Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 712913, 12 pages
doi:10.1155/2008/712913
Research Article
WKB Estimates for 2 × 2 Linear Dynamic
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 Ondřej Došlý
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.
Copyright q 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:
vΔ t Atεvt,
1.1
where vΔ is the delta derivative, vt is a 2-vector function, and
Atε Aτ a11 τ εk a12 τ
,
ε−k a21 τ a22 τ
τ tε, k is an integer.
1.2
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
2
Advances in Difference Equations
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 ε→0, or t→∞.
It is well known 13, 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 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 ε in a special way.
A time scale T is an arbitrary nonempty closed subset of the real numbers. If T has a
left-scattered minimum m, then Tk T − m, otherwise Tk 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 ∈ Tk define the delta see 10, 11 derivative f Δ t to be the
number provided 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 − f Δ tσt − s| ≤ |σt − s|
1.5
for all s ∈ U.
For any positive ε define auxilliary “slow” time scales
Tε {εt τ, t ∈ T}
1.6
with forward jump operator and graininess function
σ1 τ inf{sε ∈ Tε , sε > τ},
μ1 τ 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: f Δ t f Δt t or f Δ τ f Δτ τ.
1
see 10 for the definition of rd-differentiable function, denote
Assuming A, θj ∈ Crd
TrAτ a11 τ a22 τ,
det Aτ a11 τa22 τ − a12 τa21 τ,
TrAτ2 − 4|Aτ|
,
λτ 2a12 τ
a11 − θj Δ
Hovj t θj2 t − θj tTrAτ det Aτ − εa12 τ1 μθj τ,
a12
Q0 τ Hov1 − Hov2
,
θ1 − θ2
Q1 τ θ1 − a11 Hov2 − θ2 − a11 Hov1
,
a12 θ1 − θ2 1.8
1.9
1.10
2Hovj εa12 1 μθj a11 − θj Δ ej |θj | , 1.11
τ
1 θ − θ θ − θ
j1,2
e3−j a12
1
2
1
2
Kτ 2μtmax
Gro Hovhannisyan
3
where j 1, 2, θ1,2 t are unknown phase functions, · is the Euclidean matrix norm, and
{ej t}j1,2 are the exponential functions on a time scale 10, 11:
ej t ≡ eθj t, t0 exp
t
t0
log1 pθj sΔs
< ∞,
p
μs
p
j 1, 2.
lim
1.12
Using the ratio of Wronskians formula proposed in 15 we introduce a new definition
of adiabatic invariant of system 1.1
Jt, θ, v, ε −
v1 tθ1 t − a11 τ − v2 ta12 τv1 tθ2 t − a11 τ − v2 a12 τ
θ1 − θ2 2 teθ1 teθ2 t
,
1.13
0, A, θ ∈ C1rd Tε , and for some positive number β and any natural
Theorem 1.1. Assume a12 τ /
number m conditions
|1 μTrA Q0 μ2 det A θ1 Q0 − Hov1 |τ ≥ β,
Kτ ≤ const, ∀τ ∈ Tε ,
∞
ej Hovj m1
1 θ − θ τΔτ ≤ C0 ε ,
e
3−j
1
2
tε
∀τ ∈ Tε ,
1.14
1.15
j 1, 2,
1.16
are satisfied, where the positive parameter ε is so small that
0≤
2C0 1 Kτ m
ε ≤ 1.
β
1.17
Then for any solution vt of 1.1 and for all t1 , t2 ∈ T, the estimate
Jv, ε ≡ |Jt1 , v, ε − Jt2 , v, ε| ≤ C3 εm
1.18
is true for some positive constant C3 .
Checking condition 1.16 of Theorem 1.1 is based on the construction of asymptotic
solutions in the form of WKB series
vt C1 eθ1 t, t0 C2 eθ2 t, t0 ,
1.19
where τ tε, and
θ1,2 t ∞
εj ζj± τ,
j0
Δ
θ1,2
t ∞
k0
Δ
εk1 ζk±
τ.
1.20
Here the functions ζ0 τ, ζ0− τ are defined as
TrA
± a12 λ,
ζ0± τ 2
1 μζ0±
a11 − a22 Δ
λ∓
ζ1± τ −
τ,
2λ
2a12
1.21
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Advances in Difference Equations
where λτ is defined in 1.8, and ζk τ, ζk− τ, k 2, 3, . . . are defined by recurrence
relations
ζk± τ ∓
k−1 ζ ζ
ζk−1−j± − a11 δj,k−1 Δ
1 μζ0± ζk−1± Δ
j±
k−j±
τ ∓
μ
τ ,
2λ
a12
2λ a12
a12
j1
1.22
δjk is the Kroneker symbol δjk 1, if k j, and δkj 0 otherwise.
Denote
Z1 τ Zζ0 τ,
Z2 τ Zζ0− τ,
Zζ0 a12 1 μζ0 ζm
a12
1.23
Δ
m
ζm−j − a11 δj,m εζm1−j Δ
ζj ζm1−j εζm2−j a12 μ
τ .
a12
j1
1.24
In the next Theorem 1.2 by truncating series 1.20:
θ1 t m
εk ζk ,
θ2 t k0
m
εk ζk− ,
1.25
k0
where ζk± t, k 1, 2, . . . , m 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 Aτ.
1
0, A, θ ∈ Crd
Tε , and conditions 1.14, 1.15, 1.17, and
Theorem 1.2. Assume that a12 τ/
∞
tε
ej Zj τ Δτ ≤ C0 ,
1 e3−j θ1 − θ2 j 1, 2,
1.26
are satisfied. Then, estimate 1.18 is true.
Note that if a11 a22 , then formulas 1.21 and 1.22 are simplified:
ζ0± τ a11 τ ± a12 λτ,
ζ1± −
1 μζ0± τλΔ τ
,
2λτ
1.27
where from 1.8
a12 τa21 τ
λτ .
a12 τ
1.28
Taking m 1 in 1.25 and ζ0± t, ζ1± t as in 1.21, we have
θ1 t ζ0 t εζ1 t,
θ2 t ζ0− t εζ1− t,
1.29
which means that in 1.20 ζ2± ζ3± · · · 0, and from 1.24
Zζ0 ζ12
ζ1
a12 1 μζ0 a12
Δ
μa12 ζ1
ζ0 − a11 εζ1
a12
Δ
.
1.30
Gro Hovhannisyan
5
Example 1.3. Consider system 1.1 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
2
Z1 Z2 ζ1
a12
ζ1
a12
Δ
λτ
λ
2
− 2a12
λτ
a12 λ
τ
−1/2
λ1/2 τ a−1
ττ τ , 1.32
12 τλ
condition 1.26 under the assumption Rλ 0 turns to
∞
0
−1
a τλ−1/2 τ a−1 τλ−1/2 τ Δτ < C0 ,
τ τ
12
12
1.33
and from Theorem 1.2 we have the following corollary.
1
2
Corollary 1.4. Assume that a−1
12 ∈ C 0, ∞, λ ∈ C 0, ∞, Rλτ ≡ 0, a11 τ ≡ a22 τ, and
1.33 is satisfied. Then for ε ≤ 1/C0 estimate 1.18 with m 1 is true for all solutions vt of
system 1.1 on continuous time scale T R.
If a12 1, then 1.33 turns to
∞
t0 ε
|λ−1/2 τλ−1/2 τττ |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,
uΔΔ t w2 tεut 0,
t ∈ εZ,
1.35
which could be written in form 1.1, where
A
0
1
,
−w2 tε 0
v
u
.
uΔ
1.36
Choosing m 1 from formulas 1.27 and 1.29 we have λτ iwτ, and
θ1 t ζ0 εζ1 iwτ −
θ2 t ζ0− εζ1−
εwΔ τ iεμwΔ τ
−
,
2wτ
2
τ tε,
1.37
εwΔ τ iεμwΔ τ
.
−iwτ −
2wτ
2
From 1.13 we get
Jt, v, ε v2 t iwτv1 tv2 t − iwτv1 t
2
2wτ − εμtwΔ τ eθ1 teθ2 t
,
1.38
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Advances in Difference Equations
or
2
Jt, u, ε η θ1 θ2 μθ1 θ2 −
uΔ t w2 τu2 t
2
2wτ − εμtwΔ τ eη
,
2
2
εμwΔ
εwΔ τ μεwΔ μ
w
−
.
w
2
4w2
1.39
1.40
If we choose
wτ a b
aε2 bε3
3 2 3,
2
τ
τ
t
t
λτ a21 τ iwτ,
1.41
then all conditions of Theorem 1.2 are satisfied see proof of Example 1.5 in 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 pendulum 13:
Jt, v, ε u2t t w2 tεu2 t
.
4wtε
1.42
2. WKB series and WKB estimates
Fundamental system of solutions of 1.1 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−1 tΨΔ t−1 Ψ−1 tAtΨt − ΨΔ t.
2.2
In 16, the following theory was proved.
1
Theorem 2.1. Assume there exists a matrix function Ψt ∈ Crd
T∞ such that H ∈ Rrd , the
∇
matrix function Ψ μΨ is invertible, and the following exponential function on a time scale is
bounded:
∞
log1 pHsΔs
< ∞.
2.3
lim
eHt ∞, t exp
p
t p
μs
Then every solution of 1.1 can be represented in form 2.1 and the error vector function δt can be
estimated as
δt ≤ CeH ∞, t − 1,
where · is the Euclidean vector (or matrix) norm.
2.4
Gro Hovhannisyan
7
Remark 2.2. If μt ≥ 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
≤ Hs,
p
2.6
and by integration
∞
t
log1 pHs
Δs ≤
p
μs
p
lim
∞
t
HsΔs,
2.7
or
eH t, ∞ − 1 ≤ −1 exp
∞
t
HsΔs.
2.8
Note that from the definition
σ1 τ εσt,
μ1 τ εμt,
qΔ t εqΔτ τ.
2.9
Indeed
εσt ε inf{s, s > t} inf {εs, s > t} inf {εs, εs > εt} σ1 εt σ1 τ,
s∈T
εs∈Tε
σ1 τ εσt,
εs∈Tε
2.10
μ1 τ σ1 tε − εt εσt − t εμt,
qεσt qtε εμtqΔτ τ qtε μtqΔ t.
If a12 τ
/0, then the fundamental matrix Ψt in 2.1 is given by see 12
Ψt eθ2 t
eθ1 t
,
U1 teθ1 t U2 teθ2 t
Uj t θj t − a11 t
.
a12 t
2.11
Lemma 2.3. If conditions 1.14, 1.15 are satisfied, then
21 Kτ
Ht ≤
max
j1,2
β
ej t Hovj t ,
1
e3−j t θ1 t − θ2 t t ∈ T,
2.12
where the functions Hovj t, Kτ are defined in 1.9, 1.11.
Proof. Denote
Ω 1 μΨ−1 ΨΔ ,
M Ψ−1 AΨ − ΨΔ .
2.13
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Advances in Difference Equations
By direct calculations see 12, we get from 2.11
⎛
M
−Hov1 −
1 ⎜
⎝ e Hov
1
1
θ1 − θ2
e2
e2 Hov2 ⎞
e1 ⎟
⎠,
ΨΔ Ψ−1 Hov2
a12
a11
.
a21 Q1 a22 Q0
2.14
Using 2.14, we get
det Ω detΨΩΨ−1 det1 μΨΔ Ψ−1 1 μQ0 TrA μ2 det A a11 Q0 − a12 Q1 ,
2.15
and from 1.14
| detΩ| |1 μQ0 TrA μ2 det A a11 Q0 − a12 Q1 | ≥ β > 0,
Ω−1 Ω
Ω
Ωco ≤
≤
,
| det Ω| | det Ω|
β
H Ω−1 M,
⎞
e2 θ22 − θ2 TrA det A
2
2.16
θ
TrA
−
det
A
−
−θ
1
⎟
1
1 ⎜
θ1 0
e
−1
1
⎟
⎜
Ψ AΨ ⎠ 0 θ2 ,
θ1 − θ2 ⎝ e1 θ12 − θ1 TrA det A
θ22 − θ2 TrA det A
e2
⎛
M ≤ 2 max
j1,2
ej Hovj .
1 e3−j θ1 − θ2 So by using 1.9, we have
−1
Ψ AΨ ≤ 2 max
j1,2
ej
1 e
3−j
Hovj
θ − θ
1
2
εa12 1 μθj a11 − θj /a12 Δ τ |θj |
θ −θ
1
2
Ω 1 μΨ−1 AΨ − M ≤ 1 μΨ−1 AΨ M.
2.17
From 2.2, 2.13, 2.17, we get 2.12 in view of
H ≤ Ω−1 · M ≤
1K
Ω
M ≤
M.
β
β
2.18
Proof of Theorem 1.1. From 1.16 changing variable of integration τ εs, we get
∞
t
MsΔs ≤
∞
t
ej s Hovj s Δs ≤ 2C0 εm ,
2 max 1 j1,2
e3−j s θ1 s − θ2 s j 1, 2.
2.19
So using 2.12, we get
∞
t
HsΔs ≤
∞
t
1 Kεs
MsΔs ≤ cC0 εm .
β
2.20
Gro Hovhannisyan
9
From this estimate and 2.5, we have
∞
m
δt ≤ C e t HsΔs − 1 ≤ C eC0 cε − 1 ≤ eCC0 cεm ,
2.21
where ε is so small that 1.17 is satisfied. The last estimate follows from the inequality ex −1 ≤
ex, x ∈ 0, 1. Indeed because gx ex 1 − ex is increasing for 0 ≤ x ≤ 1, we have
gx ≥ g0.
Further from 2.1, 2.11, we have
v1 C1 δ1 eθ1 C2 δ2 eθ2 ,
v2 C1 δ1 U1 eθ1 C2 δ2 U2 eθ2 .
2.22
Solving these equation for Cj δj , we get
C1 δ1 v1 U2 − v2
,
U2 − U1 eθ1
C2 δ2 v2 − v1 U1
.
U2 − U1 eθ2
2.23
By multiplication see 1.12, we get
Jt C1 δ1 tC2 δ2 t C1 C2 C2 δ1 t C1 δ2 t δ1 tδ2 t,
Jt1 − Jt2 C2 δ1 t1 − δ1 t2 C1 δ2 t1 − δ2 t2 δ1 t1 δ2 t1 − δ1 t2 δ2 t2 ,
2.24
and using estimate 2.21, we have
|Jt1 , θ, v, ε − Jt2 , θ, v, ε| ≤ C3 εm .
2.25
Proof of Theorem 1.2. Let us look for solutions of 1.1 in the form
vt ΨtC,
2.26
where Ψ is given by 2.11, and functions θj are given via WKB series 1.20.
Substituting series 1.20 in 1.9, we get
Hovθ1 ∞
ζr εr ζj εj − TrA
r,j0
∞
ζr εr det A
r0
∞
a12 ε 1 μ ζr εr
∞
j0 ζj ε
j
a12
r0
− a11
2.27
Δ
τ,
or
Hovθ1 ≡
∞
bk τεk .
2.28
k0
To make Hovθ1 asymptotically equal zero or Hovθ1 ≡ 0 we must solve for ζk the equations
bk τ 0,
k 0, 1, 2 . . . .
2.29
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Advances in Difference Equations
By direct calculations from the first quadratic equation
b0 ζ02 − ζ0 TrA detA 0,
2.30
and the second one
b1 τ 2ζ1 ζ0 − ζ1 TrA a12 1 μζ0 ζ0 − a11
a12
Δ
0,
2.31
we get two solutions ζj± given by 1.21 and 1.22. Note that
ζ0 − a11 a22 − a11
λ,
a12
2a12
ζ1 − ζ1−
ζ0− − a11 a22 − a11
− λ,
a12
2a12
2 μTrA a11 − a22 Δ
a12 μλΔ .
2λ
2a12
2.32
Furthermore from k 1th equation
ζk−1
bk 2ζ0 − TrAζk a12 1 μζ0 a12
Δ
k−1 ζk−1−j − a11 δj,k−1 Δ
ζj ζk−j a12 μ
τ 0,
a12
j1
2.33
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
ζk ζk− 0,
k m 1, m 2, . . . ,
2.34
we get 1.25. Defining ζj± , j 1, 2, . . . , m as in 1.21 and 1.22, we have
b0 b1 · · · bm−1 bm bm3 bm4 · · · 0,
bm1 a12 1 μζ0 ζm
a12
bm2 Δ
m
ζm−j − a11 δj,m Δ
ζj ζm1−j a12 μ
τ ,
a12
j1
2.35
m
ζm1−j Δ
ζj ζm2−j a12 μ
τ .
a12
j1
Now 1.16 follows from 1.26 in view of
Hovθk εm1 bm1 bm2 ε εm1 Zk ,
k 1, 2.
2.36
Gro Hovhannisyan
11
Note that from 1.13 and the estimates
1
1 2pRθ p2 |θ|2 ≤ |2pRθ p2 |θ|2 |,
2
log |1 pθ| ≤ log 1 2pRθ p2 |θ|2 ≤ |2pRθ p2 |θ|2 |,
log |1 pθ| ≤ log
2.37
it follows
t
μs|θs|2 |eθ t, t0 | ≤ exp Rθs Δs,
2
t0
t
2Rθs
|eθ t, t0 | ≤ exp
Δs, μs > 0.
|θs|2 μs
t0
2.38
2.39
Proof of Example 1.5. From 1.37, 1.41, we have
θ1 − θ2 i2wτ − εμwΔ τ,
η1 t θ1 θ2 −
θ1 − θ2
2ia
2 Ot−3 ,
1 μθ2
t
η2 t εwΔ τ
,
w
θ1 θ2 2
2
εμwΔ
εwΔ w
−
,
2
4w2
θ2 − θ1
−2ia
2 Ot−3 ,
1 μθ1
t
τ −→ ∞,
2.40
and using 2.39, we get
eθ1 ≤ |eη | ≤ const,
1
e θ2
eθ2 ≤ |eη | ≤ const.
2
e θ1
2.41
Further for τ→∞
ζ1± −
Z1 3bεμ b2 ε2
λΔ λΔ 1 bε − 3aμ
1
2
2
∓
−
−
±
iaε
2μ
Oτ −4 ,
2λ
2
τ
2a
2aτ 2
τ3
2a2
2
ζ1
Δ
ζ1
Δ
εζ1
ζ1
μ−ε
Oτ Oτ −4 Oτ −4 ,
τ3
−4
2.42
−4
Z2 Z1 Oτ .
So if μ ε, then 1.26 and all other conditions of Theorem 1.2 are satisfied, and 1.18 is true
with m 1.
Acknowledgment
The author wants to thank Professor Ondrej Dosly for his comments that helped improving
the original manuscript.
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Advances in Difference Equations
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