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fference EquationsVolume 2008, Article ID 712913, 12 pages doi:10.1155/2008/712913 Research Article Systems on Time Scales Gro Hovhannisyan Kent State University, Stark Campus, 6000 Frank

fference Equations

Volume 2008, Article ID 712913, 12 pages

doi:10.1155/2008/712913

Research Article

Systems on Time Scales

Gro Hovhannisyan

Kent State University, Stark Campus, 6000 Frank Avenue NW, Canton, OH 44720-7599, USA

Correspondence should be addressed to Gro Hovhannisyan,ghovhann@kent.edu

Received 3 May 2008; Accepted 26 August 2008

Recommended by Ondˇrej Doˇsl ´y

We establish WKB estimates for 2× 2 linear dynamic systems with a small parameter ε on a time

scale unifying continuous and discrete WKB method We introduce an adiabatic invariant for 2×

2 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 ε

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

ε The proof is based on the truncation of WKB series and WKB estimates.

Copyrightq 2008 Gro Hovhannisyan This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Adiabatic invariant of dynamic systems on time scales

Consider the following system with a small parameter ε > 0 on a time scale:

where vΔis the delta derivative, vt is a 2-vector function, and

Atε  Aτ 



a11τ ε k a12τ

ε −k a21τ a22τ



, τ  tε, k is an integer. 1.2

WKB method1,2 is a powerful method of the description of behavior of solutions

of1.1 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 in4 8

The calculus of times scales was initiated by Aulbach and Hilger9 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 system1.1 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 ε→0, or t→∞.

It is well known13,14 that the change of adiabatic invariant of harmonic oscillator

is vanishing with the exponential speed as ε approaches zero, if the frequency is an analytic

function

In this paper, we prove that for the discrete harmonic oscillatoreven 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 ε in a special way.

A time scaleT is an arbitrary nonempty closed subset of the real numbers If T has a

left-scattered minimum m, thenTk  T − m, otherwise T k  T Here we consider the time scales with t ≥ t0, and sup T  ∞.

For t∈ T, we define forward jump operator

σt  inf{s ∈ T, s > t}. 1.3

The forward graininess function μ : T→0, ∞ is defined by

μt  σt − t. 1.4

If σt > t, we say that t is right scattered If t < ∞ and σt  t, then t is called right dense For f : T→R and t ∈ T k define the deltasee 10, 11 derivative fΔt to be the

numberprovided it exists with the property that for given any  > 0, there exist a δ > 0 and

a neighborhood U  t − δ, t δ ∩ T of t such that

|fσt − fs − fΔtσt − s| ≤ |σt − s| 1.5

for all s ∈ U.

For any positive ε define auxilliary “slow” time scales

with forward jump operator and graininess function

σ1τ  inf{sε ∈ T ε, sε > τ}, μ1τ  εμt, τ  tε. 1.7

Further frequently we are suppressing dependence on τ  tε or t To distinguish the

differentiation by t or τ we show the argument of differentiation in parenthesizes: fΔt 

fΔt t or fΔτ  fΔτ τ.

Assuming A, θ j ∈ C1

rdsee 10 for the definition of rd-differentiable function, denote

TrAτ  a11τ a22τ, det Aτ  a11τa22τ − a12τa21τ,

λτ 



TrAτ2− 4|Aτ|

2a12τ ,

1.8

Hovj t  θ2

j t − θ j tTrAτ det Aτ − εa12τ1 μθ ja11− θ j

a12

Δ

τ, 1.9

Q0τ Hov1− Hov2

θ1− θ2

, Q1τ  θ1− a11Hov2− θ2− a11Hov1

a12θ1− θ2 , 1.10

Kτ  2μtmax

j1,2



1 

 e j

e3−j



2Hovj

θ1− θ2



 εa121 μθ j

θ1− θ2

a11− θ j

a12

Δ

τ

 |θ j|



, 1.11

where j  1, 2, θ 1,2 t are unknown phase functions, · is the Euclidean matrix norm, and {e j t} j1,2are the exponential functions on a time scale10,11:

e j t ≡ e θj t, t0  exp

t

t0

lim log1 pθj sΔs

p < ∞, j  1, 2. 1.12 Using the ratio of Wronskians formula proposed in15 we introduce a new definition

of adiabatic invariant of system1.1

Jt, θ, v, ε  − v1tθ1t − a11τ − v2ta12τv1tθ2t − a11τ − v2a12τ

θ1− θ22te θ1te θ2t , 1.13

Theorem 1.1 Assume a12τ / 0, A, θ ∈ C1

rdTε , and for some positive number β and any natural

number m conditions

|1 μTrA Q0 μ2det A θ1Q0− Hov1|τ ≥ β, ∀τ ∈ T ε , 1.14

Kτ ≤ const, ∀τ ∈ Tε, 1.15

∞

tε



1 

 e j

e3−j



 Hovj

θ1− θ2



τΔτ ≤ C0ε m 1 , j  1, 2, 1.16

are satisfied, where the positive parameter ε is so small that

0≤ 2C01 Kτ

Then for any solution vt of 1.1 and for all t1, t2∈ T, the estimate

Jv, ε ≡ |Jt1, v, ε − Jt2, v, ε| ≤ C3ε m 1.18

is true for some positive constant C3.

Checking condition1.16 ofTheorem 1.1is based on the construction of asymptotic solutions in the form of WKB series

vt  C1eθ1t, t0 C2eθ2t, t0, 1.19

where τ  tε, and

θ 1,2 t  ∞

j0

ε j ζ j± τ, θΔ

1,2 t  ∞

k0

ε k 1 ζΔ

Here the functions ζ0 τ, ζ0−τ are defined as

ζ0±τ  TrA

2 ± a12λ, ζ1±τ  −1 μζ0±

2λ



λ ∓ a11− a22

2a12

Δ

where λτ is defined in 1.8, and ζ k τ, ζ k− τ, k  2, 3, are defined by recurrence

relations

ζ k± τ  ∓ 1 μζ0±

2λ

ζ

k−1±

a12

Δ

τ ∓ k−1

j1

ζ j±

2λ

ζ k−j±

a12 μ ζ k−1−j± − a11δ j,k−1

a12

Δ

τ



, 1.22

δ jkis the Kroneker symbolδ jk  1, if k  j, and δ kj 0 otherwise

Denote

Zζ0  a121 μζ0

ζ

m

a12

Δ

m

j1

ζ j



ζ m 1−j εζ m 2−j a12μ ζ m−j − a11δ j,m εζ m 1−j

a12

Δ

τ



.

1.24

In the nextTheorem 1.2by truncating series1.20:

θ1t  m

k0

ε k ζ k , θ2t  m

k0

where ζ k± t, k  1, 2, , m are given in 1.21 and 1.22, we deduce estimate 1.16 from condition1.26 below given directly in the terms of matrix Aτ.

Theorem 1.2 Assume that a12τ/0, A, θ ∈ C1

rdTε , and conditions 1.14, 1.15, 1.17, and

∞

tε



1 

 e j

e3−j



 Z j τ

θ1− θ2



Δτ ≤ C0, j  1, 2, 1.26

are satisfied Then, estimate1.18 is true.

Note that if a11  a22, then formulas1.21 and 1.22 are simplified:

ζ0±τ  a11τ ± a12λτ, ζ1±  −1 μζ0±τλΔτ

where from1.8

λτ  a12τa21τ

Taking m 1 in 1.25 and ζ0±t, ζ1±t as in 1.21, we have

θ1t  ζ0 t εζ1 t, θ2t  ζ0−t εζ1−t, 1.29 which means that in1.20 ζ2±  ζ3± · · ·  0, and from 1.24

Zζ0  ζ2

1 a121 μζ0

 ζ

1

a12

Δ

μa12ζ1

ζ

0− a11 εζ1

a12

Δ

Example 1.3 Consider system1.1 with a11  a22 Then for continuous time scale T  R we

have μ  0, and by picking m  1 in 1.25 we get by direct calculations ζ1  ζ1−and

Hovθ1  Hovθ2  Zζ0   Zζ0−. 1.31

In view of

Z1 Z2 ζ2

1 a12

ζ

1

a12

Δ



λτ

λ

2

− 2a12

 λτ

a12λ



τ  λ 1/2 τa−1

12τλ −1/2 τ ττ , 1.32

condition1.26 under the assumption Rλ  0 turns to

∞

0

a−1

12τλ −1/2 τa−1

12τλ −1/2 τ ττ Δτ < C0, 1.33 and fromTheorem 1.2we have the following corollary

Corollary 1.4 Assume that a−1

12 ∈ C10, ∞, λ ∈ C20, ∞, Rλτ ≡ 0, a11τ ≡ a22τ, and

1.33 is satisfied Then for ε ≤ 1/C0 estimate1.18 with m  1 is true for all solutions vt of

system1.1 on continuous time scale T  R.

If a12  1, then 1.33 turns to

∞

t0ε |λ −1/2 τλ −1/2 τ ττ |Δτ < C0, 1.34

and for λτ √a21  iτ −2γ it is satisfied for any real γ.

If λτ is an analytic function, then it is known (see [ 13 ]) that the change of adiabatic invariant approaches zero with exponential speed as ε approaches zero.

Example 1.5 Consider harmonic oscillator on a discrete time scale T  εZ,

uΔΔt w2tεut  0, t ∈ εZ, 1.35 which could be written in form1.1, where

A 



−w2tε 0



, v 



u

uΔ



Choosing m 1 from formulas 1.27 and 1.29 we have λτ  iwτ, and

θ1t  ζ0 εζ1  iwτ − εw 2wτΔτ−iεμwΔτ

2 , τ  tε,

θ2t  ζ0− εζ1−  −iwτ − εw 2wτΔτ iεμwΔτ

1.37

From1.13 we get

Jt, v, ε  v2t iwτv1tv2t − iwτv1t

2wτ − εμtwΔτ2e θ te θ t , 1.38

Jt, u, ε  uΔt

2 w2τu2t

2wτ − εμtwΔτ2e η , 1.39

η  θ1 θ2 μθ1θ2 −εw wΔτ μεwΔ

2

4w2 μ



w − εμwΔ

2

2

If we choose

wτ  aε2

τ2 bε3

τ3  a

t2 b

t3, λτ a21τ  iwτ, 1.41 then all conditions ofTheorem 1.2are satisfiedsee proof ofExample 1.5in the next section

for any real numbers b, a / 0, and estimate 1.18 with m  1 is true.

Note that for continuous time scale we have μ  0, and 1.39 turns to the formula of adiabatic invariant for Lorentz’s pendulum13:

Jt, v, ε  u2t t w2tεu2t

2 WKB series and WKB estimates

Fundamental system of solutions of1.1 could be represented in form

vt  ΨtC δt, 2.1 where Ψt is an approximate fundamental matrix function and δt is an error vector

function

Introduce the matrix function

Ht  1 μtΨ−1tΨΔt−1Ψ−1tAtΨt − ΨΔt. 2.2

In16, the following theory was proved

Theorem 2.1 Assume there exists a matrix function Ψt ∈ C1

rdT∞ such that H ∈ R

rd, the matrix function Ψ μΨ∇ is invertible, and the following exponential function on a time scale is bounded:

e Ht ∞, t  exp

∞

t lim log1 pHsΔs

p < ∞. 2.3

Then every solution of 1.1 can be represented in form 2.1 and the error vector function δt can be

estimated as

where · is the Euclidean vector (or matrix) norm.

Remark 2.2 If μ t ≥ 0, then from 2.4 we get

δt ≤ C e∞t HsΔs− 1. 2.5

Proof of Remark 2.2 Indeed if x ≥ 0, the function fx  x − log1 x is increasing, so fx ≥

f0, log1 x ≤ x, and from p ≥ 0, Ht ≥ 0 we get

log1 pHs

and by integration

∞

t lim log1 pHs

p Δs ≤

∞

or

e H t, ∞ − 1 ≤ −1 exp

∞

Note that from the definition

σ1τ  εσt, μ1τ  εμt, qΔt  εqΔτ τ. 2.9 Indeed

εσt  ε inf

s∈T {s, s > t}  inf

εs∈Tε {εs, s > t}  inf

εs∈Tε {εs, εs > εt}  σ1εt  σ1τ,

σ1τ  εσt, μ1τ  σ1tε − εt  εσt − t  εμt,

qεσt  qtε εμtqΔτ τ  qtε μtqΔt.

2.10

If a12τ/0, then the fundamental matrix Ψt in 2.1 is given by see 12

Ψt  e θ1t e θ2t

U1te θ1t U2te θ2t



, U j t  θj t − a11t

a12t . 2.11

Lemma 2.3 If conditions 1.14, 1.15 are satisfied, then

Ht ≤ 21 Kτβ max

j1,2



1 

e ej3−jt t Hovj t

θ1t − θ2t



, t ∈ T, 2.12

where the functions Hov j t, Kτ are defined in 1.9, 1.11.

Proof Denote

Ω  1 μΨ−1ΨΔ, M  Ψ−1AΨ − ΨΔ. 2.13

By direct calculationssee 12, we get from 2.11

M  θ 1

1− θ2

⎛

⎜−Hov1 −e2Hov2

e1

e1Hov1

e2 Hov2

⎞

⎟

⎠ , ΨΔΨ−1

a21 Q1 a22 Q0



. 2.14

Using2.14, we get

detΩ  detΨΩΨ−1  det1 μΨΔΨ−1  1 μQ0 TrA μ2det A a11Q0− a12Q1,

2.15 and from1.14

| detΩ|  |1 μQ0 TrA μ2det A a11Q0− a12Q1| ≥ β > 0,

Ω−1  | det Ω|Ωco ≤ | det Ω|Ω ≤ Ωβ , H  Ω−1M,

Ψ−1AΨ  θ 1

1− θ2

⎛

⎜

⎝

−θ2

1 θ1TrA − det A − e2θ22− θ2TrA det A

e1

e1θ2

1− θ1TrA det A

e2 θ22− θ2TrA det A

⎞

⎟

⎠



θ1 0

0 θ2



,

M ≤ 2 max

j1,2



1 

 e j

e3−j



 Hovj

θ1− θ2



.

2.16

So by using1.9, we have

Ψ−1AΨ ≤ 2 max

j1,2



1 

 e j

e3−j



 Hovj

θ1− θ2



 εa121 μθ j a11− θ j /a12Δτ

θ1− θ2



 |θ j|



Ω  1 μΨ−1AΨ − M ≤ 1 μΨ−1AΨ M.

2.17 From2.2, 2.13, 2.17, we get 2.12 in view of

H ≤ Ω−1 · M ≤ Ωβ M ≤ 1 K β M. 2.18

Proof of Theorem 1.1 From1.16 changing variable of integration τ  εs, we get

∞

t MsΔs ≤

∞

t 2 max

j1,2



1 

e ej3−js s Hovj s

θ1s − θ2s



Δs ≤ 2C0ε m , j  1, 2. 2.19

So using2.12, we get

∞

t HsΔs ≤

∞

t

1 Kεs

β MsΔs ≤ cC0ε m 2.20

From this estimate and2.5, we have

δt ≤ C e∞t HsΔs− 1≤ Ce C0cε m

− 1≤ eCC0cε m , 2.21

where ε is so small that1.17 is satisfied The last estimate follows from the inequality e x−1 ≤

ex, x ∈ 0, 1 Indeed because gx  ex 1 − e x is increasing for 0 ≤ x ≤ 1, we have

gx ≥ g0.

Further from2.1, 2.11, we have

v1 C1 δ1e θ1 C2 δ2e θ2, v2 C1 δ1U1e θ1 C2 δ2U2e θ2. 2.22

Solving these equation for C j δ j, we get

C1 δ1 v1U2− v2

U2− U1e θ1

, C2 δ2 v2− v1U1

U2− U1e θ2

By multiplicationsee 1.12, we get

Jt  C1 δ1tC2 δ2t  C1C2 C2δ1t C1δ2t δ1tδ2t,

Jt1 − Jt2  C2δ1t1 − δ1t2 C1δ2t1 − δ2t2 δ1t1δ2t1 − δ1t2δ2t2, 2.24

and using estimate2.21, we have

|Jt1, θ, v, ε − Jt2, θ, v, ε| ≤ C3ε m 2.25

Proof of Theorem 1.2 Let us look for solutions of1.1 in the form

vt  ΨtC, 2.26 whereΨ is given by 2.11, and functions θ jare given via WKB series1.20

Substituting series1.20 in 1.9, we get

Hovθ1  ∞

r,j0

ζ r ε r ζ j ε j  − TrA ∞

r0

ζ r ε r det A

a12ε



1 μ ∞

r0

ζr ε r

∞

j0 ζ j ε j − a11

a12

Δ

τ,

2.27

or

Hovθ1 ≡ ∞

k0

To make Hovθ1 asymptotically equal zero or Hovθ1 ≡ 0 we must solve for ζ kthe equations

bk τ  0, k  0, 1, 2 2.29

By direct calculations from the first quadratic equation

b0 ζ2

and the second one

b1τ  2ζ1ζ0− ζ1TrA a121 μζ0

ζ

0− a11

a12

Δ

we get two solutions ζ j±given by1.21 and 1.22 Note that

ζ0 − a11

a12  a22− a11

2a12 λ, ζ0−− a11

a12  a22− a11

2a12 − λ,

ζ1 − ζ1−  a12μλΔ 2 μTrA

2λ

a

11− a22

2a12

Δ

.

2.32

Furthermore fromk 1th equation

b k  2ζ0− TrAζ k a121 μζ0

ζ

k−1

a12

Δ

k−1

j1

ζ j



ζ k−j a12μ ζ k−1−j − a11δ j,k−1

a12

Δ

τ



 0,

2.33

we get recurrence relations1.22

In view ofTheorem 1.1, to proveTheorem 1.2it is enough to deduce condition1.16 from1.26 By truncation of series 1.20 or by taking

ζ k  ζ k−  0, k  m 1, m 2, , 2.34

we get1.25 Defining ζ j± , j  1, 2, , m as in 1.21 and 1.22, we have

b0 b1 · · ·  b m−1  b m  b m 3  b m 4  · · ·  0,

b m 1  a121 μζ0

ζ

m

a12

Δ m

j1

ζ j



ζ m 1−j a12μ ζ m−j − a11δj,m

a12

Δ

τ



,

b m 2 m

j1

ζ j



ζ m 2−j a12μ ζ m 1−j

a12

Δ

τ



.

2.35

Now1.16 follows from 1.26 in view of

Hovθk   ε m 1 b m 1 b m 2ε  ε m 1 Zk, k  1, 2. 2.36

Note that from1.13 and the estimates

log|1 pθ| ≤ log1 2pRθ p2|θ|2≤ 1

2|2pRθ p2|θ|2|,

log|1 pθ| ≤ log1 2pRθ p2|θ|2≤|2pRθ p2|θ|2|,

2.37

it follows

|e θ t, t0| ≤ exp

t

t0



Rθs μs|θs|2 2Δs, 2.38

|e θ t, t0| ≤ exp

t

t0



|θs|2 2Rθs

μs Δs, μs > 0. 2.39 Proof of Example 1.5 From1.37, 1.41, we have

θ1− θ2 i2wτ − εμwΔτ, θ1 θ2  −εw wΔτ , θ1θ2  εwΔ

2

4w2



w − εμwΔ

2

2

,

η1t  θ1− θ2

1 μθ2  2ia

t2 Ot−3, η2t  θ2− θ1

1 μθ1  −2ia

t2 Ot−3, τ −→ ∞,

2.40 and using2.39, we get



eθ1

e θ2



 ≤ |e η1| ≤ const, 

eθ2

e θ1



 ≤ |e η2| ≤ const. 2.41

Further for τ→∞

ζ1±  −λΔ

2λ ∓λΔ

2  1

τ bε − 3aμ

2aτ2 1

τ3



2μ2−3bεμ

2a − b2ε2

2a2 ± iaε2



Oτ−4,

Z1  ζ2

1 ζΔ

1 εζΔ

1 ζ1 Oτ−4  μ − ε

τ3 Oτ−4  Oτ−4, Z2 Z1 Oτ−4.

2.42

So if μ  ε, then 1.26 and all other conditions ofTheorem 1.2are satisfied, and1.18 is true

with m 1

Acknowledgment

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

References

1 M Fr¨oman and P O Fr¨oman, JWKB-Approximation Contributions to the Theory, North-Holland,

Amsterdam, The Netherlands, 1965

2 M H Holmes, Introduction to Perturbation Methods, vol 20 of Texts in Applied Mathematics, Springer,

New York, NY, USA, 1995

By direct calculationssee  12 , we get from 2. 11

M ... solve for ζ kthe equations

bk τ  0, k  0, 1, 2. 29

By... cC0ε m 2. 20

From this estimate and 2. 5, we have

δt ≤