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# Rofe-Beketov formula for symplectic systems

- Petr Zemánek
^{1}Email author

**2012**:104

https://doi.org/10.1186/1687-1847-2012-104

© Zemánek; licensee Springer. 2012

**Received:**17 March 2012**Accepted:**18 June 2012**Published:**10 July 2012

## Abstract

We establish the Rofe-Beketov formula for symplectic systems on time scales. This result generalizes the well-known d’Alembert formula (or the Reduction of Order Theorem) and the Rofe-Beketov formula published for the second order Sturm-Liouville equations on time scales. Moreover, this result is new even in the discrete time case.

**MSC:**34N05, 26E70, 39A12.

## Keywords

- Rofe-Beketov formula
- symplectic system
- time scale
- Reduction of Order Theorem

## 1 Introduction

*i.e.*, for the first order dynamic system

on a time scale $\mathbb{T}$. We unify the Rofe-Beketov formulas published recently in the literature for the second order Sturm-Liouville differential, difference, and dynamic equations and also for the linear Hamiltonian differential systems and we generalize them by establishing its dynamic counterpart for system ($\mathbb{S}$). We point out that this result is new even in the discrete time case (see Remark 4(v)) and, moreover, it can be viewed as an improvement of the corresponding Reduction of Order Theorem, see [2], Remark 6] and [3], Theorem 3.32], respectively.

*J*and ${p}^{-1}$, $q\in {\mathrm{L}}_{\mathrm{loc}}^{1}(J)$. Let

*x*be a solution of (1) without zeros in $\tilde{J}\subseteq J$ and ${t}_{0}\in \tilde{J}$. Then the second linearly independent solution

*y*of (1) on $\tilde{J}$ can be expressed with using the so-called

*d’Alembert formula*(or the

*Reduction of Order Theorem*) as

*e.g.*[4], Theorem 5.56], and these solutions are normalized,

*i.e.*,

*x*is a nontrivial solution of (1) then the function

represents the second linearly independent solution of equation (1) on *J* and it holds $w[x,y]\equiv 1$. This statement was established in [5], Lemma 2] and it is a generalization of the original Rofe-Beketov formula presented for equation (1) with $p(\cdot )\equiv 1$ in [6], Lemma 2]. An application of identity (3) can be found in the study of the relative oscillation theory and spectral properties of differential operators associated with (1), see [5, 7].

where *A*, *B*, *C* are $n\times n$ matrix-valued locally integrable complex functions and $B={B}^{\ast}$, $C={C}^{\ast}$, see [8], Theorem 6.5]. This result is recalled in the following proposition, where we denote the *Hermitian component* of the matrix *A* by Re*A*, *i.e.*, $ReA:=(A+{A}^{\ast})/2$, see *e.g.*[9], pp.268-269] or [10], Facts 3.7.27-3.7.29].

**Proposition 1** (Rofe-Beketov formula for (H))

*If*$\left(\begin{array}{c}X\\ U\end{array}\right)$

*is a*$2n\times n$

*matrix solution of*(H)

*such that*${X}^{\ast}U={U}^{\ast}X$

*and*$det({X}^{\ast}X+{U}^{\ast}U)\ne 0$,

*then the pair*

*forms the second linearly independent*$2n\times n$*matrix*-*valued solution of* (H) *satisfying the condition*${X}^{\ast}\tilde{U}-{U}^{\ast}\tilde{X}\equiv I$, *i*.*e*., *the solutions*$\left(\begin{array}{c}X\\ U\end{array}\right)$*and*$\left(\begin{array}{c}\tilde{X}\\ \tilde{U}\end{array}\right)$*are normalized*. *Moreover*, *the conditions*${\tilde{X}}^{\ast}\tilde{U}={\tilde{U}}^{\ast}\tilde{X}$*and*$det({\tilde{X}}^{\ast}\tilde{X}+{\tilde{U}}^{\ast}\tilde{U})\ne 0$*hold also true*.

*i.e.*, $\mathbb{T}=\mathbb{Z}$, the Rofe-Beketov formula for the second order Sturm-Liouville difference equation

was presented in [1], Remark 1(ii)]. In this article we complete this treatment by proving the Rofe-Beketov formula for symplectic dynamic (and consequently for difference) systems.

## 2 Preliminaries on time scales and symplectic systems

The time scale calculus was originally published by Hilger in his dissertation [11], see also [12]. It is well known that this theory provides suitable tools for a study of differential and difference (among others) equations under the unified framework. The time scale theory has been developed in the last 20 years intensively and we refer to [13, 14] for the fundamental results achieved in this field.

*time scale*$\mathbb{T}$ is any nonempty closed subset of real numbers $\mathbb{R}$. With respect to the standard terminology, the

*forward jump operator*$\sigma :\mathbb{T}\to \mathbb{T}$ and the

*backward jump operator*$\rho :\mathbb{T}\to \mathbb{T}$ are introduced as

respectively, and simultaneously we put $inf\mathrm{\varnothing}:=sup\mathbb{T}$ and $sup\mathrm{\varnothing}:=inf\mathbb{T}$. The *graininess function*$\mu :\mathbb{T}\to [0,\mathrm{\infty})$ is defined as $\mu (t):=\sigma (t)-t$.

*left-dense*and

*left-scattered*if $\rho (t)=t$ and $\rho (t)<t$, respectively. On the other hand, a point $t\in \mathbb{T}$ satisfying $t<sup\mathbb{T}$ is called

*right-dense*and

*right-scattered*if $\sigma (t)=t$ and $\sigma (t)>t$, respectively. In addition, for $a=min\mathbb{T}$ we put $\rho (a)=a$ and for $b=max\mathbb{T}$ we have $\sigma (b)=b$. We also use the following notation

*f*and $t\in {\mathbb{T}}^{\kappa}$ we define ${f}^{\mathrm{\Delta}}(t)$,

*i.e.*, the Δ-derivative of

*f*at

*t*, as the number (provided it exists) with the property that for any given $\epsilon >0$ there is a neighborhood $\mathcal{U}$ of

*t*(

*i.e.*, $\mathcal{U}=(t-\delta ,t+\delta )$ for some $\delta >0$) such that

where we used the notation ${f}^{\sigma}(t):=f(\sigma (t))$. We note that ${f}^{\mathrm{\Delta}}(b)$ is not well defined if $b=max\mathbb{T}$ exists and is left-scattered.

*fg*has the well-known form

whenever ${f}^{\mathrm{\Delta}}(t)$ exists.

*f*is called

*regressive*on $J\subseteq {\mathbb{T}}^{\kappa}$ if

where *I* denotes an appropriate identity matrix. A function *f* is called *regulated* if its right-hand limit $f({t}^{+})$ exists (finite) at all right-dense points $t\in \mathbb{T}$ and the left-hand limit $f({t}^{-})$ exists (finite) at all left-dense points $t\in \mathbb{T}$. Provided a function *f* is regulated and it is continuous at each right-dense point $t\in \mathbb{T}$, it is called *rd-continuous* (we write $f\in {\mathrm{C}}_{\mathrm{rd}}$) on $\mathbb{T}$. A function *f* is said to be *piecewise rd-continuous* ($f\in {\mathrm{C}}_{\mathrm{prd}}$) on $\mathbb{T}$ if it is regulated and if *f* is rd-continuous at all but possibly finitely many right-dense points $t\in \mathbb{T}$. A function *f* is said to be *rd-continuously* Δ*-differentiable* ($f\in {\mathrm{C}}_{\mathrm{rd}}^{\mathrm{1}}$) on $\mathbb{T}$ if ${f}^{\mathrm{\Delta}}$ exists for all $t\in {\mathbb{T}}^{\kappa}$ and ${f}^{\mathrm{\Delta}}\in {\mathrm{C}}_{\mathrm{rd}}$ on ${\mathbb{T}}^{\kappa}$. Finally, a function *f* is said to be *piecewise rd-continuously* Δ*-differentiable* ($f\in {\mathrm{C}}_{\mathrm{prd}}^{\mathrm{1}}$) on ${[a,b]}_{\mathbb{T}}$ if *f* is continuous on $\mathbb{T}$ and ${f}^{\mathrm{\Delta}}(t)$ exists at all except of possibly finitely many points $t\in {\mathbb{T}}^{\kappa}$, and ${f}^{\mathrm{\Delta}}\in {\mathrm{C}}_{\mathrm{prd}}$ on ${\mathbb{T}}^{\kappa}$.

*antiderivative*

*F*,

*i.e.*, a function satisfying ${F}^{\mathrm{\Delta}}(t)=f(t)$ for all $t\in {\mathbb{T}}^{\kappa}$. A time scale

*integral*of a function

*f*over a time scale interval $[c,d]\cap \mathbb{T}$, where $c,d\in \mathbb{T}$, is defined as

for any antiderivative *F* of *f*.

*t*)

Moreover, we note that identity (8) implies the symplecticity of the matrix $(I+\mu \mathbb{S})$ on ${\mathbb{T}}^{\kappa}$, *i.e.*, ${(I+\mu \mathbb{S})}^{\ast}\mathcal{J}(I+\mu \mathbb{S})=\mathcal{J}$. Since every symplectic matrix is invertible, it follows that the matrix-valued function $\mathbb{S}(\cdot )$ is regressive on ${\mathbb{T}}^{\kappa}$. Hence the existence of a unique solution for any (vector or matrix) initial value problem is a consequence of [12], Theorem 5.7] or [13], Theorem 5.8], see also [2], Remark 1].

The theory of symplectic difference systems was initiated in [15] as the discrete counterpart of system (H), while the study of system ($\mathbb{S}$) originates in [2]. It is a well-known fact that system ($\mathbb{S}$) includes many special cases, *e.g.*, equations (1), (4) and (5), system (H), discrete symplectic systems, and any even order Sturm-Liouville equation, see [16].

*Wronskian matrix*given by

where $Z=\left(\begin{array}{c}X\\ U\end{array}\right)$ and $\tilde{Z}=\left(\begin{array}{c}\tilde{X}\\ \tilde{U}\end{array}\right)$ are any $2n\times m$, $1\le m\le n$, solutions of ($\mathbb{S}$). It is a direct consequence of ${W}^{\mathrm{\Delta}}[Z,\tilde{Z}]\equiv 0$ that the Wronskian matrix takes a constant value on $\mathbb{T}$.

A solution $Z=\left(\begin{array}{c}X\\ U\end{array}\right)$ of ($\mathbb{S}$) is said to be a *conjoined solution* if $W[Z,Z]\equiv 0$, *i.e.*, ${X}^{\ast}(t)U(t)$ is a Hermitian matrix at one (and hence at any) $t\in \mathbb{T}$. Two solutions $Z,\tilde{Z}$ are called *normalized* if $W[Z,\tilde{Z}]\equiv I$. A solution $Z=\left(\begin{array}{c}X\\ U\end{array}\right)$ is said to be a *basis* if $rankZ=n$ on $\mathbb{T}$. It was shown in [17], Proposition 2.2] that the value of $rankZ(t)$ is also constant on $\mathbb{T}$.

## 3 Main result

The Reduction of Order Theorem for system ($\mathbb{S}$) was published in [2], Remark 6] and it is recalled in the following proposition.

**Proposition 2** (Reduction of Order Theorem for ($\mathbb{S}$))

*Let*$Z=\left(\begin{array}{c}X\\ U\end{array}\right)$

*be a conjoined basis of*($\mathbb{S}$)

*such that*

*X*

*is invertible on the time scale interval*$I:=[a,b]\cap \mathbb{T}$, $a,b\in \mathbb{T}$.

*Then*$\tilde{Z}=\left(\begin{array}{c}\tilde{X}\\ \tilde{U}\end{array}\right)$,

*where*

*is also a conjoined basis of* ($\mathbb{S}$). *Moreover*, *Z* *and*$\tilde{Z}$*are linearly independent* (*even normalized*), *i*.*e*., *they form a basis of the solution space for* ($\mathbb{S}$).

Now, we improve this proposition in the main result by dropping the invertibility of *X*, *i.e.*, we state and prove the Rofe-Beketov formula for system ($\mathbb{S}$).

**Theorem 3** (Rofe-Beketov formula for ($\mathbb{S}$))

*Let*$Z=\left(\begin{array}{c}X\\ U\end{array}\right)$

*be a conjoined basis of*($\mathbb{S}$).

*Then the pair*$\tilde{Z}=\left(\begin{array}{c}\tilde{X}\\ \tilde{U}\end{array}\right)$

*given by*

*for a fixed*${t}_{0}\in \mathbb{T}$*chosen without any restriction*, *solves also system* ($\mathbb{S}$). *Moreover*, $\tilde{Z}$*is a basis of* ($\mathbb{S}$) *and it holds*$W[Z,\tilde{Z}]\equiv I$ (*i*.*e*., *they are linearly independent and normalized*). *In addition*, $\tilde{Z}$*constitutes also a conjoined basis if*${F}^{\ast}=F$.

*Proof*The proof is based on the variation of parameters method. By differentiating identity (10)(i), and from (6), we get

*Z*is a conjoined basis, the multiplication of identity (14) by ${X}^{\sigma}$ from the left and identity (15) by ${U}^{\sigma}$ from the left yields in the sum

*G*explicitly. If we multiply identity (10)(i) from the left by $-{U}^{\ast}$ and identity (10)(ii) from the left by ${X}^{\ast}$, we obtain by adding these equations

*Z*is a conjoined basis and it has to be satisfied $W[Z,\tilde{Z}]\equiv I$, it follows

*i.e.*, *G* is determined by (11). Consequently, identity (16) yields that the matrix-valued function *F* is given by (12) for a fixed ${t}_{0}\in \mathbb{T}$ chosen without any restriction at the outset of the proof.

*G*has a full rank, and since it holds

*i.e.*, $rank\tilde{Z}=n$ (see also Remark (4)(i)). Moreover, from (10), (11), identity ${X}^{\ast}U={U}^{\ast}X$, and by the direct calculation, we obtain

*i.e.*, $\tilde{Z}$ is also a conjoined solution if ${F}^{\ast}=F$ on $\mathbb{T}$, and the proof is complete. □

**Remark 4**

- (i)The fact that $\tilde{Z}$ represents a basis of ($\mathbb{S}$), follows also from the calculation${\tilde{X}}^{\ast}\tilde{X}+{\tilde{U}}^{\ast}\tilde{U}={F}^{\ast}{G}^{-1}F+{G}^{\ast}$

- (ii)With using the block identities in (9) and identity (7), the function
*F*can be also given in the form$\begin{array}{rcl}F& =& {\int}_{{t}_{0}}^{t}\left\{{G}^{\sigma}\right[{X}^{\ast}({\mathbb{C}}^{\ast}+\mathbb{B}+\mu ({\mathbb{A}}^{\ast}\mathbb{B}+{\mathbb{C}}^{\ast}\mathbb{D}))X\\ -{X}^{\ast}({\mathbb{A}}^{\ast}+\mathbb{A}+\mu ({\mathbb{A}}^{\ast}\mathbb{A}+{\mathbb{C}}^{\ast}\mathbb{C}))U\\ +{U}^{\ast}({\mathbb{D}}^{\ast}+\mathbb{D}+\mu ({\mathbb{B}}^{\ast}\mathbb{B}+{\mathbb{D}}^{\ast}\mathbb{D}))X\\ -{U}^{\ast}({\mathbb{B}}^{\ast}+\mathbb{C}+\mu ({\mathbb{B}}^{\ast}\mathbb{A}+{\mathbb{D}}^{\ast}\mathbb{C}))U\left]G\right\}(s)\mathrm{\Delta}s.\end{array}$

*i.e.*, ${Z}^{\sigma}=(I+\mu \mathbb{S})Z$, and without the block notation (

*i.e.*, only with

*Z*, $\tilde{Z}$, and $\mathbb{S}$) the coefficient matrices in (11)-(12) can be written as $G={({Z}^{\ast}Z)}^{-1}$ and

- (iii)
It follows from identities (17) and (8) that it is satisfied ${F}^{\ast}(t)=F(t)$ for a point $t\in \mathbb{T}$ and, consequently, the pair $\tilde{Z}$ forms a conjoined basis if it holds $\mu (t)=0$. Especially if $\mathbb{T}=\mathbb{R}$, we have $\sigma (t)=t$, $\mu (t)=0$, and ${f}^{\mathrm{\Delta}}(t)={f}^{\prime}(t)$. In this setting, system ($\mathbb{S}$) has the form of (H) and the statement of Theorem 3 corresponds to Proposition 1,

*i.e.*, [8], Theorem 6.5]. - (iv)Equation (4), where $1/p,q\in {\mathrm{C}}_{\mathrm{prd}}$, can be written as the symplectic dynamic system$(\begin{array}{c}x\\ p{x}^{\mathrm{\Delta}}\end{array}{)}^{\mathrm{\Delta}}=(\begin{array}{cc}0& 1/p\\ q& \mu q/p\end{array}\left)\right(\begin{array}{c}x\\ p{x}^{\mathrm{\Delta}}\end{array}).$

*x*of (4) that the second linearly independent solution

*y*such that $p(x{y}^{\mathrm{\Delta}}-{x}^{\mathrm{\Delta}}y)\equiv 1$, can be expressed as

- (v)

with ${\mathbb{S}}_{k}=\left(\begin{array}{cc}{\mathbb{A}}_{k}& {\mathbb{B}}_{k}\\ {\mathbb{C}}_{k}& {\mathbb{D}}_{k}\end{array}\right)$, represents a solution of (${\mathbb{S}}_{\mathbb{Z}}$) such that $W[Z,\tilde{Z}]=I$, compare with [3], Theorem 3.32].

## Declarations

### Acknowledgements

The research was supported by the Czech Science Foundation under grant P201/10/1032 and also by the Operational Programme “Education for Competitiveness” under the project CZ.1.07/2.3.00/30.0009. The author wishes to thank the anonymous referees for the detailed reading of the manuscript and for their comments which helped to improve the presentation of the results.

## Authors’ Affiliations

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