WKB Estimates for Linear Dynamic Systems on Time Scales
© Gro Hovhannisyan. 2008
Received: 3 May 2008
Accepted: 26 August 2008
Published: 3 September 2008
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].
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 . 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
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.
for all .
Further frequently we are suppressing dependence on or . To distinguish the differentiation by or we show the argument of differentiation in parenthesizes: or
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
are satisfied. Then, estimate (1.18) is true.
and from Theorem 1.2 we have the following corollary.
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 ) that the change of adiabatic invariant approaches zero with exponential speed as approaches zero.
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 , the following theory was proved.
where is the Euclidean vector (or matrix) norm.
where the functions are defined in (1.9), (1.11).
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).
So if , then (1.26) and all other conditions of Theorem 1.2 are satisfied, and (1.18) is true with .
The author wants to thank Professor Ondrej Dosly for his comments that helped improving the original manuscript.
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