1 Introduction. Statement of the problem - Math-Net.Ru

×ÅÁÛØÅÂÑÊÈÉ ÑÁÎÐÍÈÊ Òîì 6 Âûïóñê 3 (2005) 

APPROXIMATION OF EXPONENTIAL SUMS IN THE PROBLEM ON THE OSCILLATOR MOTION CAUSED BY PUSHES1 Ekatherina A. Karatsuba (Moscow) Àííîòàöèÿ

The application of the Hardy-Littlewood-Vinogradov-Van der Corput theorem on the approximation of exponential sums by shorter ones to the solution of the problem on quasiperiodic pushes acting on harmonic oscillator is considered. New asymptotic formulas for the solution of the problem for the pushes of various forms in the presence and in the absence of friction are obtained. 1

Introduction. Statement of the problem

The analysis of wave processes both in classical and in quantum mechanics is often conducted using the model of harmonic oscillator. As a harmonic oscillator, it is possible to consider an object from broad spectrum of physical objects: from pendulum and electromagnetic circuit to atom. As in [5], we consider the solution of the problem on the oscillations of the harmonic oscillator caused by quasiperiodic pushes. The equation of the motion of the one-dimensional harmonic oscillator with regard to friction is the homogeneous dierential equation (see [2], for example) of the form

x¨ + αx˙ + ω02 x = 0,

(1)

where ω0 and α are positive constants. The constant ω0 is called the fundamental frequency of the oscillator vibrations, the constant α is the friction coecient. For α2 < 4ω02 , (1) describes the free (damped) oscillations of the harmonic oscillator √ (2) x0 (t) = x0 exp (−αt/2) sin (ωt − φ0 ), ω = ω02 − α2 /4, where x0 and φ0 are the given initial amplitude and phase of the oscillations. In [3] the following problem is considered. Assume that a short-term force acts on the oscillator (1), (2) in the successive time moments 0 < t0 < t1 < t2 < · · · < tn < . . . , tn → +∞. This force gives the positive increments the velocity of the oscillator motion: V0 , V1 , . . . , Vn , . . . . We shall call the sequence of couples (tk , Vk ), k = 0, 1, 2, . . . , n, . . . , the pushes (tk , Vk ). It is observed the struggle between the friction (when α > 0), which strives to diminish the amplitude of the oscillations,

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EKATHERINA KARATSUBA

and the pushes, which strive to enlarge it. The mathematical description of such a problem presents the Cauchy problem with initial conditions

x(0) = X0 ; x(0) ˙ = X1 .

(3)

For t > 0, x(t) satises the equation

x¨ + αx˙ + ω02 x =

∞ ∑

Vk δ(t − tk ),

(4)

k=0

where δ(t) is the Dirac delta-function:

∫ δ(x) = 0 , x ̸= 0 ,

+∞

δ(x)dx = 1. −∞

The solution of this problem for tn−1 < t < tn has the following form:

x(t) = x0 (t) + S(t) = x0 (t) + ω

−1

n−1 ∑

Vk exp(−α(t − tk )/2) sin(ω(t − tk )),

(5)

k=0

where x0 (t) and ω are dened by (2). In [3] the behavior of the oscillator under the action of the pushes (tk , Vk ), which are periodic and equal in value:

tk = kτ , Vk = V ; k = 0, 1, 2, . . . , n ; n → +∞, where V and τ are positive constants, is explicitly discussed. It is noted in [3] the impossibility to simplify the sum S(t) from (5) for arbitrary (tk , Vk ). The aim of my research is to study the behavior of the oscillator under the action of the pushes (tk , Vk ), which are more complicated than (kτ, V ). To simplify the sum S(t) in this case, my purpose is to approximate the oscillator motion after n pushes (tk , Vk ) by itÒs oscillations after n − N pushes of more simple form (t˜k , V˜k ). In [5] the oscillator vibrations in the absence of friction (α = 0) caused by the pushes (tk , Vk ) of the form

tk = τ (k + D)β , Vk = V ; k = 0, 1, 2, . . . , n ; n → +∞;

(6)

are considered. Here, as above, V and τ are positive constants, and the constants D > 1 and β > 0 dene more precisely the time of the occurrence of a push. It is clear from (6) that, if 0 < β < 1, then the pushes (tk , Vk ) become more frequent, if β > 1, then the pushes become less frequent. The case β = 1 corresponds to the aforementioned case of periodic pushes. The sum of n summands S(t) is studied in [5] for 0 < β < 2. Moreover, it is proved in [5], that if 0 < β < 1, then S(t), tn−1 < t < tn , is approximated with good accuracy by the function of the form

APPROXIMATION EXPONENTIAL SUMS

V ω02 τ β

207

(n − 1 + D)1−β cos (ω0 (t − tn−1 )) ;

and if 1 < β < 2, then S(t), tn−1 < t < tn , is approximated by the sum of N similar to that one summands, where N = N (n) ∼ nβ−1 . In the present paper the pushes (tk , Vk ) of the form

tk = τ (k + D) + ρτ (k + D)β , Vk = V ; k = 0, 1, 2, . . . , n; n → +∞; 0 < β < 1; tk = τ (k + D)β , Vk = V (k + D)γ ; k = 0, 1, 2, . . . , n; n → +∞; 0 < β < 2; are discussed. Here τ > 0, ρ > 0, γ > 0, D > 1, V > 0 are the constants. We get new asymptotic formulas for S(t) both in the presence α > 0, and in the absence α = 0 of friction. To this end, we apply, as in [5], the theorem on the approximation of exponential sums by shorter ones.

2

The theorem on the approximation of exponential sums

We consider the sum S,

S=

∑

φ(n) exp (2πif (n)),

(7)

a 0,

such that

C1 B 6 |A| 6 C2 B. 2

For a real number

α,

we assume

||α|| = min({α}, 1 − {α}), where

{α}

is the fractional part of

α.

Theorem Assume that the real functions

f (x)

and

φ(x)

satisfy the following

conditions on the segment [a, b]: ′′′′ ′′ 1) f (x) and φ (x) are continuous;

H, U and V such that 1 ≪ U ≪ V, 0 < b − a 6 V and

2) there exist numbers

H > 0,

U −1 ≪ f ′′ (x) ≪ U −1 , φ(x) ≪ H, ′′′ −1 −1 ′ f (x) ≪ U V , φ (x) ≪ HV −1 , ′′′′ −1 −2 f (x) ≪ U V , φ′′ (x) ≪ HV −2 . Then, if we dene the numbers

xµ

by the equation

f ′ (xµ ) = µ, we have

∑ a 0). In this case the solution of the problem (3), (4) has the form (5) with free (damped) oscillations (2). The sum S(t) from (5) is determined by the relation

S(t) = ω

−1

exp (−αt/2)Im

n−1 ∑

Vk exp (αtk /2) exp (iω(t − tk )),

k=0

√ where the frequency of the oscillator vibrations is ω = ω02 − α2 /4. To calculate the sum S(t) using the theorem/lemma is purposeless, because of the losses and the swift increasing values of tk , S(t) is approximated with good accuracy by the last summand of this sum. To verify it, we consider the sum S1 , S1 = Im

n−1 ∑

Vk exp (αtk /2) exp (iω(t − tk )).

k=0

Since from (28), (29) n−2 ∑

Vk exp (αtk /2) = V

k=0

k=0

∫

n−2

Vn−2

n−2 ∑ (k + D)γ exp (ατ (k + D)β /2)
τ β ((n − 2 + D)β−1 , for 0 6 ξ 6 1. Since β > 1, then for n → +∞, τ β ((n − 2 + D)β−1 → +∞ in a power manner and hence the value

2θ1 (αβτ )−1 D1−β exp (−α(tn−1 − tn−2 )/2) which doesn't exceed

2(αβτ )−1 D1−β exp (−αβτ (n − 2 + D)β−1 /2),

APPROXIMATION EXPONENTIAL SUMS

217

is decreasing to zero in exponential manner. Therefore we have for certain θ2 , |θ2 | < 1; for tn−1 < t < tn :

Vn−1 exp (−α(t − tn−1 )/2) (sin(ω(t − tn−1 )) ω ) +2θ2 (αβτ )−1 D1−β exp (−αβτ (n − 2 + D)β−1 /2) .

S(t) =

6

The oscillations caused by more frequent pushes with friction

We consider the oscillator vibrations with regard to the friction (α > 0) under the action of the pushes (13), (14). In this case the sum S(t) from (5) is dened by n−1 ∑ V S(t) = exp (−αt/2)Im exp (αtk /2) exp (iω(t − tk )) = ω k=0

(36)

V exp (−αt/2)Im exp (iωt)S0 , ω √ where the frequency of the oscillations is ω = ω02 − α2 /4, and the sum S0 , =

S0 =

n−1 ∑

exp ((α/2 − iω)tk ) =

k=0 n−1 ∑

( ) exp ((α/2 − iω)τ D) exp ((α/2 − iω)τ k) exp (α/2 − iω)τ ρ(k + D)β .

k=0

We denote

sk =

k ∑

exp ((α/2 − iω)τ j) =

j=0

Then

exp ((α/2 − iω)τ (k + 1)) − 1 . exp ((α/2 − iω)τ ) − 1

(37)

exp ((α/2 − iω)τ k) = sk − sk−1 , k > 1, s0 = 1.

Set

S0 = exp ((α/2 − iω)τ D)R0 , where the sum R0 ,

R0 =

n−1 ∑ k=0

( ) exp ((α/2 − iω)τ k) exp (α/2 − iω)τ ρ(k + D)β

(38)

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EKATHERINA KARATSUBA

( ) = R + exp (α/2 − iω)τ ρDβ Taking into account (37), we obtain for the sum R

R=

n−1 ∑

( ) (sk − sk−1 ) exp (α/2 − iω)τ ρ(k + D)β =

k=1 n−1 ∑ (

( ) ( ) sk exp (α/2 − iω)τ ρ(k + D)β − sk−1 exp (α/2 − iω)τ ρ(k − 1 + D)β

k=1

( ) ( )) +sk−1 exp (α/2 − iω)τ ρ(k − 1 + D)β − sk−1 exp (α/2 − iω)τ ρ(k + D)β ( ) ( ) = sn−1 exp (α/2 − iω)τ ρ(n − 1 + D)β − exp (α/2 − iω)τ ρDβ +

n−1 ∑

( ( ) ( )) sk−1 exp (α/2 − iω)τ ρ(k − 1 + D)β − exp (α/2 − iω)τ ρ(k + D)β .

k=1

From here and from (37) we nd

( ) exp ((α/2 − iω)τ n) − 1 exp (α/2 − iω)τ ρ(n − 1 + D)β − exp ((α/2 − iω)τ ) − 1 ( ) − exp (α/2 − iω)τ ρDβ + (exp ((α/2 − iω)τ ) − 1)−1 R1 −

R=

− (exp ((α/2 − iω)τ ) − 1)−1 · ( ( ) ( )) · exp (α/2 − iω)τ ρDβ − exp (α/2 − iω)τ ρ(n − 1 + D)β , where

R1 =

n−1 ∑

( ( ) exp ((α/2 − iω)τ k) exp (α/2 − iω)τ ρ(k − 1 + D)β −

k=1

= (

( )) − exp (α/2 − iω)τ ρ(k + D)β =

n−1 ∑

( ) exp ((α/2 − iω)τ k) exp (α/2 − iω)τ ρ(k + D)β ·

k=1

( ( )) ) · exp (α/2 − iω)τ ρ (k − 1 + D)β − (k + D)β − 1 . Since (k − 1 + D)β − (k + D)β = −β (k − ξ + D)β−1 , 0 6 ξ 6 1, then for k > 1 the following bound is valid:

√ ( ) (α/2 − iω) τ ρ (k − 1 + D)β − (k + D)β 6 βτ ρ α2 /4 + ω 2 (k − 1 + D)β−1 . As far as 0 < β < 1, for 1 ) 1−β ( √ k > k1 = −D + 1 + 2βτ ρ α2 /4 + ω 2 ,

APPROXIMATION EXPONENTIAL SUMS

219

we have the inequality

βτ ρ

√

α2 /4 + ω 2 (k − 1 + D)β−1 6 1/2.

If k1 > 1, we represent the sum R1 in the form of the sum of two summands and going over the inequalities, we get

|R1 | 6 2

∑

∑

( ) exp ατ (k + ρ(k + D)β /2) +

k6k1

( ) exp ατ (k + ρ(k + D)β /2) ·

k1 0) under the action of the pushes which become more frequent and at the same time more strong:

tk = τ (k + D)β , Vk = V (k + D)γ ; k = 0, 1, 2, . . . , n; n → +∞,

(40)

V >0, τ >0, γ>0, D>1, 0 x1 , where 1

x1 = −D + (ωτ β/π) 1−β , we have the following bounds: 0 < f ′ (x) 6 12 . We apply to S0 the formula of the partial summation, extracting beforehand the rst summand: ∫ n−1 S0 = − C(u)d ((u + D)γ exp (αtu /2)) + 0

C(n − 1)(n − 1 + D)γ exp (αtn−1 /2) + Dγ exp ((α/2 − iω)t0 ), (43) ∑ where C(u) = 0 x1 , then we represent the sum C(u) in the form of two sums and apply the Van der Corput lemma to the second of these sums. We have C(u) =

∑

exp (−iωtk ) +

0