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WKB Estimates for Linear Dynamic Systems on Time Scales


We establish WKB estimates for linear dynamic systems with a small parameter on a time scale unifying continuous and discrete WKB method. We introduce an adiabatic invariant for dynamic system on a time scale, which is a generalization of adiabatic invariant of Lorentz_s pendulum. As an application we prove that the change of adiabatic invariant is vanishing as approaches zero. This result was known before only for a continuous time scale. We show that it is true for the discrete scale only for the appropriate choice of graininess depending on a parameter . The proof is based on the truncation of WKB series and WKB estimates.

1. Adiabatic Invariant of Dynamic Systems on Time Scales

Consider the following system with a small parameter on a time scale:


where is the delta derivative, is a -vector function, and


WKB method [1, 2] is a powerful method of the description of behavior of solutions of (1.1) by using asymptotic expansions. It was developed by Carlini (1817), Liouville, Green (1837) and became very useful in the development of quantum mechanics in 1920 [1, 3]. The discrete WKB approximation was introduced and developed in [48].

The calculus of times scales was initiated by Aulbach and Hilger [911] to unify the discrete and continuous analysis.

In this paper, we are developing WKB approximations for the linear dynamic systems on a time scale to unify the discrete and continuous WKB theory. Our formulas for WKB series are based on the representation of fundamental solutions of dynamic system (1.1) given in [12]. Note that the WKB estimate (see (2.21) below) has double asymptotical character and it shows that the error could be made small by either or

It is well known [13, 14] that the change of adiabatic invariant of harmonic oscillator is vanishing with the exponential speed as approaches zero, if the frequency is an analytic function.

In this paper, we prove that for the discrete harmonic oscillator (even for a harmonic oscillator on a time scale) the change of adiabatic invariant approaches zero with the power speed when the graininess depends on a parameter in a special way.

A time scale is an arbitrary nonempty closed subset of the real numbers. If has a left-scattered minimum , then otherwise Here we consider the time scales with and

For , we define forward jump operator


The forward graininess function is defined by


If , we say that is right scattered. If and , then is called right dense.

For and define the delta (see [10, 11]) derivative to be the number (provided it exists) with the property that for given any there exist a and a neighborhood of such that


for all .

For any positive define auxilliary "slow" time scales


with forward jump operator and graininess function


Further frequently we are suppressing dependence on or . To distinguish the differentiation by or we show the argument of differentiation in parenthesizes: or

Assuming (see [10] for the definition of rd-differentiable function), denote


where are unknown phase functions, is the Euclidean matrix norm, and are the exponential functions on a time scale [10, 11]:


Using the ratio of Wronskians formula proposed in [15] we introduce a new definition of adiabatic invariant of system (1.1)


Theorem 1.1.

Assume and for some positive number and any natural number conditions


are satisfied, where the positive parameter is so small that


Then for any solution of (1.1) and for all , the estimate


is true for some positive constant

Checking condition (1.16) of Theorem 1.1 is based on the construction of asymptotic solutions in the form of WKB series


where and


Here the functions are defined as


where is defined in (1.8), and are defined by recurrence relations


is the Kroneker symbol ( if , and otherwise).



In the next Theorem 1.2 by truncating series (1.20):


where are given in (1.21) and (1.22), we deduce estimate (1.16) from condition (1.26) below given directly in the terms of matrix

Theorem 1.2.

Assume that and conditions (1.14), (1.15), (1.17), and


are satisfied. Then, estimate (1.18) is true.

Note that if , then formulas (1.21) and (1.22) are simplified:


where from (1.8)


Taking in (1.25) and as in (1.21), we have


which means that in (1.20) and from (1.24)


Example 1.3.

Consider system (1.1) with Then for continuous time scale we have and by picking in (1.25) we get by direct calculations and


In view of


condition (1.26) under the assumption turns to


and from Theorem 1.2 we have the following corollary.

Corollary 1.4.

Assume that and (1.33) is satisfied. Then for estimate (1.18) with is true for all solutions of system (1.1) on continuous time scale

If , then (1.33) turns to


and for it is satisfied for any real .

If is an analytic function, then it is known (see [13]) that the change of adiabatic invariant approaches zero with exponential speed as approaches zero.

Example 1.5.

Consider harmonic oscillator on a discrete time scale ,


which could be written in form (1.1), where


Choosing from formulas (1.27) and (1.29) we have and


From (1.13) we get




If we choose


then all conditions of Theorem 1.2 are satisfied (see proof of Example 1.5 in the next section) for any real numbers , and estimate (1.18) with is true.

Note that for continuous time scale we have and (1.39) turns to the formula of adiabatic invariant for Lorentz's pendulum ([13]):


2. WKB Series and WKB Estimates

Fundamental system of solutions of (1.1) could be represented in form


where is an approximate fundamental matrix function and is an error vector function.

Introduce the matrix function


In [16], the following theory was proved.

Theorem 2.1.

Assume there exists a matrix function such that the matrix function is invertible, and the following exponential function on a time scale is bounded:


Then every solution of (1.1) can be represented in form (2.1) and the error vector function can be estimated as


where is the Euclidean vector (or matrix) norm.

Remark 2.2.

If , then from (2.4) we get



Indeed if , the function is increasing, so and from we get


and by integration




Note that from the definition




If , then the fundamental matrix in (2.1) is given by (see [12])


Lemma 2.3.

If conditions (1.14), (1.15) are satisfied, then


where the functions are defined in (1.9), (1.11).




By direct calculations (see [12]), we get from (2.11)


Using (2.14), we get


and from (1.14)


So by using (1.9), we have


From (2.2), (2.13), (2.17), we get (2.12) in view of


Proof of Theorem 1.1.

From (1.16) changing variable of integration we get


So using (2.12), we get


From this estimate and (2.5), we have


where is so small that (1.17) is satisfied. The last estimate follows from the inequality Indeed because is increasing for we have

Further from (2.1), (2.11), we have


Solving these equation for , we get


By multiplication (see (1.12)), we get


and using estimate (2.21), we have


Proof of Theorem 1.2.

Let us look for solutions of (1.1) in the form


where is given by (2.11), and functions are given via WKB series (1.20).

Substituting series (1.20) in (1.9), we get




To make asymptotically equal zero or we must solve for the equations


By direct calculations from the first quadratic equation


and the second one


we get two solutions given by (1.21) and (1.22). Note that


Furthermore from th equation


we get recurrence relations (1.22).

In view of Theorem 1.1, to prove Theorem 1.2 it is enough to deduce condition (1.16) from (1.26). By truncation of series (1.20) or by taking


we get (1.25). Defining as in (1.21) and (1.22), we have


Now (1.16) follows from (1.26) in view of


Note that from (1.13) and the estimates


it follows



From (1.37), (1.41), we have


and using (2.39), we get


Further for


So if , then (1.26) and all other conditions of Theorem 1.2 are satisfied, and (1.18) is true with .


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The author wants to thank Professor Ondrej Dosly for his comments that helped improving the original manuscript.

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Correspondence to Gro Hovhannisyan.

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Hovhannisyan, G. WKB Estimates for Linear Dynamic Systems on Time Scales. Adv Differ Equ 2008, 712913 (2008).

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