Generalized Reciprocity Principle for Discrete Symplectic Systems

This paper studies transformations for conjoined bases of symplectic difference systems Y i+1 = S i Y i with the symplectic coefficient matrices S i. For an arbitrary symplectic transformation matrix P i we formulate most general sufficient conditions for S i , P i which guarantee that P i preserves oscillatory properties of conjoined bases Y i. We present examples which show that our new results extend the applicability of the discrete transformation theory.


Introduction
In this paper we investigate transformations of the symplectic difference systems [2] where S i , Y i are real partitioned matrices with n × n blocks A i , B i , C i , D i , X i , U i .The matrix S i is assumed to be symplectic, i.e.
S T i JS i = J, J = 0 I −I 0 , and I, 0 are the identity and zero matrices.
Together with system (1.1) we consider the transformed system Ỹi+1 = Si Ỹi , Ỹi = P i Y i , Si = P i+1 S i P −1 i , ( where P i is an arbitrary symplectic transformation matrix, i.e.

P T
i JP i = J, i = M, M + 1, . . .For the special case P i = J, (1.2) takes the form of the so called reciprocal system [3] 3) The main aim of the paper is to formulate the most general sufficient conditions for P i and S i such that systems (1.1), (1.2) have the same oscillatory properties.
Recall now some results from the continuous case which we are going to extend to (1.1).Consider the continuous counterpart of (1.1) -the differential Hamiltonian system Let P(t) be a 2n × 2n continuously differentiable matrix and suppose that the matrix P(t) is symplectic, i.e.P T (t)JP(t) = J.Then the transformation transforms (1.4) into another Hamiltonian system y = Ā(t)y + B(t)z, z = − C(t)y − ĀT (t)z, (1.6) where the matrices Ā(t), B(t), C(t) may be expressed via A(t), B(t), C(t) and blocks of P(t) (see [1]).The natural problem is to look for invariants of the above transformation, in particular, to ask when this transformation preserves oscillatory properties of transformed systems.
If P(t) = J in (1.5) and the matrices B(t), C(t) are nonnegative definite it has been shown in [15] that (1.4) is nonoscillatory iff (1.6) is nonoscillatory.This statement is now commonly referred as reciprocity principle for Hamiltonian systems.It has been shown that the reciprocitytype statement extends under natural additional assumptions to general transformation (1.5) (see [5]).Discrete analogs of these results based on the reciprocity principle for the discrete Hamiltonian systems [13] were presented for the first time in [3,Theorem 3].Later this principle was generalized for symplectic systems (1.1) in [7,11,12].
In this paper we formulate the most general reciprocity-type statements for symplectic system (1.1) (see Theorem 3.3).Previous versions of reciprocity-type statements in [3,7,11] are based on the assumptions that some symmetric matrices associated with S i and P i are nonnegative definite.For example, it was proved in [11,Corollary 3.6] that system (1.1) and the reciprocal system (1.3) oscillate and do not oscillate simultaneously under the assumption where the inequality A ≥ 0 (A ≤ 0) means that A = A T is nonnegative (nonpositive) definite.However, conditions of the given type impose serious restrictions on the applicability of the discrete transformation theory.For example, for the Fibonacci sequence y i+2 = y i+1 + y i rewritten in form (1.1), assumption (1.8) does not hold (see Section 4).Condition (1.8) was generalized to the case ind(A i B T i ) = ind(A T i C i ) (see Theorem 3.2 and formula (3.11) in [12]), where ind A is the number of negative eigenvalues of A = A T .However, [12,Theorem 3.2] deals with the constant transformation matrix P i = P in (1.2).The main theorem of this paper covers and explains all these special cases.The paper is organized as follows.In the next section we recall basic facts concerning oscillatory properties of (1.1) (see [3,14]).We also recall relatively new results of the comparative index theory for (1.1) (see [9][10][11][12]) and complete their by new relations between the number of focal points for solutions of (1.1) and (1.2).In Section 3 we prove the main result of the paper (see Theorem 3.3) and its corollaries.In Section 4 we provide several examples illustrating the results of Section 3.

The comparative index in the transformation theory
We will use the following notation.For a matrix A, we denote by A T , A −1 , A −T , A † , rank A, ind A, A ≥ 0, A ≤ 0, respectively, its transpose, inverse, transpose and inverse, Moore-Penrose pseudoinverse, rank (i.e., the dimension of its image), index (i.e., the number of its negative eigenvalues), positive semidefiniteness, negative semidefiniteness.We also use the notation ∆A k for the forward difference operator A k+1 − A k and the notation A i | N M for the difference A N − A M .By I and 0 we denote the identity and zero matrices of appropriate dimensions.
Oscillatory properties of discrete symplectic systems are defined using the concept of focal points of conjoined bases of (1.1).A 2n × n matrix solution Y = X U of (1.1) is said to be a conjoined basis of this system if Note that if (2.1) holds for a fixed i = i 0 , then it holds for any i ∈ Z.The concept of the multiplicity of a focal point of a conjoined basis was introduced by W. Kratz [14] as follows.
Given a conjoined basis, introduce the matrices Then obviously M i T i = 0 and it can be shown (see [14]) that the matrix P i is symmetric.The multiplicity of a forward focal point of a conjoined basis Y = X U in the interval (i, i + 1] is defined as the number m(i) The number of focal points q(i) of a conjoined basis of (1.2) can be defined similarly.Recall (see [3]) that the conjoined basis Y (M) i of (1.1) given by the initial conditions M = I is said to be the principal solution of (1.1) at M. System (1.1) is said to be nonoscillatory (see [3]), if there exists M ∈ N such that the principal solution at M of (1.1) has no focal points in the discrete interval (M, ∞), i.e., m(i) = 0 for i ∈ (M, ∞).In the opposite case (1.1) is said to be oscillatory.
Define the numbers of focal points in (M, for conjoined bases Y i and Ỹi = P i Y i . Another important notion we use is the concept of the comparative index as introduced and treated in [9][10][11][12].We define the comparative index for 2n × n matrices Y = X U , Ŷ = X Û with condition (2.1) using the notation where w(Y, Ŷ) is the Wronskian given by The comparative index is defined by For the special case Y := Y k+1 , Ŷ := S k [0 I] T the numbers µ 1 and µ 2 are actually equal to the quantities rank M k and ind P k from the definition of the multiplicity of a forward focal point (see [10,Lemma 3.1]).Based on this connection and properties of the comparative index [10] we prove the following result of the transformation theory of (1.1).
Lemma 2.1.Let Y i , Ỹi = P i Y i be conjoined bases of (1.1) and (1.2), then ) where m(i) and q(i) are the numbers of focal points in (i, i + 1] for Y i and Ỹi = P i Y i , respectively. Proof. where the Wronskians are evaluated according to (2.3).It is easy to verify that So we have rank w ) and then the second representation of u i in (2.6) follows from the identity Note that we can interchange the roles of systems (1.1) and (1.2) in Lemma 2.1.In this case we have to replace P i , Y i , S i by P −1 i , Ỹi , Si , respectively.This approach makes it possible to derive new formulas presenting the difference q(i) − m(i).
Lemma 2.2.Under the notation of Lemma 2.1 we have for u i given by (2.6).
For the most important special case P i = J we have the following corollary to Lemmas 2.1, 2.2.
Corollary 2.3.For the case P i = J the sequences u i , ũi in Lemmas 2.1, 2.2 are defined by the formulas ) where A i , B i , C i , D i are the blocks of S i in (1.1).Similarly, for the comparative indexes µ( Ỹi , ) in the left hand sides of (2.5), (2.7) for the case P i = J we have the representations ) Proof.Formulas (2.10), (2.11), (2.12), (2.13) are verified by direct computations according to the definition of the comparative index.Note that for the special case P i = J we have rank([I 0]P i [0 I] T ) = n, then u i = ũi according to (2.9).

Generalized reciprocity principle
Based on Lemmas 2.1, 2.2 we can derive connections between total numbers of focal points (2.2) of conjoined bases of (1.1), (1.2).
Remark 3.2.Note that by (2.4) and (2.9) for the partial sums S(M, N), S(M, N) in (3.1) we have the estimate It follows from (3.2) that either the partial sums S(M, N) and S(M, N) are simultaneously bounded for a fixed M ∈ Z as N → ∞, i.e. the inequalities hold for some C(M) > 0, C(M) > 0 or these sums are simultaneously unbounded.
The main result of this paper is the following.Proof.
1. Consider the proof of (i).Note first that in the definition of nonoscillation of (1.1) (see Section 2) it is possible to replace the principal solution Y proved in [8].Our purpose is to show that under assumption (3.3) the similar inequality holds for the numbers of focal points l(Y, M, N), l( Ỹ, M, N) of conjoined bases of (1.1), (1.2).Indeed, by (3.3), (3.1), and (2.4) we have So we have proved that (3.3) implies (3.5).Since l(Y, M, N), l( Ỹ, M, N) are the partial sums of the series with natural or zero members, then, by (3.5), l(Y, M 1 , N) = 0 for for some M 1 and for all N ≥ M 1 iff l( Ỹ, M 2 , N) = 0 for some M 2 and for all N > M 2 .So we see that (1.1) is nonoscillatory if and only if so is (1.2).

It is easy to see that assertion (ii) is equivalent to assertion (iii).
The proof is completed.
Note that Theorem 3.3 answers the question about the oscillation (nonoscillation) of one system ((1.1) or (1.2)) provided we posses information on oscillation (nonoscillation) of other one in all situations except the case when S(M, N), S(M, N) are unbounded (see Theorem 3.3 (iii)) and one of the systems ((1.1) or (1.2)) is oscillatory.This case demands additional information.For example, we can offer the following criterion.The following theorem formulates the simplest sufficient conditions for the boundedness of S(M, N), S(M, N) in (3.1).Theorem 3.5.Systems (1.1) and (1.2) oscillate and do not oscillate simultaneously if at least one of the sequences u i , ũi given by (2.6), (2.8) tends to zero as i → ∞, i.e. there exists M > 0 such that In particular, for P i = J we have the corollary to Theorem 3.5.
Corollary 3.6.Systems (1.1) and (1.3) oscillate and do not oscillate simultaneously if there exists and (3.9) is equivalent to (i) Note that for the case rank([I 0]P i [0 I] T ) = const, i ≥ M conditions (3.7) and (3.8) are equivalent according to (2.9).In particular, rank([I 0]P i [0 I] T ) = n for the case P i = J (see Corollary ) will be satisfied if we assume or In particular, for the case P = J from (3.11) we derive conditions (1.8) while (3.12) implies In the next section we give examples illustrating the applicability of Theorems 3.3, 3.5.

Applications
The following example illustrates Theorem 3.3 (iii).According to Theorem 3.3 (iii), if (1.1) does not oscillate and the sum S(M, N) is unbounded, then (1.2) is necessary oscillatory.
Example 4.1.Consider system (1.1) with the matrix S i = 1 0 3 1 .It is easy to verify that for any conjoined basis the number of focal points m(i) = 0, i.e. this system is nonoscillatory.For the transformation matrix P i = J the assumptions of Theorem 3.3 (iii) are satisfied by virtue of u M, N) is unbounded.The transformed system (1.3) with the matrix Si = J T S i J = 1 −3 0 1 is oscillatory.Indeed, the conjoined basis The following example presents the situation when conditions (1.8) do not hold, but (3.9) is true.
Since for the principal solution at 0 we have y 0 = 0, y 1 = 1, y i+2 = y i + y i+1 > 0, then the number of focal points of this solution is defined as m(i) = m 2 (i) = ind(−1) i , i ≥ 1, i.e. system (1.1) is oscillatory.Note that for the given system condition (1.8) does not hold, but (3.9) is true for all i, then the transformed system (1.3) is also oscillatory.Point out that for the given example conditions (3.13) are trivially satisfied by D i = 0.
Example 4.3.This example illustrates the situation when condition (3.9) does not hold, but (3.3) is true.Consider system (1.1) with the matrix This system is nonoscillatory because it is derived using the low triangular transformation matrix 1 0 −(−2) i 1 applied to conjoined bases Y i of the nonoscillatory symplectic system Y i+1 = 1 1 0 1 Y i .Indeed, the number of focal points of the conjoined basis Y i = [1 0] of the last system equals m(i) = ind(1) = 0 and low triangular transformation matrices do not change the number of focal points (see [6,Corollary 2.2]).For the matrix S i given by (4.1) we have We see that condition (3.9) is not satisfied, but the partial sum S(M, N) = ∑ N i=M (−1) i is bounded, then reciprocal system (1.3) associated with (1.1) given by (4.1) is nonoscillatory by Theorem 3.3 (i).
The last example is devoted to the so-called trigonometric difference systems [4] The principal solution at 0 for this system has the upper blocks X i given by then we can calculate the numbers of focal points of (1.1) which form the periodic sequence with the minimal period 4: So we see that system (1.1) is oscillatory.Note that the block B i in (1.1) associated with (4.2) is singular for all i and rank B i = 1.Introduce the following orthogonal transformation matrix The matrix of the transformed system (1.2) takes the form (4.2) where the angles ϕ 1,2 i have to be replaced by φ1,2 Using (4.4), (4.5) it is possible to show that u i = 0, i ≥ 0. Assume first that for the fixed j = 1, 2 we have ϕ j i = πk, then ϑ j i = 0 while θ j i > 0 because of sin(α i+1 ) > 0, sin(ϕ Then, for the given case ind(θ j i ) = ind(ϑ j i ) = 0.For the opposite case sin(ϕ j i ) = 0 we have that the signs of sin(ϕ j i ) and sin(ϕ j i − ∆α i ) are the same because of the definition of the angles in (4.2), (4.3).Then for this case ind(θ j i ) − ind(ϑ j i ) = 0. Applying Theorem 3.5 we see that system (1.2) is oscillatory.This fact can be verified by a direct computation.We have that q(0) = q(1) = 1, q(2) = q(3) = 0, q(i + 4) = q(i), i ≥ 0, where q(i) is the number of focal points of the transformed principal solution Ỹi .

Example 4 . 4 .
and illustrates recent results of the transformation theory in[6, Lemma 3.2].Consider the trigonometric difference system (1.1) for M = 0 with the orthogonal matrix The upper blocks of the transformed principal solution are Xi = with (4.3) regularizes the system (1.1) in the following sense: transformed system (1.2) has the nonsingular block Bi and, additionally, the transformed principal solution has the nonsingular upper block Xi .Moreover, the transformation with (4.3) preserves the oscillation properties of (1.1), i.e. system (1.2) is also oscillatory.Indeed, applying (2.5) we haveu i = µ(P i+1 [0 I] T , Si [0 I] T ) − µ * (P T i [0 I] T , S T i [0 I] T), where µ is the comparative index and µ * is the dual comparative index.As can be verified by a direct computationµ(P i+1 [0 I] T , Si [0 I] T ) = ind(diag(θ 1 i , i ) sin(ϕ j i − ∆α i ) sin(α i+1 ) , (4.4) µ * (P T i [0 I] T , S T i [0 I] T ) = ind(diag(ϑ 1 i , [11,proof of the first representation of the sequence u i in (2.6) is given in[11, Lemma  3.1].Consider the proof of the second one.By Property 5 in[10, p. 448]