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 [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).
Denote
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
or
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]):