Copyright © 2008 Gro Hovhannisyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish WKB estimates for 2×2 linear dynamic systems with a small parameter ε on a time scale unifying continuous and discrete WKB method. We introduce an adiabatic invariant for 2×2 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 ε 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 ε. 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:(1.1)where is the delta derivative, is a -vector function, and(1.2)

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 [4–8].

The calculus of times scales was initiated by Aulbach and Hilger [9–11] 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(1.3)The forward graininess function is defined by(1.4)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(1.5)for all .

For any positive define auxilliary “slow" time scales(1.6)with forward jump operator and graininess function(1.7)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(1.8)(1.9)(1.10)(1.11) where are unknown phase functions, is the Euclidean matrix norm, and are the exponential functions on a time scale [10, 11]:(1.12)

Using the ratio of Wronskians formula proposed in [15] we introduce a new definition of adiabatic invariant of system (1.1)(1.13)

Theorem 1.1. Assume and for some positive number and any natural number conditions (1.14) (1.15) (1.16) are satisfied, where the positive parameter is so small that (1.17) Then for any solution of (1.1) and for all , the estimate (1.18) 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(1.19)where and(1.20)Here the functions are defined as(1.21)where is defined in (1.8), and are defined by recurrence relations(1.22) is the Kroneker symbol ( if , and otherwise).

Denote(1.23)(1.24)

In the next Theorem 1.2 by truncating series (1.20):(1.25)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 (1.26) are satisfied. Then, estimate (1.18) is true.

Note that if , then formulas (1.21) and (1.22) are simplified:(1.27)where from (1.8)(1.28)Taking in (1.25) and as in (1.21), we have(1.29)which means that in (1.20) and from (1.24)(1.30)

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(1.31)In view of(1.32)condition (1.26) under the assumption turns to(1.33) 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 (1.34) 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 ,(1.35)which could be written in form (1.1), where(1.36)

Choosing from formulas (1.27) and (1.29) we have and(1.37)From (1.13) we get(1.38)or(1.39)(1.40)If we choose(1.41)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]):(1.42)

2. WKB Series and WKB Estimates

Fundamental system of solutions of (1.1) could be represented in form(2.1)where is an approximate fundamental matrix function and is an error vector function.

Introduce the matrix function(2.2) 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: (2.3) Then every solution of (1.1) can be represented in form (2.1) and the error vector function can be estimated as (2.4) where is the Euclidean vector (or matrix) norm.

Remark 2.2. If , then from (2.4) we get(2.5)

Proof. Indeed if , the function is increasing, so and from we get(2.6)and by integration(2.7)or(2.8)

Note that from the definition(2.9)Indeed(2.10) If , then the fundamental matrix in (2.1) is given by (see [12])(2.11)

Lemma 2.3. If conditions (1.14), (1.15) are satisfied, then (2.12) where the functions are defined in (1.9), (1.11).

Proof. Denote(2.13)By direct calculations (see [12]), we get from (2.11)(2.14)Using (2.14), we get(2.15)and from (1.14)(2.16)So by using (1.9), we have(2.17)

From (2.2), (2.13), (2.17), we get (2.12) in view of(2.18)

Proof of Theorem 1.1. From (1.16) changing variable of integration we get(2.19)

So using (2.12), we get(2.20)From this estimate and (2.5), we have(2.21)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(2.22)Solving these equation for , we get(2.23)By multiplication (see (1.12)), we get(2.24)and using estimate (2.21), we have(2.25)

Proof of Theorem 1.2. Let us look for solutions of (1.1) in the form(2.26)where is given by (2.11), and functions are given via WKB series (1.20).

Substituting series (1.20) in (1.9), we get(2.27) or(2.28)To make asymptotically equal zero or we must solve for the equations(2.29)

By direct calculations from the first quadratic equation(2.30)and the second one(2.31)we get two solutions given by (1.21) and (1.22). Note that(2.32) Furthermore from th equation(2.33)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(2.34)we get (1.25). Defining as in (1.21) and (1.22), we have(2.35)Now (1.16) follows from (1.26) in view of(2.36)

Note that from (1.13) and the estimates(2.37)it follows(2.38)(2.39)

Proof. From (1.37), (1.41), we have(2.40)and using (2.39), we get(2.41)Further for (2.42)So if , then (1.26) and all other conditions of Theorem 1.2 are satisfied, and (1.18) is true with .

Acknowledgment

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

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