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Rofe-Beketov formula for symplectic systems
Advances in Difference Equations volume 2012, Article number: 104 (2012)
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.
In this article, we solve the open problem presented in , Remark 1(iv)] concerning the Rofe-Beketov formula for symplectic systems on time scales (see Theorem 3), i.e., for the first order dynamic system
on a time scale . 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 (). 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 , Remark 6] and , Theorem 3.32], respectively.
Let us consider the second order Sturm-Liouville differential equation
where on J and , . Let x be a solution of (1) without zeros in and . Then the second linearly independent solution y of (1) on can be expressed with using the so-called d’Alembert formula (or the Reduction of Order Theorem) as
see, e.g., Theorem 5.56], and these solutions are normalized, i.e.,
The Rofe-Beketov formula improves identity (2) and it gives a similar result without the assumption for . More precisely, if x is a nontrivial solution of (1) then the function
represents the second linearly independent solution of equation (1) on J and it holds . This statement was established in , Lemma 2] and it is a generalization of the original Rofe-Beketov formula presented for equation (1) with in , 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].
In the year 2005, identity (3) was generalized for the linear Hamiltonian differential system
where A, B, C are matrix-valued locally integrable complex functions and , , see , Theorem 6.5]. This result is recalled in the following proposition, where we denote the Hermitian component of the matrix A by ReA, i.e., , see e.g., pp.268-269] or , Facts 3.7.27-3.7.29].
Proposition 1 (Rofe-Beketov formula for (H))
Ifis amatrix solution of (H) such thatand, then the pair
where we put
forms the second linearly independentmatrix-valued solution of (H) satisfying the condition, i.e., the solutionsandare normalized. Moreover, the conditionsandhold also true.
The Rofe-Beketov formula was also established for the second order Sturm-Liouville dynamic equation
where denotes a time scale, see , Theorem 1]. In addition, since this result was new in the discrete time case, i.e., , the Rofe-Beketov formula for the second order Sturm-Liouville difference equation
was presented in , 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 , see also . 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.
A time scale is any nonempty closed subset of real numbers . With respect to the standard terminology, the forward jump operator and the backward jump operator are introduced as
respectively, and simultaneously we put and . The graininess function is defined as .
Let be a point such that . It is said to be left-dense and left-scattered if and , respectively. On the other hand, a point satisfying is called right-dense and right-scattered if and , respectively. In addition, for we put and for we have . We also use the following notation
For a complex-valued function f and we define , i.e., the Δ-derivative of f at t, as the number (provided it exists) with the property that for any given there is a neighborhood of t (i.e., for some ) such that
where we used the notation . We note that is not well defined if exists and is left-scattered.
The rule for the differentiation of a product fg has the well-known form
and it also holds
A complex function f is called regressive on if
where I denotes an appropriate identity matrix. A function f is called regulated if its right-hand limit exists (finite) at all right-dense points and the left-hand limit exists (finite) at all left-dense points . Provided a function f is regulated and it is continuous at each right-dense point , it is called rd-continuous (we write ) on . A function f is said to be piecewise rd-continuous () on if it is regulated and if f is rd-continuous at all but possibly finitely many right-dense points . A function f is said to be rd-continuously Δ-differentiable () on if exists for all and on . Finally, a function f is said to be piecewise rd-continuously Δ-differentiable () on if f is continuous on and exists at all except of possibly finitely many points , and on .
It is also well known that for any rd-continuous function on there exists an antiderivative F, i.e., a function satisfying for all . A time scale integral of a function f over a time scale interval , where , is defined as
for any antiderivative F of f.
In this article, we are interested in the symplectic system on time scales
where the matrix-valued complex function satisfies
With using the block notation and , system () can be written in the form
where are , , matrix-valued complex functions and the coefficients are matrix-valued complex functions such that on . Identity (8) can be also written in this block notation as (we omit the argument t)
Moreover, we note that identity (8) implies the symplecticity of the matrix on , i.e., . Since every symplectic matrix is invertible, it follows that the matrix-valued function is regressive on . Hence the existence of a unique solution for any (vector or matrix) initial value problem is a consequence of , Theorem 5.7] or , Theorem 5.8], see also , Remark 1].
The theory of symplectic difference systems was initiated in  as the discrete counterpart of system (H), while the study of system () originates in . It is a well-known fact that system () includes many special cases, e.g., equations (1), (4) and (5), system (H), discrete symplectic systems, and any even order Sturm-Liouville equation, see .
We associate with system () the Wronskian matrix given by
where and are any , , solutions of (). It is a direct consequence of that the Wronskian matrix takes a constant value on .
A solution of () is said to be a conjoined solution if , i.e., is a Hermitian matrix at one (and hence at any) . Two solutions are called normalized if . A solution is said to be a basis if on . It was shown in , Proposition 2.2] that the value of is also constant on .
3 Main result
The Reduction of Order Theorem for system () was published in , Remark 6] and it is recalled in the following proposition.
Proposition 2 (Reduction of Order Theorem for ())
Letbe a conjoined basis of () such that X is invertible on the time scale interval, . Then, where
is also a conjoined basis of (). Moreover, Z andare linearly independent (even normalized), i.e., they form a basis of the solution space for ().
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 ().
Theorem 3 (Rofe-Beketov formula for ())
Letbe a conjoined basis of (). Then the pairgiven by
for a fixedchosen without any restriction, solves also system (). Moreover, is a basis of () and it holds (i.e., they are linearly independent and normalized). In addition, constitutes also a conjoined basis if.
Proof The proof is based on the variation of parameters method. By differentiating identity (10)(i), and from (6), we get
Since solves system () and has to solve the same system, it follows from (13) that
Similarly, we obtain
Since Z is a conjoined basis, the multiplication of identity (14) by from the left and identity (15) by from the left yields in the sum
Analogously, multiplying identity (14) by from the left and identity (15) by from the left, we get the dynamic equation
Nevertheless, we can get the form of G explicitly. If we multiply identity (10)(i) from the left by and identity (10)(ii) from the left by , we obtain by adding these equations
Since Z is a conjoined basis and it has to be satisfied , 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 chosen without any restriction at the outset of the proof.
Now, from the facts that the value of is constant and the matrix G has a full rank, and since it holds
it follows that the solution forms a basis of (), i.e., (see also Remark (4)(i)). Moreover, from (10), (11), identity , and by the direct calculation, we obtain
i.e., is also a conjoined solution if on , and the proof is complete. □
The fact that represents a basis of (), follows also from the calculation
since the condition is equivalent with . This condition is obviously satisfied at and hence it holds true for any .
With using the block identities in (9) and identity (7), the function F can be also given in the form
On the other hand, with using identity (7) for system (), i.e., , and without the block notation (i.e., only with Z, , and ) the coefficient matrices in (11)-(12) can be written as and
It follows from identities (17) and (8) that it is satisfied for a point and, consequently, the pair forms a conjoined basis if it holds . Especially if , we have , , and . In this setting, system () has the form of (H) and the statement of Theorem 3 corresponds to Proposition 1, i.e., , Theorem 6.5].
Equation (4), where , can be written as the symplectic dynamic system
Theorem 3 yields for a nontrivial solution x of (4) that the second linearly independent solution y such that , can be expressed as
where we put
As mentioned in the introduction, Theorem 3 is new even in the discrete time case, i.e., for . In this case, system () can be written as the discrete symplectic system
where we put , , , and . Hence, if is a conjoined basis of (), then the pair given by (10), where
with , represents a solution of () such that , compare with , Theorem 3.32].
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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.
The author declares that he has no competing interests.
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Zemánek, P. Rofe-Beketov formula for symplectic systems. Adv Differ Equ 2012, 104 (2012). https://doi.org/10.1186/1687-1847-2012-104
- Rofe-Beketov formula
- symplectic system
- time scale
- Reduction of Order Theorem