- Gro Hovhannisyan
^{1}Email author

**2008**:712913

https://doi.org/10.1155/2008/712913

© Gro Hovhannisyan. 2008

**Received: **3 May 2008

**Accepted: **26 August 2008

**Published: **3 September 2008

## Abstracts

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

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

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

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

Theorem 1.1.

is true for some positive constant

is the Kroneker symbol ( if , and otherwise).

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.

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

Example 1.3.

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

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.

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.

## 2. WKB Series and WKB Estimates

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

In [16], the following theory was proved.

Theorem 2.1.

where is the Euclidean vector (or matrix) norm.

Remark 2.2.

Proof.

Lemma 2.3.

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

Proof.

Proof of Theorem 1.1.

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

Proof of Theorem 1.2.

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

we get recurrence relations (1.22).

Proof.

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

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

## References

- Fröman M, Fröman PO:
*JWKB-Approximation. Contributions to the Theory*. North-Holland, Amsterdam, The Netherlands; 1965:viii+138.MATHGoogle Scholar - Holmes MH:
*Introduction to Perturbation Methods, Texts in Applied Mathematics*.*Volume 20*. Springer, New York, NY, USA; 1995:ix+337.View ArticleGoogle Scholar - Birkhoff GD: Quantum mechanics and asymptotic series.
*Bulletin of the American Mathematical Society*1933, 39(10):681-700. 10.1090/S0002-9904-1933-05716-6MathSciNetView ArticleGoogle Scholar - Braun PA: WKB method for three-term recursion relations and quasienergies of an anharmonic oscillator.
*Theoretical and Mathematical Physics*1979, 37(3):1070-1081.View ArticleGoogle Scholar - Costin O, Costin R: Rigorous WKB for finite-order linear recurrence relations with smooth coefficients.
*SIAM Journal on Mathematical Analysis*1996, 27(1):110-134. 10.1137/S0036141093248037MATHMathSciNetView ArticleGoogle Scholar - Dingle RB, Morgan GJ:
methods for difference equations—I.
*Applied Scientific Research*1967, 18: 221-237.MATHMathSciNetView ArticleGoogle Scholar - Geronimo JS, Smith DT: WKB (Liouville-Green) analysis of second order difference equations and applications.
*Journal of Approximation Theory*1992, 69(3):269-301. 10.1016/0021-9045(92)90003-7MATHMathSciNetView ArticleGoogle Scholar - Wilmott P: A note on the WKB method for difference equations.
*IMA Journal of Applied Mathematics*1985, 34(3):295-302. 10.1093/imamat/34.3.295MATHMathSciNetView ArticleGoogle Scholar - Aulbach B, Hilger S: Linear dynamic processes with inhomogeneous time scale. In
*Nonlinear Dynamics and Quantum Dynamical Systems (Gaussig, 1990), Mathematical Research*.*Volume 59*. Akademie, Berlin, Germany; 1990:9-20.Google Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introduction with Application*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar - Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus.
*Results in Mathematics*1990, 18(1-2):18-56.MATHMathSciNetView ArticleGoogle Scholar - Hovhannisyan G:Asymptotic stability for
linear dynamic systems on time scales.
*International Journal of Difference Equations*2007, 2(1):105-121.MathSciNetGoogle Scholar - Littlewood JE: Lorentz's pendulum problem.
*Annals of Physics*1963, 21(2):233-242. 10.1016/0003-4916(63)90107-6MATHMathSciNetView ArticleGoogle Scholar - Wasow W: Adiabatic invariance of a simple oscillator.
*SIAM Journal on Mathematical Analysis*1973, 4(1):78-88. 10.1137/0504009MATHMathSciNetView ArticleGoogle Scholar - Hovhannisyan G, Taroyan Y:Adiabatic invariant for
-connected linear oscillators.
*Journal of Contemporary Mathematical Analysis*1997, 31(6):47-57.MathSciNetGoogle Scholar - Hovhannisyan G: Error estimates for asymptotic solutions of dynamic equations on time scales. In
*Proceedings of the 6th Mississippi State–UBA Conference on Differential Equations and Computational Simulations, 2007, San Marcos, Tex, USA, Electronic Journal of Differential Equations Conference*.*Volume 15*. Southwest Texas State University; 159-162.Google Scholar

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