Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 11, 1-29; http://www.math.u-szeged.hu/ejqtde/

q-DOMINANT AND q-RECESSIVE MATRIX SOLUTIONS FOR LINEAR QUANTUM SYSTEMS

DOUGLAS R. ANDERSON AND LISA M. MOATS

Abstract. In this study, linear second-order matrix q-difference equations are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary conditions. A generalized Wronskian is introduced and a Lagrange identity and Abel’s formula are established. Two reduction-of-order theorems are given. The analysis and characterization of qdominant and q-recessive solutions at infinity are presented, emphasizing the case when the quantum system is disconjugate.

1. Introduction

Quantum calculus has been utilized since at least the time of Pierre de Fermat [10, Chapter B.5] to augment mathematical understanding gained from the more traditional continuous calculus and other branches of the discipline [3]. In this study we will analyze a second-order linear self-adjoint matrix q-difference system, especially in the case that admits q-dominant and q-recessive solutions at infinity. Historically, dominant and recessive solutions of linear matrix differential systems of the form

(P X′)′(t) + Q(t)X(t) = 0

were introduced and extensively studied in a series of classic works by W. T. Reid [5, 6, 7, 8, 9], and in matrix difference systems of the form

Δ (P (t)ΔX(t − 1)) + Q(t)X(t) = 0

by Ahlbrandt [1], Ahlbrandt and Peterson [2], and recently by Ma [4]; there the forward difference operator ΔX(t) := X(t + 1) − X(t) was used. We introduce here an analysis of the quantum (q-difference) system

(1.1)  D^{q} (P D_{q}X) (t) + Q(t)X(t) = 0,

EJQTDE, 2007 No. 11, p. 1 where the real scalar q > 1 and the q-derivatives are given, respectively, by the difference quotients

2000 Mathematics Subject Classification. 34A30, 39A10, 34C10.

Key words and phrases. quantum calculus, self-adjoint matrices, linear systems, q-difference equations.

(D_{q}y)(t) = \frac{y(qt) − y(t)}{(q − 1)t} and (D^{q}y)(t) = \frac{y(t) − y(t/q)}{(1 − 1/q)t} = (D_{q}y)(t/q).

We will be particularly interested in the case where invertible solutions of (1.1) exist, and their characterization as q-dominant and/or q-recessive solutions at infinity.

The analysis of (1.1) and its solutions will unfold as follows. In Section 2 we explore (1.1), show how it is formally a self-adjoint equation, introduce a generalized Wronskian and establish a Lagrange identity and Abel’s formula. Section 3 contains two reduction of order theorems, followed in Section 4 by the notion of a prepared basis. In the main section, Section 5, we give definitions of q-dominant and q-recessive solutions, a connection to disconjugacy, and the construction of q-recessive solutions. Finally, future directions are touched on in Section 6, where a Pólya factorization of (1.1) leads to a variation of parameters result.

2. Self-Adjoint Matrix Equations

Let q > 1 be a real scalar, and let P and Q be Hermitian n × n-matrix-valued functions such that P (t) > 0 (positive definite) for all

t ∈ (0, ∞)_{q} := {…, q^{−2}, q^{−1}, 1, q, q^{2}, …}.

(A matrix M is Hermitian iff M^{∗} = M, where ^{∗} indicates conjugate transpose.) In this section we are concerned with the second-order matrix q-difference equation

(2.1) LX = 0, where LX(t) := D^{q} (P D_{q}X) (t) + Q(t)X(t) = 0, t ∈ (0, ∞)_{q},

which will be shown to be (formally) self-adjoint.

Theorem 2.1. Let a ∈ (0, ∞)_{q} be fixed and X_{a}, X′_{a} be given constant n×n matrices. Then the initial value problem

LX(t) = D^{q} (P D_{q}X) (t) + Q(t)X(t) = 0, X(a) = X_{a}, D_{q}X(a) = X′_{a}

has a unique solution.

Proof. For a ∈ (0, ∞)_{q} fixed, expanding out (2.1) we obtain LX(a) = 0 in the form

LX(a) = \frac{q}{(q − 1)^{2}a^{2}} [P (a) (X(qa) − X(a)) − qP (a/q) (X(a) − X(a/q))] + Q(a)X(a);

since P is invertible and

X(a) = X_{a}, X(qa) = X_{a} + a(q − 1)X′_{a},

EJQTDE, 2007 No. 11, p. 2 the term X(a/q) can be solved for uniquely and the unique solution X can be constructed to the left of a ∈ (0, ∞)_{q}. In the same way, LX(qa) = 0 given by

LX(qa) = \frac{1}{q(q − 1)^{2}a^{2}} [P (qa) (X(q^{2}a)−X(qa))−qP (a) (X(qa)−X(a))]+Q(qa)X(qa),

and again the term X(q^{2}a) can be solved for uniquely and the unique solution X can be constructed to the right of a ∈ (0, ∞)_{q}. □

In view of the theorem just proven, the following definition is now possible. Definition 2.2. The unique solution of the initial value problem

LX = 0, X(a) = 0, D_{q}X(a) = P^{−1}(a)

is called the principal solution of (2.1) (at a), while the unique solution of the initial value problem

LX = 0, X(a) = −I, D_{q}X(a) = 0

is called the associated (coprincipal) solution of (2.1) (at a). Definition 2.3. For matrix functions X and Y, the function W (X, Y) given by

W (X, Y)(t) = X^{∗}(t)P (t)D_{q}Y (t) − [P (t)D_{q}X(t)]^{∗}Y (t), t ∈ (0, ∞)_{q} is the (generalized) Wronskian matrix of X and Y. Lemma 2.4. The product rule for D^{q} is given by

D^{q}(XY)(t) = X(t/q)D^{q}Y (t) + (D^{q}X)(t)Y (t) = X(t)D^{q}Y (t) + (D^{q}X)(t)Y (t/q), and for D_{q} is given by

D_{q}(XY)(t) = X(qt)D_{q}Y (t) + (D_{q}X)(t)Y (t) = X(t)D_{q}Y (t) + (D_{q}X)(t)Y (qt), for matrix functions X and Y defined on (0, ∞)_{q}. Proof. The proof is straightforward using the definitions of D^{q} and D_{q} and is omitted.

□

Theorem 2.5 (Lagrange Identity). The Wronskian matrix W (X, Y) satisfies

D^{q}W (X, Y)(t) = X^{∗}(t)(LY)(t) − (LX)^{∗}(t)Y (t), t ∈ (0, ∞)_{q}

for matrix functions X and Y defined on (0, ∞)_{q}.

EJQTDE, 2007 No. 11, p. 3 Proof. For matrix functions X and Y, using the product rule for D^{q} derivatives we have

D^{q}W (X, Y)(t) = D^{q} [X^{∗}P D_{q}Y − (P D_{q}X)^{∗}Y] (t)
= X^{∗}(t)D^{q}(P D_{q}Y)(t) + (D^{q}X^{∗})(t)P (t/q)(D^{q}Y)(t)
−(P D_{q}X)^{∗}(t/q)(D^{q}Y)(t) − D^{q}(P D_{q}X)^{∗}(t)Y (t)
= X^{∗}(t)(LY − QY)(t) + (D^{q}X^{∗})(t)P (t/q)D^{q}Y (t)
−(D^{q}X^{∗})(t)P (t/q)D^{q}Y (t) − (LX − QX)^{∗}(t)Y (t)
= X^{∗}(t)(LY)(t) − (LX)^{∗}(t)Y (t)

on (0, ∞)_{q}. □

Definition 2.6. Let a, b ∈ (0, ∞)_{q} with a < b. We define the q-inner product of n×n matrix functions M and N on [a, b]_{q} to be

(2.2)  ⟨M, N⟩ = (1 − \frac{1}{q}) ∑_{t∈(a,b]_{q}} tM^{∗}(t)N(t), a, b ∈ (0, ∞)_{q}.

Since a = q^{α}, b = q^{β}, and t = q^{τ} for integers α ≤ τ ≤ β, the q-inner product is given by the expression

⟨M, N⟩ = (q − 1) ∑_{τ=α+1}^{β} q^{τ−1}M^{∗}(q^{τ})N(q^{τ}).

Corollary 2.7 (Self-Adjoint Operator). The operator L in (2.1) is formally selfadjoint with respect to the q-inner product (2.2); that is, the identity

⟨LX, Y ⟩ = ⟨X, LY ⟩

holds provided X, Y satisfy W (X, Y)(t)|_{a}^{b} = 0, called the self-adjoint boundary conditions.

Proof. Let the matrix functions X and Y satisfy W (X, Y)(t)|_{a}^{b} = 0. From Definition 2.3 and Theorem 2.5 we see that Green’s formula holds, namely

(1 − \frac{1}{q}) ∑_{t∈(a,b]_{q}} t (D^{q}W (X, Y)) (t) = W (X, Y)(t)|_{a}^{b} = ⟨X, LY ⟩ − ⟨LX, Y ⟩,

and the proof is complete. □

Another immediate corollary of the Lagrange identity is Abel’s matrix formula. EJQTDE, 2007 No. 11, p. 4 Corollary 2.8 (Abel’s Formula). If X, Y are solutions of (2.1) on (0, ∞)_{q}, then

W (X, Y)(t) ≡ C, t ∈ (0, ∞)_{q},

where C is a constant matrix.

Corollary 2.9. If X, Y are solutions of (2.1) on (0, ∞)_{q}, then either W (X, Y)(t) = 0 for all t ∈ (0, ∞)_{q}, or W (X, Y)(t) ≠ 0 for all t ∈ (0, ∞)_{q}.

From Abel’s formula we get that if X is a solution of (2.1) on (0, ∞)_{q}, then

W (X, X)(t) ≡ C, t ∈ (0, ∞)_{q},

where C is a constant matrix. With this in mind we make the following definition.

Definition 2.10. Let X and Y be matrix functions and W (X, Y) be given as in (2.3).

(i) The matrix function X is a prepared (conjoined, isotropic) solution of (2.1) iff X is a solution of (2.1) and

W (X, X)(t) ≡ 0, t ∈ (0, ∞)_{q}.

(ii) The matrix functions X and Y are normalized prepared bases of (2.1) iff X, Y are two prepared solutions of (2.1) with

W (X, Y)(t) ≡ I, t ∈ (0, ∞)_{q}.

Theorem 2.11. Any two prepared solutions of (2.1) on (0, ∞)_{q} are linearly independent iff their Wronskian is nonzero.

Theorem 2.12. Equation (2.1) on (0, ∞)_{q} has two linearly independent solutions, and every solution of (2.1) on (0, ∞)_{q} is a linear combination of these two solutions.

Theorem 2.13 (Converse of Abel’s Formula). Assume X is a solution of (2.1) on (0, ∞)_{q} such that X^{−1} exists on (0, ∞)_{q}. If Y satisfies W (X, Y)(t) ≡ C, where C is a constant matrix, then Y is also a solution of (2.1).

Proof. Suppose that X is a solution of (2.1) such that X^{−1} exists on (0, ∞)_{q}, and assume Y satisfies W (X, Y)(t) ≡ C, where C is a constant matrix. By the Lagrange identity (Theorem 2.5) we have

0 ≡ D^{q}W (X, Y)(t) = X^{∗}(t)(LY)(t) − (LX)(t)Y^{∗}(t) = X^{∗}(t)(LY)(t), t ∈ (0, ∞)_{q}.

As (X^{∗})^{−1} exists on (0, ∞)_{q}, (LY)(t) = 0 on (0, ∞)_{q}. Thus Y is also a solution of (2.1). □

EJQTDE, 2007 No. 11, p. 5 Theorem 2.14. Assume that X is a solution of (2.1) on (0, ∞)_{q}. Then the following are equivalent:

(i) X is a prepared solution; (ii) (X^{∗}P D_{q}X)(t) is Hermitian for all t ∈ (0, ∞)_{q}; (iii) (X^{∗}P D_{q}X)(a) is Hermitian for some a ∈ (0, ∞)_{q}.

Proof. Use the Wronskian W and Abel’s formula. □

Note that one can easily get prepared solutions of (2.1) by taking initial conditions at a ∈ (0, ∞)_{q} so that X^{∗}(a)P (a)D_{q}X(a) is Hermitian.

In the Sturmian theory for equations of the form (2.1) the matrix function X^{∗}(t)P (t)X(qt) is important. We note the following result.

Lemma 2.15. Let X be a solution of (2.1). If X is prepared, then

X^{∗}(t)P (t)X(qt) is Hermitian for all t ∈ (0, ∞)_{q}.

Conversely, if there is an a ∈ (0, ∞)_{q} such that X^{∗}(a)P (a)X(qa) is Hermitian, then X is a prepared solution of (2.1). Moreover, if X is an invertible prepared solution, then

P (t)X(qt)X^{−1}(t), P (t)X(t)X^{−1}(qt), and Z(t) := (P (D_{q}X)X^{−1}) (t)

are all Hermitian for all t ∈ (0, ∞)_{q}.

Proof. Let X be a solution of (2.1). Then the relation

(2.3)  X^{∗}(t)P (t)X(qt) = (X^{∗}P X)(t) + (q − 1)t(X^{∗}P D_{q}X)(t)

proves the first two statements of this lemma. Now assume that X is an invertible prepared solution of (2.1). Then

X^{∗}(t)P (t)X(qt) = X^{∗}(qt)P (t)X(t) and (X^{∗}P D_{q}X)(t) = ((D_{q}X)^{∗}P X) (t) (2.4)

on (0, ∞)_{q} by (2.3) and Theorem 2.14. We multiply the first equation in (2.4) from the left with X^{∗−1}(t) and from the right with X^{−1}(t) to obtain that P (t)X(qt)X^{−1}(t) is Hermitian. To see that P (t)X(t)X^{−1}(qt) is Hermitian, we multiply the first equation in (2.4) with X^{∗−1}(qt) from the left and with X^{−1}(qt) from the right. Multiplying the second equation in (2.4) with X^{∗−1}(t) from the left and with X^{−1}(t) from the right shows that Z is Hermitian. □

Lemma 2.16. Assume that X is a prepared solution of (2.1) on (0, ∞)_{q}. Then the following are equivalent:

EJQTDE, 2007 No. 11, p. 6 (i) X^{∗}(qt)P (t)X(t) = X^{∗}(t)P (t)X(qt) > 0 on (0, ∞)_{q}; (ii) X is invertible and P (t)X(qt)X^{−1}(t) > 0 on (0, ∞)_{q}; (iii) X is invertible and P (t)X(t)X^{−1}(qt) > 0 on (0, ∞)_{q}.

Proof. First note that X^{∗}(qt)P (t)X(t) > 0 for t ∈ (0, ∞)_{q} implies that X(t) is invertible for t ∈ (0, ∞)_{q}. Since X is a prepared solution of (2.1), by Lemma 2.15 we have (2.5) P (t)X(qt)X^{−1}(t) = X^{∗−1}(t)X^{∗}(qt)P (t), P (t)X(t)X^{−1}(qt) = X^{∗−1}(qt)X^{∗}(t)P (t)

for all t ∈ (0, ∞)_{q}. We multiply the right-hand side of the first equation in (2.5) from the right with (XX^{−1}) (t) to obtain the equivalence of (i) and (ii). For the equivalence of (i) and (iii), multiply the right-hand side of the second equation in (2.5) from the right with (XX^{−1}) (qt). The other implications are similar. □

3. Reduction of Order Theorems

In this section we establish two related reduction of order theorems; first, we need the following preparatory lemma, which allows us to q-differentiate an inverse matrix.

Lemma 3.1. Let t ∈ (0, ∞)_{q}, and assume X is invertible on (0, ∞)_{q}. Then

D^{q}X^{∗−1}(t) = −X^{∗−1}(t/q)(D^{q}X)^{∗}(t)X^{∗−1}(t) = −X^{∗−1}(t)(D^{q}X)^{∗}(t)X^{∗−1}(t/q)

for t ∈ (0, ∞)_{q}.

Proof. Use the product rules given in Lemma 2.4 on the equation XX^{−1} = I. □

Remark 3.2. Throughout this work it is to be understood that

0 ≡ ∑_{s∈[a,a)_{q}} M(s) ≡ ∑_{s∈(a,a]_{q}} M(s), a ∈ (0, ∞)_{q}

for any matrix function M defined on (0, ∞)_{q}.

Theorem 3.3 (Reduction of Order I). Let a ∈ (0, ∞)_{q}, and assume X is a prepared solution of (2.1) with X invertible on [a, ∞)_{q}. Then a second prepared solution Y of (2.1) is given by

Y (t) := (q − 1)X(t) ∑_{s∈[a,t)_{q}} s (X^{∗}(s)P (s)X(qs))^{−1} , t ∈ [a, ∞)_{q}

such that X, Y are normalized prepared bases of (2.1).

EJQTDE, 2007 No. 11, p. 7 Proof. For Y defined above, by the product rule in Lemma 2.4 for D_{q} we have

D_{q}Y = P^{−1}X^{∗−1} + (D_{q}X)X^{−1}Y.

For W (X, Y) given in Definition 2.3,

W (X, Y) = X^{∗}P D_{q}Y − (P D_{q}X)^{∗}Y
= X^{∗}P (P^{−1}X^{∗−1} + (D_{q}X)X^{−1}Y) − (P D_{q}X)^{∗}Y
= I + X^{∗}P (D_{q}X)X^{−1}Y − (D_{q}X)^{∗}P Y
= I + (X^{∗}P D_{q}X − (D_{q}X)^{∗}P X)X^{−1}Y = I

since X^{∗}P D_{q}X is Hermitian by Theorem 2.14 (ii). By Theorem 2.13, W (X, Y) = I guarantees that Y is a solution of (2.1). To see that Y is prepared, note that

Y^{∗}P D_{q}Y = Y^{∗}P (P^{−1}X^{∗−1} + (D_{q}X)X^{−1}Y) = Y^{∗}X^{∗−1} + Y^{∗} (P (D_{q}X)X^{−1}) Y
= (X^{−1}Y)^{∗} + Y^{∗}ZY = (q − 1) ∑_{s∈[a,t)_{q}} s (X^{∗}(s)P (s)X(qs))^{−1} + Y^{∗}ZY,

which is Hermitian by Lemma 2.15 since X is prepared and Z is Hermitian. Consequently, X, Y are normalized prepared bases for (2.1). □

Lemma 3.4. Assume X, Y are normalized prepared bases of (2.1). Then U := XE +Y F is a prepared solution of (2.1) for constant n×n matrices E, F if and only if F^{∗}E is Hermitian. If F = I, then X, U are normalized prepared bases of (2.1) if and only if E is a constant Hermitian matrix.

Proof. Assume X, Y are normalized prepared bases of (2.1). Then by Theorem 2.14 and Definition 2.3,

X^{∗}P D_{q}X = (D_{q}X)^{∗}P X, Y^{∗}P D_{q}Y = (D_{q}Y)^{∗}P Y, X^{∗}P D_{q}Y − (D_{q}X)^{∗}P Y = I.

By linearity U := XE + Y F is a solution of (2.1). Checking appropriate Wronskians,

W (U, U) = U^{∗}P D_{q}U − (D_{q}U)^{∗}P U
= (E^{∗}X^{∗} + F^{∗}Y^{∗})P ((D_{q}X)E + (D_{q}Y)F)
− (E^{∗}(D_{q}X)^{∗} + F^{∗}(D_{q}Y)^{∗}) P (XE + Y F)
= E^{∗} (X^{∗}P D_{q}X − (D_{q}X)^{∗}P X) E + F^{∗} (Y^{∗}P D_{q}Y − (D_{q}Y)^{∗}P Y) F

+E^{∗} (X^{∗}P D_{q}Y − (D_{q}X)^{∗}P Y) F + F^{∗} (Y^{∗}P D_{q}X − (D_{q}Y)^{∗}P X) E

= 0 + 0 + E^{∗}IF − F^{∗}IE,

EJQTDE, 2007 No. 11, p. 8 and

W (X, U) = X^{∗}P D_{q}U − (D_{q}X)^{∗}P U
= X^{∗}P [(D_{q}X)E + (D_{q}Y)F] − (D_{q}X)^{∗}P [XE + Y F] = F.

Clearly the first claim holds. If F = I, then W (X, U) = I, and U = XE + Y is a prepared solution of (2.1) if and only if E is a constant Hermitian matrix. □

Theorem 3.5 (Reduction of Order II). Let a ∈ (0, ∞)_{q}, and assume X is a prepared solution of (2.1) with X invertible on [a, ∞)_{q}. Then U is a second n × n matrix solution of (2.1) iff U satisfies the first-order matrix equation

(3.1)  D_{q}(X^{−1}U)(t) = (X^{∗}(t)P (t)X(qt))^{−1}F, t ∈ [a, ∞)_{q},

for some constant n × n matrix F iff U is of the form

(3.2) U(t) = X(t)E + (q − 1)X(t) ( ∑_{s∈[a,t)_{q}} s (X^{∗}(s)P (s)X(qs))^{−1}) F, t ∈ [a, ∞)_{q},

where E and F are constant n × n matrices. In the latter case,

(3.3)  E = X^{−1}(a)U(a),  F = W (X, U)(a),

such that U is a prepared solution of (2.1) iff F^{∗}E = E^{∗}F.

Proof. Assume X is a prepared solution of (2.1) with X invertible on [a, ∞)_{q}. Let U be any n × n matrix solution of (2.1); we must show U is of the form (3.2). Using the Wronskian from Definition 2.3, set

F := W (X, U)(a) = (X^{∗}P D_{q}U − (D_{q}X)^{∗}P U) (a).

Since

D_{q}(X^{−1}U)(t) = −X^{−1}(qt)(D_{q}X)(t)X^{−1}(t)U(t) + X^{−1}(qt)D_{q}U(t)

and X is prepared we have that

(X^{∗}(t)P (t)X(qt))^{−1} F = X^{−1}(qt)(D_{q}U)(t) − (X^{∗}(t)P (t)X(qt))^{−1} (D_{q}X)^{∗}(t)P (t)U(t)
= D_{q}(X^{−1}U)(t) + X^{−1}(qt)(D_{q}X)(t)X^{−1}(t)U(t)
−X^{−1}(qt)P^{−1}(t)X^{∗−1}(t)(D_{q}X)^{∗}(t)P (t)U(t)
= D_{q}(X^{−1}U)(t) + X^{−1}(qt)P^{−1}(t)P (t)(D_{q}X)(t)X^{−1}(t)U(t)
−X^{−1}(qt)P^{−1}(t) (P (D_{q}X)X^{−1})^{∗} U
= D_{q}(X^{−1}U)(t).

EJQTDE, 2007 No. 11, p. 9 Multiplying by the variable and summing both sides from a to t,

X^{−1}(t)U(t) − X^{−1}(a)U(a) = (q − 1) ( ∑_{s∈[a,t)_{q}} s(X^{∗}(s)P (s)X(qs))^{−1}) F;

recovering U yields

U(t) = X(t)E + (q − 1)X(t) ( ∑_{s∈[a,t)_{q}} s(X^{∗}(s)P (s)X(qs))^{−1}) F

provided E = X^{−1}(a)U(a).

Conversely, assume U is given by (3.2). By Theorem 3.3 and linearity U is a solution of (2.1) on [a, ∞)_{q}. Setting t = a in (3.2) leads to E in (3.3). By the constancy of the Wronskian, W (X, U)(t) ≡ W (X, U)(a); suppressing the a, and using (3.2) and the fact that X is prepared,

W (X, U) = X^{∗}P D_{q}U − (D_{q}X)^{∗}P U = X^{∗}P [(D_{q}X)E + P^{−1}X^{∗−1}F ] − (D_{q}X)^{∗}P U
= X^{∗}P (D_{q}X)E + F − (D_{q}X)^{∗}P XE = F.

From Lemma 3.4, U is a prepared solution of (2.1) iff F^{∗}E is Hermitian. □

4. Prepared Bases

Let X be an n × p matrix function defined on (0, ∞)_{q}, and define the 2n × p matrix X by

(4.1)  X (t) = [\begin{matrix}X(t) \\ X(qt) \end{matrix}] , t ∈ (0, ∞)_{q};

we also define the block matrix

P(t) = [\begin{matrix}0 & P (t) \\ −P (t) & 0 \end{matrix}] , t ∈ (0, ∞)_{q}.

It follows that

(4.2)  W (X, X)(t) = \frac{1}{(q − 1)t} (X^{∗}PX) (t), t ∈ (0, ∞)_{q}.

EJQTDE, 2007 No. 11, p. 10 Theorem 4.1. Assume X is an n×p matrix solution of (2.1). Then X has constant rank on (0, ∞)_{q}. Furthermore, if X is a prepared solution of (2.1) and rank X = p, then p ≤ n.

Proof. Assume X is an n × p matrix solution of (2.1). Let a ∈ (0, ∞)_{q}, and suppose X (a)v = 0 for some vector v ∈ ℂ^{p}. Then

X(a)v = 0, X(qa)v = 0

by assumption; since X solves (2.1), as in the proof of Theorem 2.1 we have that

X(a/q)v = 0, X(q^{2}a)v = 0

as well, so that

X (a/q)v = 0, X (qa)v = 0.

Therefore X has constant rank on (0, ∞)_{q}. Now suppose X is an n × p prepared solution of (2.1) with rank X = p. Since X is prepared, W (X, X) ≡ 0 on (0, ∞)_{q}. By (4.2),

X^{∗}PX ≡ 0 on (0, ∞)_{q}.

As P is invertible, rank (PX) = p, and it follows from the previous line that the nullity of X^{∗} is at least p. Since

rank X^{∗} + nullity X^{∗} = 2n,

we have that

2p = p + p ≤ p + nullity X^{∗} = 2n,

putting p ≤ n. □

Definition 4.2. An n × n solution X of (2.1) is a prepared basis for (2.1) iff X is a prepared solution of (2.1) and rank X = n on (0, ∞)_{q}, where X is given in (4.1).

Theorem 4.3. Assume X, Y are n × n prepared solutions of (2.1). If W (X, Y) is invertible, then X and Y are both prepared bases of (2.1).

Proof. Assume X, Y are n × n prepared solutions of (2.1) with W (X, Y) invertible. Note that by Abel’s Formula and the definitions above,

constant ≡ W (X, Y)(t) = \frac{1}{(q − 1)t} (X^{∗}PY) (t), t ∈ (0, ∞)_{q}.

Let a ∈ (0, ∞)_{q}, and suppose Y(a)v = 0 for some vector v ∈ ℂ^{n}. Then W (X, Y)v = 0, so that by the assumption of invertibility v = 0. Hence rank Y(a) = n and due to constant rank by the theorem above, rank Y ≡ n. Thus Y is a prepared basis. In the

EJQTDE, 2007 No. 11, p. 11 same manner v^{∗}X^{∗}(a) = 0 implies v^{∗}W (X, Y) = 0 implies v = 0, and rank X (a) = rank X (t) = n and X is a prepared basis as well. □

5. q-Dominant and q-Recessive Solutions

In this main section we seek to introduce the notions of q-dominant and q-recessive solutions for the q-difference equation (2.1) when the equation has an invertible solution; in particular, we ultimately will be able to construct an (essentially) unique q-recessive solution for (2.1) in the event that it admits invertible solutions. Note that throughout the rest of the paper we assume a ∈ (0, ∞)_{q}.

Definition 5.1. A solution V of (2.1) is q-dominant at infinity iff V is a prepared basis and there exists an a such that V is invertible on [a, ∞)_{q} and

∑_{s∈[a,∞)_{q}} sΥ^{−1}(s), Υ(s) := V^{∗}(s)P (s)V (qs)

converges to a Hermitian matrix with finite entries.

Lemma 5.2. Assume the self-adjoint equation LX = 0 has a q-dominant solution V at ∞. If X is any other n × n solution of (2.1), then

lim_{t→∞} V^{−1}(t)X(t) = K

for some n × n constant matrix K.

Proof. Since V is a q-dominant solution at ∞ of (2.1), there exists an a such that V is invertible on [a, ∞)_{q}. By the second reduction of order theorem, Theorem 3.5,

X(t) = V (t)V^{−1}(a)X(a) + (q − 1)V (t) ( ∑_{s∈[a,t)_{q}} sΥ^{−1}(s)) W (V, X)(a).

Multiplying on the left by V^{−1}(t) we have

V^{−1}(t)X(t) = V^{−1}(a)X(a) + (q − 1) ( ∑_{s∈[a,t)_{q}} sΥ^{−1}(s)) W (V, X)(a).

Since V is q-dominant at ∞, the following limit exists:

lim_{t→∞} V^{−1}(t)X(t) = K := V^{−1}(a)X(a) + (q − 1) ( ∑_{s∈[a,∞)_{q}} sΥ^{−1}(s)) W (V, X)(a).

The proof is complete. □ EJQTDE, 2007 No. 11, p. 12 Definition 5.3. A solution U of (2.1) is q-recessive at infinity iff U is a prepared basis and whenever X is any other n × n solution of (2.1) such that W (X, U) is invertible, X is eventually invertible and

lim_{t→∞} X^{−1}(t)U(t) = 0.

Lemma 5.4. If U is a solution of (2.1) which is q-recessive at ∞, then for any invertible constant matrix K, the solution UK of (2.1) is q-recessive at ∞ as well.

Proof. The proof follows from the definition. □

Lemma 5.5. If U is a solution of (2.1) which is q-recessive at ∞, and V is a prepared solution of (2.1) such that W (V, U) is invertible, then V is q-dominant at ∞.

Proof. Note that by the assumptions and Theorem 4.3, V is a prepared basis. By the definition of q-recessive, W (V, U) invertible implies that V is invertible on [a, ∞)_{q} for some a ∈ (0, ∞)_{q}, and

(5.1)  lim_{t→∞} V^{−1}(t)U(t) = 0.

Let K := W (V, U); by assumption K is invertible, and by Definition 2.3

K = Υ(t)V^{−1}(qt)D_{q}U(t) − ((D_{q}V)^{∗}P V)(t)V^{−1}(t)U(t)

for all t ∈ [a, ∞)_{q}. Since V is prepared,

Υ^{−1}(t)K = V^{−1}(qt)D_{q}U(t) − V^{−1}(qt)(D_{q}V)(t)V^{−1}(t)U(t)
= D_{q} (V^{−1}U) (t).

Multiply by s, sum both sides from a to ∞, and use (5.1) to see that

∑_{s∈[a,∞)_{q}} sΥ^{−1}(s) = \frac{−1}{(q − 1)}V^{−1}(a)U(a)K^{−1}

converges to a Hermitian matrix, as Υ is Hermitian. Thus V is q-dominant at ∞. □

Theorem 5.6. Assume (2.1) has a solution V which is q-dominant at ∞. Then

U(t) := (q − 1)V (t) ∑_{s∈[t,∞)_{q}} s(V^{∗}(s)P (s)V (qs))^{−1} = (q − 1)V (t) ∑_{s∈[t,∞)_{q}} sΥ^{−1}(s)

is a solution of (2.1) which is q-recessive at ∞ and W (V, U) = −I.

EJQTDE, 2007 No. 11, p. 13 Proof. Since V is q-dominant at ∞, U is a well-defined function and can be written as

U(t) = (q − 1)V (t) [ ∑_{s∈[a,∞)_{q}} sΥ^{−1}(s) − ( ∑_{s∈[a,t)_{q}} sΥ^{−1}(s)) I] , t ∈ [a, ∞)_{q};

by the second reduction of order theorem, Theorem 3.5, U is a solution of (2.1) of the form (3.2) with

E = (q − 1) ∑_{s∈[a,∞)_{q}} sΥ^{−1}(s), F = −I.

From (3.3), W (V, U) = F = −I. Since

E^{∗}F = (1 − q) ∑_{s∈[a,∞)_{q}} sΥ^{−1}(s)

is Hermitian, U is a prepared solution of (2.1), and W (−V, U) = I implies that U and −V are normalized prepared bases. Let X be an n × n matrix solution of LX = 0 such that W (X, U) is invertible. By the second reduction of order theorem,

X(t) = V (t) [V^{−1}(a)X(a) + (q − 1) ∑_{s∈[a,t)_{q}} sΥ^{−1}(s)W (V, X)]
(5.2)  = V (t)C_{1} + U(t)C_{2},

where

C_{1} := V^{−1}(a)X(a) + (q − 1) ∑_{s∈[a,∞)_{q}} sΥ^{−1}(s)W (V, X)

and

C_{2} := −W (V, X).

Note that

W (X, U) = C_{1}^{∗}W (V, U) + C_{2}^{∗}W (U, U) = −C_{1}^{∗}.

As W (X, U) is invertible by assumption, C_{1} is invertible. From (5.2),

lim_{t→∞} V^{−1}(t)X(t) = lim_{t→∞} (C_{1} + V^{−1}(t)U(t)C_{2})
= C_{1} + (q − 1) lim_{t→∞} ∑_{s∈[t,∞)_{q}} sΥ^{−1}(s)C_{2} = C_{1}

EJQTDE, 2007 No. 11, p. 14 is likewise invertible. Consequently for large t, X(t) is invertible. Lastly,

lim_{t→∞} X^{−1}(t)U(t) = lim_{t→∞} [V (t)C_{1} + U(t)C_{2}]^{−1} U(t)
= lim_{t→∞} [C_{1} + V^{−1}(t)U(t)C_{2}]^{−1} V^{−1}(t)U(t)
= [C_{1} + 0]^{−1} 0 = 0.

Therefore U is a q-recessive solution at ∞. □

Theorem 5.7. Assume (2.1) has a solution U which is q-recessive at ∞, and U(a) is invertible for some a ∈ (0, ∞)_{q}. Then U is uniquely determined by U(a), and (2.1) has a solution V which is q-dominant at ∞.

Proof. Assume U(a) is invertible; let V be the unique solution of the intial value problem

LV = 0, V (a) = 0, D_{q}V (a) = I.

Then V is a prepared basis and

W (V, U) = W (V, U)(a) = (V^{∗}P D_{q}U)(a) − (P D_{q}V)^{∗}(a)U(a) = −P (a)U(a)

is invertible. It follows from Lemma 5.5 that V is q-dominant at ∞. Let Γ be an arbitrary but fixed n × n constant matrix. Let X solve the initial value problem

LX = 0, X(a) = I, D_{q}X(a) = Γ.

By Theorem 5.2,

lim_{t→∞} V^{−1}(t)X(t) = K,

where K is an n × n constant matrix; note that K is independent of the q-recessive solution U. Using the initial conditions at a, by uniqueness of solutions it is easy to see that there exist constant n × n matrices C_{1} and C_{2} such that

U(t) = X(t)C_{1} + V (t)C_{2},

where C_{1} = U(a) is invertible. Consequently, using the q-recessive nature of U, we have

0 = lim_{t→∞} V^{−1}(t)U(t) = lim_{t→∞} (V^{−1}(t)X(t)U(a) + C_{2}) = KU(a) + C_{2},

so that C_{2} = −KU(a). Thus the initial condition for D_{q}U is

D_{q}U(a) = (Γ − K)U(a),

and the q-recessive solution U is uniquely determined by its initial value U(a). □ EJQTDE, 2007 No. 11, p. 15 Theorem 5.8. Assume (2.1) has a solution U which is q-recessive at ∞ and a solution V which is q-dominant at ∞. If U and ∑_{s∈[t,∞)_{q}} s(V^{∗}(s)P (s)V (qs))^{−1} are both invertible for large t ∈ (0, ∞)_{q}, then there exists an invertible constant matrix K such that

U(t) = (q − 1)V (t) ( ∑_{s∈[t,∞)_{q}} s(V^{∗}(s)P (s)V (qs))^{−1}) K

for large t. In addition, W (U, V) is invertible and

lim_{t→∞} V^{−1}(t)U(t) = 0.

Proof. For sufficiently large t ∈ (0, ∞)_{q} define

Y (t) = (q − 1)V (t) ∑_{s∈[t,∞)_{q}} s(V^{∗}(s)P (s)V (qs))^{−1}.

By Theorem 5.6, Y is also a q-recessive solution of (2.1) at ∞ and W (V, Y) = −I. Because U and ∑_{s∈[t,∞)_{q}} s(V^{∗}(s)P (s)V (qs))^{−1} are both invertible for large t ∈ (0, ∞)_{q}, Y is likewise invertible for large t, and

lim_{t→∞} V^{−1}(t)Y (t) = 0

by the q-recessive nature of Y. Choose a ∈ (0, ∞)_{q} large enough to ensure that U and Y are invertible in [a, ∞)_{q}. By Lemma 5.4 the solution given by

X(t) := Y (t)Y^{−1}(a)U(a), t ∈ [a, ∞)_{q}

is yet another q-recessive solution at ∞. Since U and X are q-recessive solutions at ∞ and U(a) = X(a), we conclude from the uniqueness established in Theorem 5.7 that X ≡ U. Thus

U(t) = Y (t)Y^{−1}(a)U(a), t ∈ [a, ∞)_{q}
= (q − 1)V (t) ( ∑_{s∈[t,∞)_{q}} s(V^{∗}(s)P (s)V (qs))^{−1}) K,

where K := Y^{−1}(a)U(a) is an invertible constant matrix. □

The next result, when the domain is ℤ instead of (0, ∞)_{q}, relates the convergence of infinite series, the convergence of certain continued fractions, and the existence of recessive solutions; for more see [2] and the references therein.

EJQTDE, 2007 No. 11, p. 16 Theorem 5.9 (Connection Theorem). Let X and V be solutions of (2.1) determined by the initial conditions

X(a) = I, D_{q}X(a) = P^{−1}(a)K, and V (a) = 0, D_{q}V (a) = P^{−1}(a),

respectively, where a ∈ (0, ∞)_{q} and K is a constant Hermitian matrix. Then X, V are normalized prepared bases of (2.1), and the following are equivalent:

(i) V is q-dominant at ∞; (ii) V is invertible for large t ∈ (0, ∞)_{q} and lim_{t→∞} V^{−1}(t)X(t) exists as a Her-

mitian matrix Ω(K) with finite entries; (iii) there exists a solution U of (2.1) which is q-recessive at ∞, with U(a) invert-

ible.

If (i), (ii), and (iii) hold then

(D_{q}U)(a)U^{−1}(a) = D_{q}X(a) − (D_{q}V)(a)Ω(K) = −P^{−1}(a)Ω(0).

Proof. Since V (a) = 0, V is a prepared solution of (2.1). Also,

W (X, X) = W (X, X)(a) = (X^{∗}P D_{q}X − (D_{q}X)^{∗}P X)(a) = IK − K^{∗}I = 0

as K is Hermitian, making X a prepared solution of (2.1) as well. Checking

W (X, V) = W (X, V)(a) = (X^{∗}P D_{q}V − (D_{q}X)^{∗}P V)(a) = I − 0 = I,

we see that X, V are normalized prepared bases of (2.1). Now we show that (i) implies (ii). If V is a q-dominant solution of (2.1) at ∞, then there exists a t_{1} ∈ (a, ∞)_{q} such that V (t) is invertible for t ∈ [t_{1}, ∞)_{q}, and the sum

∑_{s∈[t_{1},∞)_{q}} s(V^{∗}(s)P (s)V (qs))^{−1}

converges to a Hermitian matrix with finite entries. By the second reduction of order theorem,

(5.3)  X(t) = V (t)E + (q − 1)V (t) ( ∑_{s∈[t_{1},t)_{q}} s (V^{∗}(s)P (s)V (qs))^{−1}) F,

where

E = V^{−1}(t_{1})X(t_{1}), F = W (V, X)(t_{1}) = −W (X, V)^{∗} = −I.

EJQTDE, 2007 No. 11, p. 17 Since X is prepared, E^{∗}F = −E^{∗} is Hermitian, whence E is Hermitian. As a result, by (5.3) we have

lim_{t→∞} V^{−1}(t)X(t) = E − (q − 1) ∑_{s∈[t_{1},∞)_{q}} s (V^{∗}(s)P (s)V (qs))^{−1}

converges to a Hermitian matrix with finite entries, and (ii) holds. Next we show that (ii) implies (iii). If V is invertible on [t_{1}, ∞)_{q} and

(5.4)  lim_{t→∞} V^{−1}(t)X(t) = Ω

exists as a Hermitian matrix, then from (5.3) and (5.4) we have

Ω = lim_{t→∞} V^{−1}(t)X(t) = E − (q − 1) ∑_{s∈[t_{1},∞)_{q}} s (V^{∗}(s)P (s)V (qs))^{−1} ;

in other words,

(q − 1) ∑_{s∈[t_{1},∞)_{q}} s (V^{∗}(s)P (s)V (qs))^{−1} = E − Ω.

Define

(5.5)  U(t) := X(t) − V (t)Ω.

Then

W (U, U) = W (X − V Ω, X − V Ω)
= W (X, X) − W (X, V)Ω − Ω^{∗}W (V, X) + Ω^{∗}W (V, V)Ω
= −Ω + Ω^{∗} = 0,

and U(a) = X(a) = I, making U a prepared basis for (2.1). If X_{1} is an n × n matrix solution of LX = 0 such that W (X_{1}, U) is invertible, then

(5.6)  X_{1}(t) = V (t)C_{1} + U(t)C_{2}

for some constant matrices C_{1} and C_{2} determined by the initial conditions at a. It follows that

W (X_{1}, U) = W (V C_{1} + UC_{2}, U) = C_{1}^{∗}W (V, U) + C_{2}^{∗}W (U, U)
= C_{1}^{∗}W (V, U) = C_{1}^{∗}W (V, U)(a) = −C_{1}^{∗}

by (5.5), so that C_{1} is invertible. From (5.4) and (5.5) we have that

lim_{t→∞} V^{−1}(t)U(t) = lim_{t→∞} [V^{−1}(t)X(t) − Ω] = 0,

EJQTDE, 2007 No. 11, p. 18 resulting in

lim_{t→∞} V^{−1}(t)X_{1}(t) = lim_{t→∞} [C_{1} + V^{−1}(t)U(t)C_{2}] = C_{1},

which is invertible. Thus X_{1}(t) is invertible for large t ∈ (0, ∞)_{q}, and

lim_{t→∞} X_{1}^{−1}(t)U(t) = lim_{t→∞}[V (t)C_{1} + U(t)C_{2}]^{−1}U(t)
= lim_{t→∞}[C_{1} + V^{−1}(t)U(t)C_{2}]^{−1}V^{−1}(t)U(t) = C_{1}^{−1}(0) = 0.

Hence U is a q-recessive solution of (2.1) at ∞ and (iii) holds. Finally we show that (iii) implies (i). If U is a q-recessive solution of (2.1) at ∞ with U(a) invertible, then

W (V, U) = W (V, U)(a) = −U(a)

is also invertible. Hence by Lemma 5.5, V is a q-dominant solution of (2.1) at ∞.

To complete the proof, assume (i), (ii), and (iii) hold. It can be shown via initial conditions at a that

U(t) = X(t)U(a) + V (t)C

for some suitable constant matrix C. By (ii),

lim_{t→∞} V^{−1}(t)X(t) = Ω(K),

and thus

V^{−1}(t)U(t) = V^{−1}(t)X(t)U(a) + C.

As U is a q-recessive solution at ∞ by (iii),

0 = lim_{t→∞} (V^{−1}(t)X(t)U(a) + C) = Ω(K)U(a) + C,

yielding

U(t) = [X(t) − V (t)Ω(K)] U(a).

An application of the quantum derivative D_{q} at a yields

(D_{q}U)(a)U^{−1}(a) = D_{q}X(a) − (D_{q}V)(a)Ω(K).

Now let Y be the unique solution of the initial value problem

LY = 0, Y (a) = I, D_{q}Y (a) = 0.

Using the initial conditions at a we see that

X(t) = Y (t) + V (t)K.

Consequently,

lim_{t→∞} V^{−1}(t)X(t) = lim_{t→∞} V^{−1}(t)Y (t) + K

EJQTDE, 2007 No. 11, p. 19 implies, by (ii) and the fact that X = Y when K = 0, that

Ω(K) = Ω(0) + K.

Therefore

D_{q}X(a) − (D_{q}V)(a)Ω(K) = −(D_{q}V)(a)Ω(0) = −P^{−1}(a)Ω(0).

Thus the proof is complete. □

We will also be interested in analyzing the self-adjoint vector q-difference equation

(5.7)  Lx = 0, where Lx(t) := D^{q} (P D_{q}x) (t) + Q(t)x(t), t ∈ [a, ∞)_{q},

where x is an n × 1 vector-valued function defined on (0, ∞)_{q}. We will see interesting relationships between the so-called unique two-point property (defined below) of the nonhomogeneous vector equation Lx = h, disconjugacy of Lx = 0, and the construction of q-recessive solutions at infinity to the matrix equation LX = 0. The following theorem can be proven by modifying the proof of Theorem 2.1.

Theorem 5.10. Let h be an n × 1 vector function defined on [a, ∞)_{q}. Then the nonhomogeneous vector initial value problem

(5.8)  Ly = D^{q}(P D_{q}y) + Qy = h, y(a) = y_{a}, D_{q}y(a) = y′_{a}

has a unique solution.

Definition 5.11. Assume h is an n × 1 vector function defined on [a, ∞)_{q}. Then the vector dynamic equation Lx = h has the unique two-point property on [a, ∞)_{q} provided given any a ≤ t_{1} < t_{2} in (0, ∞)_{q}, if u and v are solutions of Lx = h with u(t_{1}) = v(t_{1}) and u(t_{2}) = v(t_{2}), then u ≡ v on [a, ∞)_{q}.

Theorem 5.12. If the homogeneous vector equation (5.7) has the unique two-point property on [a, ∞)_{q}, then the boundary value problem

Lx = h, x(t_{1}) = α, x(t_{2}) = β,

where a ≤ t_{1} < t_{2} in (0, ∞)_{q} and α, β ∈ ℂ^{n}, has a unique solution on [a, ∞)_{q}.

Proof. If t_{2} = qt_{1}, then the boundary value problem is an initial value problem and the result holds by Theorem 5.10. Assume t_{2} > qt_{1}. Let X(t, t_{1}) and Y (t, t_{1}) be the unique n × n matrix solutions of (2.1) determined by the initial conditions

X(t_{1}, t_{1}) = 0, D_{q}X(t_{1}, t_{1}) = I, and Y (t_{1}, t_{1}) = I, D_{q}Y (t_{1}, t_{1}) = 0;

EJQTDE, 2007 No. 11, p. 20 then a general solution of (5.7) is given by

(5.9)  x(t) = X(t, t_{1})γ + Y (t, t_{1})δ,

for γ, δ ∈ ℂ^{n}, as x(t_{1}) = δ and D_{q}x(t_{1}) = γ. By the unique two-point property the homogeneous boundary value problem

Lx = 0, x(t_{1}) = 0, x(t_{2}) = 0

has only the trivial solution. For x given by (5.9), the boundary condition at t_{1} implies that δ = 0, and the boundary condition at t_{2} yields

X(t_{2}, t_{1})γ = 0;

by uniqueness and the fact that x is trivial, γ = 0 is the unique solution, meaning X(t_{2}, t_{1}) is invertible. Next let v be the solution of the initial value problem

Lv = h, v(t_{1}) = 0, D_{q}v(t_{1}) = 0.

Then the general solution of Lx = h is given by

x(t) = X(t, t_{1})γ + Y (t, t_{1})δ + v(t).

We now show that the boundary value problem

Lx = h, x(t_{1}) = α, x(t_{2}) = β

has a unique solution. The boundary condition at t_{1} implies that δ = α. The condition at t_{2} leads to the equation

β = X(t_{2}, t_{1})γ + Y (t_{2}, t_{1})α + v(t_{2});

since X(t_{2}, t_{1}) is invertible, this can be solved uniquely for γ. □

Corollary 5.13. If the homogeneous vector equation (5.7) has the unique two-point property on [a, ∞)_{q}, then the matrix boundary value problem

LX = 0, X(t_{1}) = M, X(t_{2}) = N

has a unique solution, where M and N are given constant n × n matrices.

Proof. Modify the proof of Theorem 5.12 to get existence and uniqueness. □ EJQTDE, 2007 No. 11, p. 21 Theorem 5.14. Assume the homogeneous vector equation (5.7) has the unique twopoint property on [a, ∞)_{q}. Further assume U is a solution of (2.1) which is q-recessive at ∞ with U(a) invertible. For each fixed s ∈ (a, ∞)_{q}, let Y (t, s) be the solution of the boundary value problem

LY (t, s) = 0, Y (a, s) = I, Y (s, s) = 0.

Then the q-recessive solution U(t)U^{−1}(a) is uniquely determined by

(5.10)  U(t)U^{−1}(a) = lim_{s→∞} Y (t, s).

Proof. Assume U is a solution of (2.1) which is q-recessive at ∞ with U(a) invertible. Let V be the unique solution of the initial value problem

LV = 0, V (a) = 0, D_{q}V (a) = P^{−1}(a).

By the connection theorem, Theorem 5.9, V is invertible for large t. By checking boundary conditions at a and s for s large, we get that

Y (t, s) = −V (t)V^{−1}(s)U(s)U^{−1}(a) + U(t)U^{−1}(a).

Then

W (V, U) = W (V, U)(a) = (V^{∗}P D_{q}U − (D_{q}V)^{∗}P U) (a) = −U(a)

is invertible, and by the q-recessive nature of U,

lim_{t→∞} V^{−1}(t)U(t) = 0.

As a result,

lim_{s→∞} Y (t, s) = 0 + U(t)U^{−1}(a),

and the proof is complete. □

Definition 5.15. A prepared vector solution x of (5.7) has a generalized zero at a ∈ (0, ∞)_{q} iff x(a/q) ≠ 0 and x^{∗}(a/q)P (a/q)x(a) ≤ 0. Equation (5.7) is disconjugate on [a, ∞)_{q} iff no nontrivial prepared vector solution of (5.7) has two generalized zeros in [a, ∞)_{q}.

Definition 5.16. A prepared basis X of (2.1) has a generalized zero at a ∈ (0, ∞)_{q} iff X(a) is noninvertible or X^{∗}(a/q)P (a/q)X(a) is invertible but X^{∗}(a/q)P (a/q)X(a) ≤ 0.

Lemma 5.17. If a prepared basis X of (2.1) has a generalized zero at a ∈ (0, ∞)_{q}, then there exists a vector γ ∈ ℂ^{n} such that x = Xγ is a nontrivial prepared solution of (5.7) with a generalized zero at a.

EJQTDE, 2007 No. 11, p. 22 Proof. The proof follows from Definitions 5.15 and 5.16. □

Theorem 5.18. If the vector equation (5.7) is disconjugate on [a, ∞)_{q}, then the matrix equation (2.1) has a solution U which is q-recessive at ∞ with U(t) invertible for t ∈ [qa, ∞)_{q}.

Proof. Let X be the solution of the initial value problem

LX = 0, X(a) = 0, D_{q}X(a) = I;

then X is a prepared solution of (2.1). If X is not invertible on [qa, ∞)_{q}, then there exists a t_{1} > a such that X(t_{1}) is singular. But then there exists a nontrivial vector δ ∈ ℂ^{n} such that X(t_{1})δ = 0. If x(t) := X(t)δ, then x is a nontrivial prepared solution of (5.7) with

x(a) = 0, x(t_{1}) = 0,

a contradiction of disconjugacy. Hence X is invertible in [qa, ∞)_{q}. We next claim that

(5.11)  X^{∗}(t)P (t)X(qt) > 0, t ∈ [qa, ∞)_{q};

if not, there exists t_{2} ∈ [qa, ∞)_{q} such that

X^{∗}(t_{2})P (t_{2})X(qt_{2}) ≯ 0.

It follows that there exists a nontrivial vector γ such that x(t) := X(t)γ is a nontrivial prepared vector solution of Lx = 0 with a generalized zero at qt_{2}. Using the initial condition for X, however, we have x(a) = 0, another generalized zero, a contradiction of the assumption that the vector equation (5.7) is disconjugate on [a, ∞)_{q}. Thus (5.11) holds. Define the matrix function

V (t) := X(t) [I + (q − 1) ∑_{s∈[qa,t)_{q}} s (X^{∗}(s)P (s)X(qs))^{−1}] , t ∈ [qa, ∞)_{q}.

By Theorem 3.5, V is a prepared solution of LV = 0 with W (X, V) = I. Note that V is also invertible on [qa, ∞)_{q}, so that by the reduction of order theorem again,

X(t) = V (t) [I − (q − 1) ∑_{s∈[qa,t)_{q}} s (V^{∗}(s)P (s)V (qs))^{−1}] , t ∈ [qa, ∞)_{q}.

Consequently,

I = [V^{−1}X][X^{−1}V](t) = [I − (q − 1) ∑_{s∈[qa,t)_{q}} sΥ^{−1}(s)] [I + (q − 1) ∑_{s∈[qa,t)_{q}} sΞ^{−1}(s)] ,

where

Υ(s) := V^{∗}(s)P (s)V (qs), Ξ(s) := X^{∗}(s)P (s)X(qs) > 0.

Since the second factor is strictly increasing by (5.11) and bounded below by I, the first factor is positive definite and strictly decreasing, ensuring the existence of a limit, in other words, we have

0 ≤ I − (q − 1) ∑_{s∈[qa,∞)_{q}} sΥ^{−1}(s) < I − (q − 1) ∑_{s∈[qa,t)_{q}} sΥ^{−1}(s) ≤ I.

It follows that

0 ≤ ∑_{s∈[qa,t)_{q}} sΥ^{−1}(s) < ∑_{s∈[qa,∞)_{q}} sΥ^{−1}(s) ≤ I, t ∈ [qa, ∞)_{q},

and V is a q-dominant solution of (2.1) at ∞. Set

U(t) := (q − 1)V (t) ∑_{s∈[t,∞)_{q}} sΥ^{−1}(s).

By Theorem 5.6, U is a q-recessive solution of (2.1) at ∞. Since

U(t) = (q − 1)V (t) [ ∑_{s∈[qa,∞)_{q}} sΥ^{−1}(s) − ∑_{s∈[qa,t)_{q}} sΥ^{−1}(s)] ,

V is invertible on [qa, ∞)_{q}, and the difference in brackets is positive definite on [qa, ∞)_{q}, we get that U is invertible on [qa, ∞)_{q} as well, and the conclusion of the theorem follows. □

Corollary 5.19. Assume the vector equation (5.7) is disconjugate on [a, ∞)_{q}, and K is a constant Hermitian matrix. Let U, V be the matrix solutions of LX = 0 satisfying the initial conditions

U(a) = I, D_{q}U(a) = P^{−1}(a)K, and V (a) = 0, D_{q}V (a) = P^{−1}(a).

Then V is invertible in [qa, ∞)_{q}, V is a q-dominant solution of (2.1) at ∞, and

lim_{t→∞} V^{−1}(t)U(t)

exists as a Hermitian matrix.

EJQTDE, 2007 No. 11, p. 24 Proof. By Theorem 5.18, the matrix equation (2.1) has a solution U which is qrecessive at ∞ with U(t) invertible for t ∈ [qa, ∞)_{q}. Thus (iii) of the connection theorem, Theorem 5.9 holds; by (i), then, V is a q-dominant solution of (2.1) at ∞, and by (ii),

lim_{t→∞} V^{−1}(t)U(t)

exists as a Hermitian matrix. Since V (a) = 0 and the vector equation (5.7) is disconjugate on [a, ∞)_{q},

V^{∗}(t)P (t)V (qt) > 0, t ∈ [qa, ∞)_{q}.

In particular, V is invertible in [qa, ∞)_{q}. □

Theorem 5.20. If the vector equation (5.7) is disconjugate on [a, ∞)_{q}, then Lx(t) = h(t) has the unique two-point property in [a, ∞)_{q}. In particular, every boundary value problem of the form

Lx(t) = h(t),  x(τ_{1}) = α,  x(τ_{2}) = β,

where τ_{1}, τ_{2} ∈ [a, ∞)_{q} with τ_{1} < τ_{2}, and where α, β are given n-vectors, has a unique solution.

Proof. By Theorem 5.18, disconjugacy of (5.7) implies the existence of a prepared, invertible matrix solution of (2.1). Thus by Theorem 5.12, it suffices to show that (5.7) has the unique two-point property in [a, ∞)_{q}. To this end, assume u, v are solutions of Lx = 0, and there exist points s_{1}, s_{2} ∈ (0, ∞)_{q} such that a ≤ s_{1} < s_{2} and

u(s_{1}) = v(s_{1}), u(s_{2}) = v(s_{2}).

If s_{2} = qs_{1}, then u and v satisfy the same initial conditions and u ≡ v by uniqueness; hence we assume s_{2} > qs_{1}. Setting x = u − v, we see that x solves the initial value problem

Lx = 0, x(τ_{1}) = 0, x(τ_{2}) = 0.

Since Lx = 0 is disconjugate and x is a prepared solution with two generalized zeros, it must be that x ≡ 0 in [a, ∞)_{q}. Consequently, u = v and the two-point property holds. □

Corollary 5.21 (Construction of the Recessive Solution). Assume the vector equation (5.7) is disconjugate on [a, ∞)_{q}. For each s ∈ (a, ∞)_{q}, let U(t, s) be the solution of the boundary value problem

LU(·, s) = 0, U(a, s) = I, D_{q}U(s, s) = 0.

EJQTDE, 2007 No. 11, p. 25 Then the solution U with U(a) = I which is q-recessive at ∞ is given by

U(t) = lim_{s→∞} U(t, s),

satisfying

(5.12)  U^{∗}(t)P (t)U(qt) > 0, t ∈ [a, ∞)_{q}.

Proof. By Theorem 5.18 and Theorem 5.20, LX = 0 has a q-recessive solution and Lx = h has the unique two-point property. The conclusion then follows from Theorem 5.14, except for (5.12). From the initial condition U(s, s) = 0 and the fact that Lx = 0 is disconjugate, it follows that

U^{∗}(t, s)P (t)U(qt, s) > 0

holds in [a, s/q)_{q}. Again from Theorem 5.14,

lim_{s→∞} U(t, s) = U(t)U^{−1}(a) = U(t),

so that U invertible on [a, ∞)_{q} and (5.12) holds. □

Remark 5.22. In an analogous way we could analyze the related (formally) selfadjoint quantum (h-difference) system

(5.13)  D^{h} (P D_{h}X) (t) + Q(t)X(t) = 0, t ∈ (0, ∞)_{h} := {h, 2h, 3h, ⋯ },

where the real scalar h > 0 and the h-derivatives are given, respectively, by the difference quotients

(D_{h}y)(t) = \frac{y(t + h) − y(t)}{h} and (D^{h}y)(t) = \frac{y(t) − y(t − h)}{h} = (D_{h}y)(t − h).

In the case where invertible solutions of (5.13) exist, their characterization as h-dominant and/or h-recessive solutions at infinity can be developed in parallel with the previous results on the q-equation (2.1). As q approaches 1 in the limit or h approaches zero in the limit, we can recover results from classical ordinary differential equations.

6. Future Directions

In this section we lay the groundwork for possible further exploration of the nonhomogeneous equation by introducing the Pólya factorization for the self-adjoint matrix q-difference operator L defined in (2.1), which in turn leads to a variation of parameters result.

EJQTDE, 2007 No. 11, p. 26 Theorem 6.1 (Pólya Factorization). If (2.1) has a prepared solution X > 0 (positive definite) on an interval I ⊂ (0, ∞)_{q} such that X^{∗}(t)P (t)X(qt) > 0 for t ∈ I, then for any matrix function Y defined on (0, ∞)_{q} we have on the interval I a Pólya factorization

LY = M_{1}^{∗}D^{q} {M_{2}D_{q}(M_{1}Y)} , M_{1}(t) := X^{−1}(t) > 0, M_{2}(t) := X^{∗}(t)P (t)X(qt) > 0.

Proof. Assume X > 0 is a prepared solution of (2.1) on I ⊂ (0, ∞)_{q} such that M_{2} > 0 on I, and let Y be a matrix function defined on (0, ∞)_{q}. Then X is invertible and

LY =^{Thm 2.5} (X^{∗})^{−1}D^{q}W (X, Y)
=^{Def 2.3} (X^{∗})^{−1}D^{q} {X^{∗}P D_{q}Y − (D_{q}X)^{∗}P Y }
= M_{1}^{∗}D^{q} {X^{∗}[P D_{q}Y − X^{∗−1}(D_{q}X)^{∗}P Y]}
=^{Thm 2.15} M_{1}^{∗}D^{q} {X^{∗}[P D_{q}Y − P (D_{q}X)X^{−1}Y]}
= M_{1}^{∗}D^{q} {M_{2}[X^{−1}(qt)D_{q}Y − X^{−1}(qt)(D_{q}X)X^{−1}Y]}
= M_{1}^{∗}D^{q} {M_{2}[X^{−1}(qt)D_{q}Y + (D_{q}X^{−1})Y]}
= M_{1}^{∗}D^{q} {M_{2}D_{q}(X^{−1}Y)}
= M_{1}^{∗}D^{q} {M_{2}D_{q}(M_{1}Y)} ,

for M_{1} and M_{2} as defined in the statement of the theorem. □

Theorem 6.2 (Variation of Parameters). Let H be an n × n matrix function defined on [a, ∞)_{q}. If the homogeneous matrix equation (2.1) has a prepared solution X with X(t) invertible for t ∈ [a, ∞)_{q}, then the nonhomogeneous equation LY = H has a solution Y given by

Y (t) = X(t)X^{−1}(a)Y (a) + (q − 1)X(t) ∑_{s∈[a,t)_{q}} s (X^{∗}(s)P (s)X(qs))^{−1} W (X, Y)(a)

+\frac{(q − 1)^{2}}{q} X(t) ∑_{s∈[a,t)_{q}} s ((X^{∗}(s)P (s)X(qs))^{−1} ∑_{τ∈(a,s]_{q}} τX^{∗}(τ)H(τ)) .

Proof. Let Y be a matrix function defined on (0, ∞)_{q}, and assume X is a prepared solution of (2.1) invertible on [a, ∞)_{q}. As in Theorem 6.1, we factor LY to get

H(t) = LY (t) = X^{∗−1}(t)D^{q} (X^{∗}(t)P (t)X(qt)D_{q}(X^{−1}Y)(t)) .

EJQTDE, 2007 No. 11, p. 27 Multiplying by sX^{∗} and summing over (a, t]_{q} we arrive at

X^{∗}(t)P (t)X(qt)D_{q}(X^{−1}Y)(t) − W (X, Y)(a) = (1 − \frac{1}{q}) ∑_{s∈(a,t]_{q}} sX^{∗}(s)H(s),

where

W (X, Y)(a) = X^{∗}(a)P (a)X(qa)D_{q}(X^{−1}Y)(a)

since X is prepared. This leads to

D_{q}(X^{−1}Y)(t) = (X^{∗}(t)P (t)X(qt))^{−1} (W (X, Y)(a) + (1 − \frac{1}{q}) ∑_{s∈(a,t]_{q}} sX^{∗}(s)H(s)) ,

which is then multiplied by (q − 1)s and summed over [a, t)_{q} to obtain the form for Y given in the statement of the theorem. Clearly the right-hand side of the form of Y above reduces to Y (a) at a, and since X is an invertible prepared solution, by Theorem 2.15 the quantum derivative reduces to D_{q}Y (a) at a. □

Corollary 6.3. Let H be an n × n matrix function defined on [a, ∞)_{q}. If the homogeneous matrix equation (2.1) has a prepared solution X with X(t) invertible for t ∈ [a, ∞)_{q}, then the nonhomogeneous initial value problem

(6.1)  LY = D^{q}(P D_{q}Y) + QY = H, Y (a) = Y_{a}, D_{q}Y (a) = Y′_{a}

has a unique solution.

Proof. By Theorem 6.2, the nonhomogeneous initial value problem (6.1) has a solution. Suppose Y_{1} and Y_{2} both solve (6.1). Then X = Y_{1} − Y_{2} solves the homogeneous initial value problem

LX = 0, X(a) = 0, D_{q}X(a) = 0;

by Theorem 2.1, this has only the trivial solution X = 0. □

References

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[2] C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Academic Publishers, Dordrecht, 1996.

[3] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002.

[4] M. Ma, Dominant and recessive solutions for second-order self-adjoint linear difference systems, Appl. Math. Lett., 18 (2005) 179–185.

[5] W. T. Reid, Oscillation criteria for linear differential systems with complex coefficients, Pacific

J. Math., 6 (1956) 733–751.

EJQTDE, 2007 No. 11, p. 28

[6] W. T. Reid, Principal solutions of non-oscillatory self-adjoint linear differential systems, Pacific J. Math., 8 (1958) 147–169.

[7] W. T. Reid, Ordinary Differential Equations, Wiley, New York, 1971.

[8] W. T. Reid, Riccati Differential Equations, Academic Press, New York, 1972.

[9] W. T. Reid, Sturmian Theory for Ordinary Differential Equations, Springer-Verlag, New York, 1980.

[10] G. F. Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, McGraw-Hill, New York, 1992.

(Received March 15, 2007)

Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562 USA

    E-mail address: andersod@cord.edu, lmmoats@cord.edu

EJQTDE, 2007 No. 11, p. 29