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 [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 . 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  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  we introduce a new definition of adiabatic invariant of system (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
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
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)
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.
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 ) that the change of adiabatic invariant approaches zero with exponential speed as approaches zero.
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 ():