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WKB Estimates for Linear Dynamic Systems on Time Scales
Advances in Difference Equations volume 2008, Article number: 712913 (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
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 ():
2. WKB Series and WKB Estimates
Fundamental system of solutions of (1.1) could be represented in form
where is an approximate fundamental matrix function and is an error vector function.
Introduce the matrix function
In , the following theory was proved.
Assume there exists a matrix function such that the matrix function is invertible, and the following exponential function on a time scale is bounded:
Then every solution of (1.1) can be represented in form (2.1) and the error vector function can be estimated as
where is the Euclidean vector (or matrix) norm.
If , then from (2.4) we get
Indeed if , the function is increasing, so and from we get
and by integration
Note that from the definition
If , then the fundamental matrix in (2.1) is given by (see )
If conditions (1.14), (1.15) are satisfied, then
where the functions are defined in (1.9), (1.11).
By direct calculations (see ), we get from (2.11)
Using (2.14), we get
and from (1.14)
So by using (1.9), we have
From (2.2), (2.13), (2.17), we get (2.12) in view of
Proof of Theorem 1.1.
From (1.16) changing variable of integration we get
So using (2.12), we get
From this estimate and (2.5), we have
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
Solving these equation for , we get
By multiplication (see (1.12)), we get
and using estimate (2.21), we have
Proof of Theorem 1.2.
Let us look for solutions of (1.1) in the form
where is given by (2.11), and functions are given via WKB series (1.20).
Substituting series (1.20) in (1.9), we get
To make asymptotically equal zero or we must solve for the equations
By direct calculations from the first quadratic equation
and the second one
we get two solutions given by (1.21) and (1.22). Note that
Furthermore from th equation
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
we get (1.25). Defining as in (1.21) and (1.22), we have
Now (1.16) follows from (1.26) in view of
Note that from (1.13) and the estimates
From (1.37), (1.41), we have
and using (2.39), we get
So if , then (1.26) and all other conditions of Theorem 1.2 are satisfied, and (1.18) is true with .
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The author wants to thank Professor Ondrej Dosly for his comments that helped improving the original manuscript.
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Hovhannisyan, G. WKB Estimates for Linear Dynamic Systems on Time Scales. Adv Differ Equ 2008, 712913 (2008). https://doi.org/10.1155/2008/712913
- Direct Calculation
- Vector Function
- Recurrence Relation
- Matrix Function
- Fundamental Matrix