Volume 2008, Article ID 712913,12pages 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 Ondˇrej Doˇsl ´y
We establish WKB estimates for 2×2 linear dynamic systems with a small parameterεon 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 asε 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.
Copyrightq2008 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 parameterε >0 on a time scale:
vΔt Atεvt, 1.1
wherevΔis the delta derivative,vtis a 2-vector function, and
Atε Aτ
a11τ εka12τ
ε−ka
21τ a22τ
, τ tε, kis an integer. 1.2
WKB method1,2is a powerful method of the description of behavior of solutions of1.1by using asymptotic expansions. It was developed by Carlini1817, Liouville, Green
1837and became very useful in the development of quantum mechanics in 19201,3. The discrete WKB approximation was introduced and developed in4–8.
The calculus of times scales was initiated by Aulbach and Hilger9–11to unify the discrete and continuous analysis.
are based on the representation of fundamental solutions of dynamic system1.1given in
12. Note that the WKB estimatesee2.21belowhas double asymptotical character and it shows that the error could be made small by eitherε→0,ort→∞.
It is well known13,14that the change of adiabatic invariant of harmonic oscillator is vanishing with the exponential speed asεapproaches 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 scalethe change of adiabatic invariant approaches zero with the power speed when the graininess depends on a parameterεin a special way.
A time scaleTis an arbitrary nonempty closed subset of the real numbers. IfThas a left-scattered minimumm, thenTk T−m,otherwiseTk T.Here we consider the time scales witht≥t0,and supT∞.
Fort∈T, we define forward jump operator
σt inf{s∈T, s > t}. 1.3 The forward graininess functionμ:T→0,∞is defined by
μt σt−t. 1.4
Ifσt> t, we say thattis right scattered. Ift <∞andσt t, thentis called right dense. For f : T→R and t ∈ Tk define the deltasee10, 11 derivative fΔt to be the numberprovided it existswith the property that for given any >0,there exist aδ >0 and a neighborhoodU t−δ, t δ∩Toftsuch that
|fσt−fs−fΔtσt−s| ≤|σt−s| 1.5
for alls∈U.
For any positiveεdefine auxilliary “slow” time scales
Tε{εtτ, t∈T} 1.6
with forward jump operator and graininess function
σ1τ inf{sε∈Tε, sε > τ}, μ1τ εμt, τ tε. 1.7
Further frequently we are suppressing dependence on τ tε or t. To distinguish the differentiation bytorτ we show the argument of differentiation in parenthesizes: fΔt
fΔttorfΔτ fΔττ.
AssumingA, θj∈Crd1 see10for the definition of rd-differentiable function, denote
TrAτ a11τ a22τ, detAτ a11τa22τ−a12τa21τ,
λτ
TrAτ2−4|Aτ| 2a12τ
,
1.8
Hovjt θ2jt−θjtTrAτ detAτ−εa12τ1 μθj
a
11−θj a12
Δ
τ, 1.9
Q0τ Hovθ1−Hov2 1−θ2
, Q1τ θ1−
a11Hov2−θ2−a11Hov1
a12θ1−θ2
, 1.10
Kτ 2μtmax
j1,2
1 ej
e3−j
2Hovj θ1−θ2
εa121 μθj θ1−θ2
a
11−θj a12
Δ
τ
|θj|
wherej 1,2, θ1,2tare unknown phase functions,·is the Euclidean matrix norm, and {ejt}j1,2are the exponential functions on a time scale10,11:
ejt≡eθjt, t0 exp
t
t0
lim
pμs
log1 pθjsΔs
p <∞, j 1,2. 1.12
Using the ratio of Wronskians formula proposed in15we introduce a new definition of adiabatic invariant of system1.1
Jt, θ, v, ε −v1tθ1t−a11τ−v2ta12τv1tθ2t−a11τ−v2a12τ
θ1−θ22teθ1teθ2t
, 1.13
Theorem 1.1. Assumea12τ/0, A, θ∈C1rdTε,and for some positive numberβand any natural
numbermconditions
|1 μTrA Q0 μ2detA θ1Q0−Hov1|τ≥β, ∀τ∈Tε, 1.14
Kτ≤const, ∀τ∈Tε, 1.15
∞
tε
1 ej
e3−j
Hovj θ1−θ2
τΔτ≤C0εm 1, j1,2, 1.16
are satisfied, where the positive parameterεis so small that
0≤ 2C01 Kτ
β εm≤1. 1.17
Then for any solutionvtof 1.1and for allt1, t2∈T, the estimate
Jv, ε≡ |Jt1, v, ε−Jt2, v, ε| ≤C3εm 1.18
is true for some positive constantC3.
Checking condition1.16ofTheorem 1.1is based on the construction of asymptotic solutions in the form of WKB series
vt C1eθ1t, t0 C2eθ2t, t0, 1.19
whereτtε,and
θ1,2t
∞
j0
εjζj
±τ, θ1Δ,2t
∞
k0
εk 1ζΔ
k±τ. 1.20
Here the functionsζ0 τ, ζ0−τare defined as
ζ0±τ Tr2A±a12λ, ζ1±τ −1 2μζλ 0±
λ∓a11−a22
2a12
Δ
where λτ is defined in 1.8, and ζk τ, ζk−τ, k 2,3, . . . are defined by recurrence
In the nextTheorem 1.2by truncating series1.20:
θ1t
condition1.26below given directly in the terms of matrixAτ.
Theorem 1.2. Assume thata12τ/0, A, θ∈C1rdTε,and conditions1.14,1.15,1.17, and
are satisfied. Then, estimate1.18is true.
Example 1.3. Consider system1.1witha11 a22.Then for continuous time scaleTRwe
condition1.26under the assumptionRλ 0 turns to
∞
and fromTheorem 1.2we have the following corollary.
Corollary 1.4. Assume thata−121 ∈C10,∞, λ∈ C20,∞, Rλτ ≡0, a
Ifλτis an analytic function, then it is known (see [13]) that the change of adiabatic invariant approaches zero with exponential speed asεapproaches zero.
Example 1.5. Consider harmonic oscillator on a discrete time scaleTεZ,
or
Jt, u, ε uΔt
2 w2τu2t
2wτ−εμtwΔτ2eη
, 1.39
ηθ1 θ2 μθ1θ2−
εwΔτ
w
μεwΔ2
4w2 μ
w− εμwΔ
2
2
. 1.40
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 numbersb, a /0, and estimate1.18withm1 is true.
Note that for continuous time scale we haveμ0,and1.39turns to the formula of adiabatic invariant for Lorentz’s pendulum13:
Jt, v, ε u2tt w2tεu2t
4wtε . 1.42
2. WKB series and WKB estimates
Fundamental system of solutions of1.1could 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 functionΨt ∈ C1
rdT∞ such thatH ∈ Rrd,the
matrix function Ψ μΨ∇ is invertible, and the following exponential function on a time scale is bounded:
eHt∞, t exp ∞
t plimμs
log1 pHsΔs
p <∞. 2.3
Then every solution of 1.1can be represented in form2.1and the error vector functionδtcan be estimated as
δt ≤ CeH∞, t−1, 2.4
Remark 2.2. Ifμt≥0, then from2.4we get
δt ≤ Ce∞tHsΔs−1
. 2.5
Proof ofRemark 2.2. Indeed ifx≥0, the functionfx x−log1 xis increasing, sofx≥
f0, log1 x≤x,and fromp≥0, Ht ≥0 we get
log1 pHs
p ≤ Hs, 2.6
and by integration
∞
t plimμs
log1 pHs
p Δs≤
∞
t HsΔs, 2.7
or
eHt,∞−1≤ −1 exp ∞
t HsΔs. 2.8
Note that from the definition
σ1τ εσt, μ1τ εμt, qΔt εqΔττ. 2.9
Indeed
εσt εinf
s∈T{s, s > t}εsinf∈Tε{εs, s > t}εsinf∈Tε{εs, εs > εt}σ1εt σ1τ, σ1τ εσt, μ1τ σ1tε−εtεσt−t εμt,
qεσt qtε εμtqΔττ qtε μtqΔt.
2.10
Ifa12τ/0, then the fundamental matrixΨtin2.1is given bysee12
Ψt eθ1t eθ2t U1teθ1t U2teθ2t
, Ujt θjta−a11t
12t
. 2.11
Lemma 2.3. If conditions1.14,1.15are satisfied, then
Ht ≤ 21 βKτmax
j1,2
1 ejt
e3−jt
θ1Hovt−jtθ2t, t∈T, 2.12
where the functionsHovjt, Kτare defined in1.9,1.11.
Proof. Denote
By direct calculationssee12, we get from2.11
Proof ofTheorem 1.1. From1.16changing variable of integrationτεs,we get
From this estimate and2.5, we have
δt ≤ Ce∞tHsΔs−1
≤ CeC0cεm−1≤eCC
0cεm, 2.21
whereεis so small that1.17is satisfied. The last estimate follows from the inequalityex−1≤
ex, x ∈ 0,1. Indeed because gx ex 1−ex is increasing for 0 ≤ x ≤ 1,we have
gx≥g0.
Further from2.1,2.11, we have
v1 C1 δ1eθ1 C2 δ2eθ2, v2 C1 δ1U1eθ1 C2 δ2U2eθ2. 2.22
Solving these equation forCj δj, we get
C1 δ1
v1U2−v2
U2−U1eθ1
, C2 δ2
v2−v1U1
U2−U1eθ2
. 2.23
By multiplicationsee1.12, 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| ≤C3εm. 2.25
Proof ofTheorem 1.2. Let us look for solutions of1.1in the form
vt ΨtC, 2.26
whereΨis given by2.11, and functionsθjare given via WKB series1.20. Substituting series1.20in1.9, we get
Hovθ1
∞
r,j0
ζrεrζjεj−TrA ∞
r0
ζrεr detA
a12ε
1 μ
∞
r0
ζrεr
∞
j0ζjεj−a11
a12
Δ
τ,
2.27
or
Hovθ1≡
∞
k0
bkτεk. 2.28
To make Hovθ1asymptotically equal zero or Hovθ1≡0 we must solve forζkthe equations
By direct calculations from the first quadratic equation
b0ζ20−ζ0TrA detA 0, 2.30
and the second one
b1τ 2ζ1ζ0−ζ1TrA a121 μζ0
Furthermore fromk 1th equation
bk 2ζ0−TrAζk a121 μζ0
we get recurrence relations1.22.
In view ofTheorem 1.1, to proveTheorem 1.2it is enough to deduce condition1.16
Note that from1.13and the estimates
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.
3 G. D. Birkhoff, “Quantum mechanics and asymptotic series,”Bulletin of the American Mathematical Society, vol. 39, no. 10, pp. 681–700, 1933.
4 P. A. Braun, “WKB method for three-term recursion relations and quasienergies of an anharmonic oscillator,”Theoretical and Mathematical Physics, vol. 37, no. 3, pp. 1070–1081, 1979.
5 O. Costin and R. Costin, “Rigorous WKB for finite-order linear recurrence relations with smooth coefficients,”SIAM Journal on Mathematical Analysis, vol. 27, no. 1, pp. 110–134, 1996.
6 R. B. Dingle and G. J. Morgan, “WKB methods for difference equations—I,”Applied Scientific Research, vol. 18, pp. 221–237, 1967.
7 J. S. Geronimo and D. T. Smith, “WKBLiouville-Greenanalysis of second order difference equations and applications,”Journal of Approximation Theory, vol. 69, no. 3, pp. 269–301, 1992.
8 P. Wilmott, “A note on the WKB method for difference equations,”IMA Journal of Applied Mathematics, vol. 34, no. 3, pp. 295–302, 1985.
9 B. Aulbach and S. Hilger, “Linear dynamic processes with inhomogeneous time scale,” inNonlinear Dynamics and Quantum Dynamical Systems (Gaussig, 1990), vol. 59 ofMathematical Research, pp. 9–20, Akademie, Berlin, Germany, 1990.
10 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application, Birkh¨auser, Boston, Mass, USA, 2001.
11 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
12 G. Hovhannisyan, “Asymptotic stability for 2×2 linear dynamic systems on time scales,”International Journal of Difference Equations, vol. 2, no. 1, pp. 105–121, 2007.
13 J. E. Littlewood, “Lorentz’s pendulum problem,”Annals of Physics, vol. 21, no. 2, pp. 233–242, 1963.
14 W. Wasow, “Adiabatic invariance of a simple oscillator,”SIAM Journal on Mathematical Analysis, vol. 4, no. 1, pp. 78–88, 1973.
15 G. Hovhannisyan and Y. Taroyan, “Adiabatic invariant forN-connected linear oscillators,”Journal of Contemporary Mathematical Analysis, vol. 31, no. 6, pp. 47–57, 1997.
