we consider the symmetric three-term recurrence equation

S k1 x k2 − T k1 x k1  S T

where x k ∈ Rn for k ∈ 0, N  1Z, the real n × n matrices S k and T k are defined on

0, NZ with T k being symmetric and S k being invertible The discrete intervals are defined

bya, bZ : a, b ∩ Z Traditionally, the recurrence equation T is studied in the literature; see, for example,1, Chapter 5 or 2 4, as a generalization of the Jacobi difference equation

ΔR k Δx k  Q T

k x k1



 P k x k1  Q k Δx k , k ∈ 0, N − 1Z, J

whereΔx k: xk1 − x kis the forward difference and where the matrices Pk , Q k , R k∈ Rn×nfor

k ∈ 0, NZwith P k and R kbeing symmetric Jacobi equationJ arises in the discrete calculus

of variations as the Euler equation for the second variation; see, for example,1, Section 4.2

or5 When the forward differences in J are expanded, then J becomes the three-term recurrence equationT in which the matrices Sk: Rk  Q T

k are invertible for all k ∈ 0, NZ and T k: Rk R k−1 Q k−1 Q T

k−1 P k−1are symmetric; see1, Section 3.6 orProposition 2.4 In the same reference, it is shown that Jacobi difference equations J and recurrence equations

T can be embedded into discrete symplectic systems seeSection 2for the details

x k1 Ak x k Bk u k , u k1 Ck x k Dk u k , k ∈ 0, NZ, S

where for k ∈ 0, NZ

ST

kJ Sk  J, Sk:



Ak Bk

Ck Dk



, J :



0 I

−I 0



that is, the 2n × 2n matrices S k are symplectic However, the transition fromJ and T into

S in reference 1 requires that both Sk and R k be invertible This invertibility assumption

essentially means that these equations are first transformed into a linear Hamiltonian system

Δx k  A k x k1  B k u k , Δu k  C k x k1 − A T

for which it is required that I − A k  R−1

k S kbe invertible so that the solutions ofH exist in the backward time And then the linear Hamiltonian systemH is written as the symplectic systemS

Recently in 6, the authors proposed to study the Jacobi equations J as discrete symplectic systemsS in a direct way which bypasses the Hamiltonian system H This new

approach requires that only the matrices S k be invertible while the matrices R kare allowed

to be singular, which yields more general results for J obtained, for example, through the theory of symplectic systems S In the present paper, we continue in this direction and we show that the three-term recurrence equations T naturally possess a symplectic structure Theorem 3.1and Corollary 3.2 More precisely, we show that symmetric three-term recurrencesT and Jacobi equations J and symplectic systems S with Bkinvertible are completely equivalent Therefore, the general theory of discrete symplectic systems recently developed, for example, in 7 18 can be applied to obtain, in particular, the Riccati equations and inequalities, and the oscillation and Sturmian theorems including multiplicities of focal points for the symmetric three-term recurrence equations T

The paper is divided as follows In the next section, we present an overview of the known transformations between the equationsT , J , and system S InSection 3we prove the main results about the symplectic structure of the recurrence equationT In Sections

4 6, we present recent results from the theory of discrete symplectic systems S adopted for the setting of recurrence equationsT These results include the Riccati equations and inequalities as being a part of the Reid roundabout theorems in Section 4, the Sturmian separation and comparison theorems inSection 5, and the oscillation theorems and Rayleigh principle inSection 6 In Section 7, we make some final comments about the results of this paper

2 Known Results

We first present known transformations between the three-term recurrence equation T , Jacobi equationJ , and the symplectic system S We adopt the following notation

Notation 2.1Three-term recurrence T The matrices Sk , T k , and the vectors x kinT have

the following properties: S k , T k ∈ Rn×nare defined on0, N  1Z with S k invertible and T k symmetric; x k∈ Rnare defined on0, N  1Z.

Notation 2.2Jacobi equation J The matrices Pk , Q k , R k , and the vectors x kinJ have the

following properties: P k , Q k , R k∈ Rn×nare defined on0, NZ, P k and R kare symmetric, and

the matrix S k: Rk  Q T

k is invertible; x k∈ Rn for k in 0, N  1Z.

Note that the coefficients T0, S N1 , and T N1 are not explicitly needed in T and the coefficient PN is not needed in J However, it will be convenient to use them when we transform T into J or system S and vice versa For example, we can now

define x N2 : S−1

N1 T N1 x N1 − S T

N x N , so that the recurrence in T is satisfied also at

t  N.

Notation 2.3Symplectic system S The matrices Ak ,Bk ,Ck ,Dk , and the vectors x k , u k in

S have the following properties: Ak , B k , C k , D k ∈ Rn×n are defined on0, NZ and satisfy

1.1 ; xk , u k∈ Rn are defined for k in 0, N  1Z.

The following two known results are verified by straightforward calculations They can be found in1, Section 3.6

Proposition 2.4 Jacobi J to three-term recurrence T Assume that Pk , Q k , R k , S k: Rk Q T

k

satisfy the conditions in Notation 2.2 and set R N1: I Then the Jacobi equation J is the symmetric

three-term recurrence equationT , whose coefficients

S k: Rk  Q T

k , T k1: Rk1  R k  Q k  Q T

k  P k , k ∈ 0, NZ 2.1

satisfy the conditions in Notation 2.1

Note that the choice of R N1 : I inProposition 2.4 is arbitrary, that is, any matrix

R N1∈ Rn×nwill do the job

Proposition 2.5 Three-term recurrence T to Jacobi J , Rk invertible Assume that Sk ,

T k satisfy the conditions in Notation 2.1 Let R k be any symmetric and invertible matrices for

k ∈ 0, N  1Z Then the symmetric three-term recurrence equationT is the Jacobi equation J ,

whose coefficients

R k , Q k: ST

k − R k , P k: Tk1 − S T

k − S k  R k − R k1 , k ∈ 0, NZ 2.2

satisfy the conditions in Notation 2.2

Remark 2.6 When the matrix S k is also symmetric, we may take R k : SkinProposition 2.5.

It follows that the recurrence equationT is then transformed into the Jacobi equation J , whose coefficients

R k: Sk , Q k: 0, Pk: Tk1 − S k1 − S k , k ∈ 0, NZ 2.3 satisfy the conditions inNotation 2.2 Note that in this case the matrix Rk  S kis invertible

The invertibility condition on R k in Proposition 2.5 or Remark 2.6 means that the resulting Jacobi equation can be written as a linear Hamiltonian systemH , which in turn can be written as a symplectic systemS This is shown in 1, Example 3.17

Proposition 2.7 Jacobi J to symplectic S , Rk invertible Assume that Pk , Q k , R k , S k :

R k  Q T

k satisfy the conditions in Notation 2.2 with R k being invertible Then the Jacobi equationJ

is the symplectic systemS , whose coefficients

Ak: S−1

k R k , Ck:P k − Q k R−1k Q T k

S−1k R k , k ∈ 0, NZ,

Bk: S−1

k , Dk:P k − Q k R−1k Q T k

S−1k  I  Q k R−1k , k ∈ 0, NZ,

2.4

with u k: Sk x k1 −R k x k on 0, NZand u N1: PN Q N S N x N1 −S T

N x N satisfy the conditions

in Notation 2.3

It is interesting to observe that by using the identity Q T

k  S k −R kone can eliminate the

inverse of R kin the coefficients 2.4 to obtain the coefficients inProposition 2.8below This was actually the motivation for the investigation of Jacobi systems as discrete symplectic systems in6 In this latter reference, the authors showed that it is possible to treat Jacobi equationJ directly as a symplectic system S by bypassing the Hamiltonian system H ; see6, Corollary 5.2

Proposition 2.8 Jacobi J to symplectic S Assume that Pk , Q k , R k , S k: Rk  Q T

k satisfy the conditions in Notation 2.2 Then the Jacobi equationJ is the symplectic system S , whose coefficients

Ak: S−1

k R k , Ck: Pk S−1k R k − Q k S−1k Q T k , k ∈ 0, NZ,

Bk: S−1

k , Dk:P k  Q k  Q T

k  R k



S−1k , k ∈ 0, NZ,

2.5

with u k : Sk x k1 − R k x k on 0, NZ and u N1 : PN  Q N  S N x N1 − S T

N x N satisfy the conditions in Notation 2.3 Moreover, the resulting symplectic systemS is Hamiltonian if and only

if the matrix R k is invertible.

The resulting symplectic system in Proposition 2.8 has Bk  S−1

k invertible This turns out to be a characterizing property of symplectic systemsS corresponding to Jacobi equationsJ ; see 6, Corollary 5.3

Proposition 2.9 Symplectic S to Jacobi J Assume that Ak ,Bk ,Ck ,Dk satisfy the conditions

in Notation 2.3 with Bk being invertible Then the symplectic systemS is the Jacobi equation J ,

whose coefficients

R k: B−1

k Ak , Q k:I −A T k

BT−1

k , P k: Dk −I B−1

k AT

k −IBT−1

k , k ∈ 0, NZ, 2.6

with S k  R k  Q T

k  B−1

k satisfy the conditions in Notation 2.2

Next we turn our attention back to the symmetric three-term recurrence equationsT

By combining the transformations in Propositions2.5and2.8we get the following

Proposition 2.10 Three-term recurrence T to symplectic S , Rk invertible Assume that

S k , T k satisfy the conditions in Notation 2.1 Let R k be any symmetric and invertible matrices for

k ∈ 0, N  1Z Then the symmetric three-term recurrence equationT is the symplectic system S ,

whose coefficients

Ak: S−1

k R k , Ck: T k1 − R k1 S−1

k R k − S T

k , k ∈ 0, NZ,

Bk: S−1

k , Dk: T k1 − R k1 S−1

k , k ∈ 0, NZ,

2.7

with u k: Sk x k1 −R k x k on 0, NZand u N1: TN1 −R N1 x N1 −S T

N x N satisfy the conditions

in Notation 2.3

3 Main Results

The need to have R kinvertible inProposition 2.10is artificial, because R kis not furnished by the three-term recurrence equationT and furthermore, R−1

k is not even present in equations

2.7 which define the coefficients of the corresponding symplectic system S However, the

invertibility of R k is a requirement inherited from Proposition 2.5 that was derived in 1,

Section 3.6 Therefore, an important question naturally surfaces: is it possible to obtain the

result of Propositions2.5and2.10without any assumption on R k?

The following new result provides an answer to the above question, that is, it shows that the recurrence equationsT are naturally special cases of symplectic systems S for any

choice of matrices R k and without any assumption on the invertibility of T k

Theorem 3.1 Three-term recurrence T to symplectic S , Rkarbitrary Assume that Sk , T k

satisfy the conditions in Notation 2.1 Let R k be any symmetric matrices for k ∈ 0, N  1Z Then the symmetric three-term recurrence equationT is the symplectic system S , whose coefficients are

given by2.7 with uk: Sk x k1 − R k x k on 0, NZand u N1: TN1 − R N1 x N1 − S T

N x N and they satisfy the conditions in Notation 2.3

Proof GivenT with the data as inNotation 2.1, we set uk : Sk x k1 − R k x k for k ∈ 0, NZ and u N1: TN1 − R N1 x N1 − S T

N x N Then

x k1  S−1

k R k x k  S−1

k u k Ak x k Bk u k , k ∈ 0, NZ,

u k1  S k1 x k2 − R k1 x k1 T

 T k1 − R k1 x k1 − S T

k x k

 Ck x k Dk u k , k ∈ 0, N − 1Z.

3.1

Therefore, with the coefficients Ak,Bk,Ck, andDk defined by 2.7 , we have that the pair

x, u satisfies the first equation in system S for all k ∈ 0, NZand the second equation in

S for all k ∈ 0, N − 1Z However, the definition of u N1yields that the second equation in

S holds also at k  N It remains to show that the matrix Sk, defined in1.1 through Ak,Bk,

Ck, andDkin2.7 , is symplectic We have after easy calculations that for every k ∈ 0, NZ,

ST

kJ Sk



Ak Bk

Ck Dk

T

0 I

−I 0



Ak Bk

Ck Dk



where we used the symmetry of R k The proof is complete

Note that the matrices R kin the above theorem are arbitrary, and hence, one can choose them to simplify the formulas of the coefficients in 2.7 One choice standing out is when

R k  T k In this case, the result ofTheorem 3.1reduces to the following

Corollary 3.2 Three-term recurrence T to symplectic S Assume that Sk , T k satisfy the conditions in Notation 2.1 Then the symmetric three-term recurrence equationT is the symplectic

systemS , whose coefficients

Ak: S−1

k T k , Bk: S−1

k , Ck: −ST

k , Dk: 0, k ∈ 0, NZ, 3.3

with u k: Sk x k1 − T k x k on 0, NZand u N1: −ST

N x N satisfy the conditions in Notation 2.3

The above result has an important consequence By usingProposition 2.9, we can now

transform any symmetric three-term recurrence equationT into a Jacobi equation J by the procedure described inCorollary 3.2andProposition 2.9 However, compared with the result

inProposition 2.5, we do not need the matrices Rkto be invertible, but they can be arbitrary

Corollary 3.3 Three-term recurrence T to Jacobi J Assume that Sk , T k satisfy the conditions

in Notation 2.1 Let R k be any symmetric matrices for k ∈ 0, N 1Z Then the symmetric three-term recurrence equationT is the Jacobi equation J , whose coefficients are given by equations 2.2 and

they satisfy the conditions in Notation 2.2 When R k: Tk , the coefficients of J reduce to

R k: Tk , Q k: ST

k − T k , P k: Tk − S k − S T

Note that if the matrices R k are invertible, then Corollary 3.3 is a consequence of Proposition 2.5 This is also the case for the second part of theCorollary 3.3if T kare invertible When combining the transformations in Propositions2.9and2.4, we get the following result

Corollary 3.4 Symplectic S to three-term recurrence T Assume that Ak ,Bk ,Ck ,Dk satisfy the conditions in Notation 2.3 withBk being invertible and setAN1 BN1: I Then the symplectic

systemS is the the symmetric three-term recurrence equation T , whose coefficients

S k: B−1

k , T k1: B−1

k1A−1

k1 DkB−1

with T0 B−1

0 A0satisfy the conditions in Notation 2.1

In the last part of this section, we pay attention to the quadratic functionals associated with the equations T , J , and system S In particular, we show that these functionals transform exactly in the same way as their corresponding systems Consider the quadratic functionals

FT x :N

k0

x T k1 T k1 x k1 − x T

k S k x k1 − x T

FJ x :N

k0

x T k1 P k x k1  2x T

k1 Q k x k  Δx T

Fx, u :N

k0

x T kCT

kAk x k  2x T

kCT

kBk u k  u T

kDT

where x  {x k}N1

k0 satisfies x0  0  x N1 In addition, in functionalF the pair x, u solves

the first equation in systemS , that is, xk1  Ak x k Bk u k for k ∈ 0, NZ The following result is from6, Proposition 3.7

Proposition 3.5 Quadratic functionals for J and S Assume that

i either P k , Q k , and R k satisfy the conditions in Notation 2.2 withAk ,Bk ,Ck ,Dk , and u k

being given by2.5 ofProposition 2.8

ii or A k ,Bk ,Ck , andDk satisfy the conditions in Notation 2.3 withBk being invertible and

P k , Q k , R k are given by2.6

ThenFJ x  Fx, u for every x  {x k}N1

k0 with x0  0  x N1

As a consequence of the results in Theorem 3.1 and Corollary 3.4, and in Proposition 2.4andCorollary 3.3, we get the transformations between the functionalsFTand

F, and FTandFJ

Proposition 3.6 Quadratic functionals for T and S Assume that

i either S k , T k satisfy the conditions in Notation 2.1 withAk ,Bk ,Ck ,Dk , and u k being given

by2.7 ofTheorem 3.1

ii or A k ,Bk ,Ck , andDk satisfy the conditions in Notation 2.3 withBk being invertible and

T k , S k are given by3.5

ThenFT x  Fx, u for every x  {x k}N1

k0 with x0  0  x N1

Proposition 3.7 Quadratic functionals for J and T Assume that

i either P k , Q k , and R k satisfy the conditions in Notation 2.2 with T k , S k being given by

2.1

ii or T k , S k satisfy the conditions in Notation 2.1 with S k being invertible and P k , Q k , R k are given by3.4

ThenFT x  F J x for every x  {x k}N1

k0 with x0  0  x N1

4 Applications in Reid Roundabout Theorems

In the previous section, we proved that symmetric three-term recurrence equationsT and Jacobi difference equations J are completely equivalent, that is, any result for one equation

T or J can be translated via the transformations inProposition 2.4andCorollary 3.3to a result for the other equation This equivalence is carried over via discrete symplectic systems

S with Bk being invertible, utilizing the transformations in Propositions 2.8 and2.9 for the passage between equationJ and system S and the transformations inTheorem 3.1 andCorollary 3.4for the passage between equation T and system S In this section, we give some applications of this equivalence For example, we derive the Riccati equation and Riccati inequality which are naturally associated with the symmetric three-term recurrence equationsT —the results which have not been known in the literature for T

Consider the quadratic functional FT x defined in 3.6 subject to sequences x  {x k}N1

k0 satisfying x0  0  x N1 Note that due to x N1  0, the functional F T does not

depend on the matrix T N1, as we mentioned at the beginning ofSection 2 We say that the functionalFT is positive definite ifFT x > 0 for every x  {x k}N1

k0 with x0  0  x N1and

x / 0 We say that FTis nonnegative ifFT x ≥ 0 for every x  {x k}N1

k0 with x0 0  x N1 The positivity of the functionalFT was first characterized in2, Theorem 4; see also

1, Theorem 5.13 and 4, Corollary 2, in terms of the properties of the so-called conjoined bases ofT These are the n × n matrix solutions X  {Xk}N1

k0 ofT such that XT

k S k X k1is symmetric and rankXT

k X T k1  n for some and hence for any index k ∈ 0, NZ A special conjoined basis X of T , determined by the initial conditions X0  0 and X1  S−1

0 , is called the principal solution ofT

Proposition 4.1 Reid roundabout theorem—positivity Assume that S k , T k satisfy the conditions in Notation 2.1 Then the following statements are equivalent.

i The functional F T is positive definite.

ii The principal solution X of T has X k invertible for all k ∈ 1, N  1Z and satisfies

X T

k S k X k1 > 0 for all k ∈ 1, NZ.

iii There exists a conjoined basis X of T such that X k is invertible for all k ∈ 0, N  1Z and satisfying X T

k S k X k1 > 0 for all k ∈ 0, NZ Proof See1, Theorem 5.13 or 4, Corollary 2

A similar result holds for the nonnegativity ofFT

Proposition 4.2 Reid roundabout theorem—nonnegativity Assume that S k , T k satisfy the conditions in Notation 2.1 Then the following statements are equivalent.

i The functional F T is nonnegative.

ii The principal solution X of T has X k invertible for all k ∈ 1, NZ and satisfies

X T

k S k X k1 > 0 for all k ∈ 1, N − 1Zand X T

N S N X N1 ≥ 0.

Proof See4, Theorem 2

In the following result, we add three more equivalent conditions toProposition 4.1and one more equivalent condition toProposition 4.2in terms of solutions of the discrete Riccati matrix equation and inequality corresponding toT , thus completing the above results to

their full standard forms For symmetric matrices W  {W k}N1

k0 , define the Riccati operator

R W k: Wk1 S−1k T k  W k  S T

This Riccati operator is obtained from the Riccati operatorRW k: Wk1AkBk W k −Ck

Dk W k for symplectic system S ; see, for example, 7,14–16,19 That is, if the coefficients

ofS are given by formulas 3.3 , then RWk  RW k The equation RW k  0 for k ∈

0, NZis called the discrete Riccati matrix equation If T k  W kis invertible, then we may solve

the equation RW k  0 for W k1and obtain thesymmetric Riccati equation corresponding

to the recurrence equationT , that is,

W k1  S T

The Riccati equation RE has been studied in the literature by many authors; see, for example, the references discussed in 20, page 12 However, its natural connection to the recurrence equationT is established for the first time in this paper In addition, the discrete Riccati inequalityRW kAkBk W k −1≤ 0 derived in 14, Theorem 1 for symplectic systems

S yields throughCorollary 3.2a new Riccati inequality

W k1  S T

for the recurrence equationT Equivalently, the Riccati equation RE and inequality RI can be obtained from the Riccati equation and inequality for the Jacobi equation

ΔW k − P k  W k − Q k R k  W k −1

W k − Q T k



 0, ≤ 0 k ∈ 0, NZ, 4.2

in which the coefficients are given by the formulas in 3.4 Note that discrete Riccati equations obtained from symplectic system S corresponding to three-term recurrence equationsT with Rkbeing invertibleas in Propositions2.5and2.10 are considered in 20,

1, Section 6.1, and 16, Section 4 InTheorem 4.3below, we do not require any condition

on R k The following result is a complement ofProposition 4.1

Theorem 4.3 Reid roundabout theorem—positivity continued Assume that S k , T k satisfy the conditions in Notation 2.1 Then each of the conditions (i)–(iii) of Proposition 4.1 is equivalent to any of the following statements.

iv There exists a symmetric solution W k on 0, N  1Z of the Riccati equationRE for

k ∈ 1, NZsuch that W0  0 and T k  W k > 0 for all k ∈ 1, NZ.

v There exists a symmetric solution W k on 0, N  1Zof the Riccati equationRE such that

T k  W k > 0 for all k ∈ 0, NT.

vi There exists a symmetric solution W k on 0, N1Zof the Riccati inequalityRI satisfying

T k  W k > 0 for all k ∈ 0, NT.

Proof With the coefficients of system S given by 3.3 , the equivalence of conditions i and

v is established in 16, Theorem 7 and the equivalence of i and vi in 14, Theorem 1 Conditionii ofProposition 4.1implies conditioniv by setting W0 : 0, Wk: Sk X k1 X−1

k −

T k for k ∈ 1, NZ, and W N1: −ST

N X N X−1

N1 i.e., we set W k: U k X−1

k on1, N  1Z, where

U k: Sk X k1 −T k X k for k ∈ 0, NZand U N1: −ST

N X N accordingly to the definition of u kin Theorem 3.1 Then Tk  W k X k T−1 X T

k S k X k1 X−1

k > 0 for all k ∈ 1, NZ Hence, condition

iv is satisfied Finally, if we assume that condition iv holds, then the proof of the positivity

ofFTis similar to the proof of4, Corollary 3

Note that conditionvi ofTheorem 4.3yields that−W k1 ≥ S T

k T k W k −1S k > 0, which

implies that the matrices W kinTheorem 4.3vi are negative definite for k ∈ 1, N  1Z The second result of this section is concerned with the nonnegativity of the functional

FT

Theorem 4.4 Reid roundabout theorem—nonnegativity continued Assume that S k , T k

satisfy the conditions in Notation 2.1 Then each of the conditions (i)–(ii) of Proposition 4.2 is equivalent to the following statement.

iii There exists a symmetric solution W k on 0, N  1Z of the Riccati equationRE for

k ∈ 1, N − 1Zsuch that W0  0, T k  W k > 0 for all k ∈ 1, N − 1Z, and T N  W N ≥ 0.

Proof The proof is similar to implicationsii ⇒ iv ⇒ i from the proof ofTheorem 4.3 The details are here therefore omitted Alternatively, see the proof of4, Theorem 3

By comparing the Riccati equation conditions for the positivity and nonnegativity

ofFT inTheorem 4.3iv andTheorem 4.4iii , we can see that the Riccati equation for the positivity of FT is satisfied on the closed interval including k  N , while the Riccati

equation for the nonnegativity ofFT is satisfied on the open interval excluding k  N This

phenomenon resembles the situation in the continuous time setting in21 or 22, Section

6.2, that is, for Jacobi differential equations or Hamiltonian systems and their corresponding

Riccati differential equations

5 Applications in Sturmian Theory

In 1, Sections 5.3 and 5.6, several Sturmian comparison and separation theorems are presented for the symmetric three-term recurrence equationsT However, these results do not involve the multiplicities of focal points for conjoined bases ofT Therefore, our next aim is to extend the Sturmian separation and comparison theorems for symmetric three-term recurrence equationsT in this direction

Following18, we say that a conjoined basis X of T has a focal point in the point k1

if X k1 is singular and then def X k1 : dim Ker Xk1is its multiplicity, while the conjoined

basis X has a focal point in the interval k, k  1 Zif the matrix X k T S k X k1is not nonnegative

definite and then ind X T

k S k X k1 is its multiplicity Here ind A is defined as the number of

negative eigenvalues of thesymmetric matrix A The number of focal points in the interval

k, k  1Zincluding multiplicities is then m k : def Xk1  ind X T

k S k X k1 We will always count the focal points of conjoined bases ofT including their multiplicities This definition

of multiplicities is motivated by the appearance of the symmetric matrix X T

k S k X k1in the Reid roundabout theoremPropositions4.1and4.2 Therefore, the conditions ii in Propositions 4.1and4.2can be reformulated as follows