ASYMPTOTIC EXPANSIONS OF RESONANCES FOR WAVEGUIDES COUPLED THROUGH CONVERGING WINDOWS Текст научной статьи по специальности «Математика»

Chelyabinsk Physical and Mathematical Journal. 2023. Vol. 8, iss. 1. P. 72-82.

DOI: 10.47475/2500-0101-2023-18106

ASYMPTOTIC EXPANSIONS OF RESONANCES FOR WAVEGUIDES COUPLED THROUGH CONVERGING WINDOWS

E.S. Trifanova", A.S. Bagmutovb, V.G. Katasonovc, I.Y. Popovd

ITMO University, St. Petersburg, Russia

"[email protected], [email protected], [email protected], [email protected]

Two-dimensional waveguides coupled through small windows are considered. First terms of the asymptotic expansion of resonances are obtained and studied for the case when the distance between the windows decreases. Method of matching of the asymptotic expansions of solutions of boundary value problems is used.

Keywords: resonance, asymptotics, coupled waveguides, scattering, low-dimensional system.

1. Introduction

The problem of resonances for scattering system is intensively studied due to its importance for physical applications (see, e.g., [1-6]. There are several approaches to resonances: Lax — Phillips scattering theory [7], complex scaling [8; 9], perturbation theory [10; 11], functional model [12], asymptotic method [13-15], operator extensions theory model [16-19]. Last decade a new wave of interest to resonance problem for waveguide with perforated boundary and to homogenization is stimulated by nanotechnology progress (see, e.g., [20-24]). For these purposes, it is, particularly, interesting to clarify the resonance behaviour in the case when the distance between the coupling windows vanishes. An additional interest to the question is given by its relation to the choice of regularization when fitting the operator extension theory model [25]. In the present paper, we investigate the problem in the framework of the asymptotic approach.

We consider a system of two plane waveguides , Q- connected by two apertures (Fig. 1). The wave function satisfies the Helmholtz equation with the Neumann boundary conditions:

, 2 du

An + k2 u = 0, —

dn

= 0.

dQ

The lower boundary of the continuous spectrum is zero. The presence of coupling windows leads to appearance of resonances (quasi-eigenvalues) close to the second (third, etc.) threshold value.

Systems of coupled waveguides have been studied for a long time. Estimations and asymptotics for bound states close to the lower bound of the continuous spectrum were obtained in [26; 27]. In the present paper, we use the technique similar to that in [28-31]. The method is based on matching of the asymptotic expansions of the solutions of boundary value problems [32]. Particularly, asymptotic estimates of the resonance (quasi eigenvalue) close to the second threshold n2/d+ (d+ is the width of the widest waveguide)

The work was supported by Russian Science Foundation (grant number 22-11-00046

Fig. 1. Geometry of the system

of the continuous spectrum of the Neumann Laplacian for two-dimensional strips coupled through a small window were obtained in [29].

In the present article, we obtain the first terms of the asymptotic expansion of the resonance close to the second threshold assuming that the distance between the openings decreases along with their diameters.

2. The problem description

Let d+ and d- be the waveguide widths, d+ > d-. Let a be small parameter and the radius of the apertures are ai = au1, a2 = au2. Choose the coordinate system as shown in Fig. 1. We assume that the distance e between the centers (x1,0) and (x2,0) of the apertures decreases when a ^ 0 and the character of e decreasing is related to the decreasing of a:

e = Ix1 - x21 = ma6, 0 <5< 1. (2)

One can note that 5 =1 corresponds to simple similarity in position and size of the apertures.

For resonance value close to the second threshold n2/d+, we construct the following asymptotic expansion in the form of series in powers of ln-1 a

f n2

Y a = \l dp - = T1 ln 1 a + T2 ln 2 a + ... Our goal is to find the first coefficients t1 and t2 of this asymptotic expansion (3).

(3)

3. Construction of asymptotic expansion

To find a formal asymptotic solution to the problem (1), we need the expression for the Green function in a single waveguide with the Neumann boundary conditions:

G± (Xi ,X2 ) =

1 nnyi nny2

cos —;— cos —-— exp

n=0

d± Y± (¿no + 1) d

■±

L±

(- Y±d±|xi - X2^, (4)

where X1 = (x1,y1) and X2 = (x2,y2) are the Cartesian coordinates, 5n0 is Kronecker symbol and

Y± = J —- k2 Yn y d± k -

The asymptotic expansions of the Green function (4) in the neighborhood of singularity (0, 0) (x ^ x1 = 0) have the form:

G+ ((x, 0); (x1,0); ka) - = - - ln |x| + g+(x),

V ' d+ Vn /d + — ka n

(x, 0); (x1,0); k^---ln |x| + g-(x),

where functions g±(x\) have no singularities in the waveguide.

We also need another asymptotic behavior of the Green function near the next opening for distance e (2) between the windows tending to zero:

G+((x, 0); (x 2, 0); ka) ~ — - - ln a + h+(x — x2),

V / Ya n

—

G- ^(x, 0); (x2, 0); ka) ~--ln a + h-(x — x2),

where functions h±(x) have no singularities in the waveguide. The main result of the present paper is as follows.

Theorem 1. There are two resonances k^ close to the second threshold n2/d+ which have the following asymptotic expansions in the form of series in powers of ln-1 a :

Ya = \ l dp - ka = T1 ln 1 a + T2 ln 2 a +

2

where

n

T1

(1 + S)d+'

= (g+(0)+ g- (0)+ln 2 --ln m n2 ( 2 1

t2 = ----tt^ ln —+ - ln m

2 d+(1 - Ô)2 \ U n

U1 = U2 = U.

The goal of the paper is proving of the theorem. We will use matching of asymptotic expansions of the corresponding solutions (quasi-eigenfunctions). To construct the asymptotic expansion for quasi-eigenfunction ^a(X), we perform a partition of the domain. For this purpose, we introduce four disks. Their centers are the windows centers and their radii are r(a) and 2r(a) where r(a) is chosen in such a way that

au1 < r(a) < 2r(a) < e/2 = — ad. Then, we construct the asymptotic expansion for quasi-eigenfunction ^a(X) in the form

^ ) = \±Ya(a 1G±(X, (x1, 0), ka) + a2G±(X, (x2, 0), ka)) , X e \ S^; (?) [v0'2(x/a) + v\'2(x/a) ln-1 a + ..., X e S2>'2(a).

In accordance with the matching method for the leading terms, we need to find such "bonding" functions, satisfying the Neumann boundary conditions, that the terms of the corresponding orders in the asymptotic expansions of the solution coincide in the domains

\ s^) n s%a), \ s^) n

Functions v0 '2 and v1 '2 should satisfy the Laplace equation with the Neumann boundary conditions. Further, to determine the form of these functions, we compare the coefficients for the corresponding powers of a in the expansion (7).

The expansion (7) in the neighborhood of the first window (x 1 = 0) takes the form

Ya ai(d+Ta - 1 ln a - 1 ln |x/a| + g+(x)) +

d+Ya - f ln a + h+(x))

v0 (x/a) + v^x/a) ln-1 a + ...,

a ( - 1 ln a - 1 ln |x/a| + g- (x)) +

Ya

+a2^ — f ln a + h (x)j

X g H+ \ S;(a);

X G S2;(a) ;

X G H- \ S;(a).

Using (6), we equate the coefficients with a0 in (8):

1 TiN il T1 in

----ai + ----a2 = — ai +--a2,

d+ n J \a+ n J n n

and obtain two equations corresponding to x ^ x1 and x ^ x2

> — ^)ai + (d+ — ^ ( a2 = 0, — ai + (d+ — a2 = 0.

System (9) should have nonzero solution, then

Ti

(1 + ¿)d+'

We choose the constant functions as v

i,2

:iq)

vi(x/a) = — (ai + ia2); v0(x/a) = — (iai + a2 )

Ti

For finding t2, we equate the coefficients with ln a in (8):

(—Tn ln |x| + Tig+(0) — (2) ai + (Tih+ (0) — Ç2) a2 =

= — T1 On |a| + ai + ^

— (—ln |x| + T!g-(0) — T2) ai — (Tih-(0) + Ç2) a2

= Tn On |x| + ai + ^

or

(rig+(0) + n ln (±) - T2) + (rih+(0) - ) «2 = Cl, - (Tig-(0) + n ln (±) - T2) ai - (Tih-(0) + Ç2) «2 = Ci.

It is known that there exist suitable functions V1

i,2

Tl«l,2 V i,2

i,2 v1,

X^ (x/a) + Ci,2, x > 0; ^X¿,2(x/a) + Ci,2, x < 0,

where

and

X0i,2(C) = Xo (^ - x1,2) , Xo(Z) = ln (z + Vë-T) ,

X0i,2(C) = ln |zI + ln 2 - ln |^i,2| + o(1), z ^

Then asymptotics for vj'2(x/a) should be as follows

i 2 Ti

vi (x/a) = ai,2 (ln |x/a| + ln(2/^i ,2)) + Ci,2. n

Then one comes to a system of linear equations

g+(0) + g-(0) + 2l^¿j] ti - ^ + [(h+ (0) + h- (0)) Ti - ^] «2 = 0, [(h+(0) + h-(0)) Ti - ^] «1 +

g+(0) + g-(0) + 2ln(±)) ti - ^

ai +

a2 = 0.

11)

Let us introduce the notation

Ai = g+(0) + g-(0) + 2ln(2/^i), A2 = g+(0) + g-(0) + 2ln(2/^)

B = h+ (0) + h- (0), p = —.

n

Then system (11) takes the form

i [AiTi - ß] ai + [Bti - öß] «2 = 0, { [Bti - öß] ai + [A2Ti - ß] «2 = 0.

System (12) has a nonzero solution if

'12)

AiTi - ß Bti - öß Bti - öß A2Ti - ß

0.

For apertures of equal width, one has u1 = u2, A1 = A2 = A. Hence, two values are possible for P:

p+ A + B A — B P+ = -, , f Ti, P = --rti,

and

T+ =

1 + ö

n A + B 2 1 + ö

■Ti, t2

1 - ö

n A - B 2 1- ö

■Ti.

Substituting (10), one obtains

t

+

n2 A + B 2d+ (1 + ¿)2 ,

n

2 A B

2d+ (1 - ¿)2'

Using h±(0) = g±(0) — n lnm, one, finally, comes to the following expressions

t+ =

n

d+(1 + ¿)2

n

21

g+(0) + g- (0) +ln---ln m

21 ln —|— ln m

2 d+(1 — 5)2 V ^ n

where = = w.

Thus, the proof of the theorem is complete.

4. Discussion

Fig. 2. The dependence of the resonance ka on a for different 6 (a) and w2/w1 (b)

(b)

Fig. 2 (a) represents the behavior of resonance fca when two equal apertures are decreasing and distance e between them is decreasing as e = ma6 for different 5. One

can see that is real and the speed of convergence 8 of the apertures does not affect the shape of the resonance curve.

Fig. 2 (b) shows the resonance curve for the case 8 = 0.5 for different values of the ratio of the apertures sizes o^/o^. If the apertures are not equal the both values of fca have nonzero imaginary part.

In Fig. 3, the resonance curves for varied for different values of 8 and fixed a

are shown.

3.130 He(U

Fig. 3. The dependence of the resonance ka on w2/w1 for different 6

The calculations for plotting the graphs (Figs. 2 and 3) are performed numerically and are based on the analytical arguments given in the Appendix.

5. Appendix

Here we propose a way to calculate the remainder g+(0) of the series for the Green function (5). Namely, (5) gives us:

g+(x) = G+ ((x, 0); (0,0); h) — - = + - ln |x|

e

—d+fci|x| -

2d+fci + n

^ exp ( — Wn2 — (hd+/n)2|x| )

E-V . o „ , -¿ — ln |x|

n=2

\J n2 — (hd+/n)2

Let s = kd+/n (s is close to 1), t = n|x|. Then

g+(0) = + 1 ln n + 1 lim

2d+k% n n

E

n=2

^ exp ( - t\/n? - s2

yjn2 - s2

ln t

To find this limit, one uses the Taylor series for lnt, 0 <t< 2. Then

œ exp ( - t\jn2 - s2 ) œ

y pv , J- - ln t = t - 1 + y

Vn2 - ^

n=2

n2 s

n=2

exp ( - tVn2 - s2) (1 - t)

Vn2 - s2

n2 s

n

t -1 + y

n=2

n

exp

(- tVn2 - s2) - (1 - t)n

+

+y

n=2

n

\J1 - s2/n2

1

exp

- tVn2 - °2

n2 s

Obviously, the last series in (13) converges uniformly for t e [0,1), I > 0.

Let us consider the first series in (13) and prove its uniform convergence. Let

^n(x) = exp ^ - xVn2 - s2) - (1 - x)n.

:i3)

We are interested in the upper estimate of the function ^n(x):

max >^n = ^n(xo).

x

If |x0(n)| ^ C > 0 for all n, then max ^n(x) ^ *^n(C) and the series n<£n(t)

converges uniformly (the first series in (13)).

If x0(n) has a subsequence converging to zero (£n ^ 0), then

1 - ^e-«-7" = (1 - £n)n-1 ^ n2

1 — ^ (1 — Cn)e-A = 1, A = inVn^—V2 + n ln(l — Cn), ^

n2

A ^ 0 ^ n ln(1 — Cn) + nCn ^ 0 ^

—1 nC + R2(Cn)n ^ 0,

where R2(Cn) is the remainder of the Taylor series for ln(1 — £n). Since R2(£n) < 0, then Cn = o{1/y/n) and using d^n/dx(£n) = 0, one obtains

Vn(Cn) = (1 - Cn)

n1

1

\J1 - s2/n2

- 1 + Cn

= o

then ^=2 n^n(t) converges uniformly.

n

1

1

1

2

Due to the uniform convergence, the limit (5) gives one

lim

£

n=2

exp ^ — ¿Vn2 — s2j

V n2 —

- ln t

-1 +

n=2

n

a/1 — s2/n2

1

i+£

n=2

nVn2 — s2 (n + V n2 — S2 )

To calculate this series numerically with an arbitrary predetermined error, we could estimate the remainder:

£

n=m+1

1

du

nVn2 — s2(n + Vn2 — s2) " J uVu2 — s2(u + a/u2 — s2

Vu2 — s2 (u + V u2 — s2 )

ln 2 — ln + — s2/m2

Finally,

g+(0)

1 1, 1 v: + - ln n + - >

n=2

n ^v^n2 — s2(n + Vn2 — s2)

where s = kd+/n.

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Article received 11.07.2022.

Corrections received 12.01.2023.

Челябинский физико-математический журнал. 2023. Т. 8, вып. 1. С. 72-82.

УДК 517.955.8:517.956.2 DOI: 10.47475/2500-0101-2023-18106

АСИМПТОТИЧЕСКИЕ РАЗЛОЖЕНИЯ РЕЗОНАНСОВ ДЛЯ ВОЛНОВОДОВ, СВЯЗАННЫХ ЧЕРЕЗ СБЛИЖАЮЩИЕСЯ ОТВЕРСТИЯ

Е. С. Трифанова", А. С. Багмутов6, В. Г. Катасоновс, И. Ю. Попов^

Университет ИТМО, Санкт-Петербург, Россия

"[email protected], [email protected], [email protected], [email protected]

Рассмотрены двумерные волноводы, связанные через малые отверстия. Получены и исследованы первые члены асимптотических разложений резонансов для случая, когда расстояние между отверстиями уменьшается. Используется метод согласования асимптотических разложений решений краевых задач.

Ключевые слова: резонанс, асимптотика, связанные волноводы, 'рассеяние, низкоразмерная система.

Поступила в редакцию 11.07.2022. После переработки 12.01.2023.

Сведения об авторах

Трифанова Екатерина Станиславовна, кандидат физико-математических наук, доцент образовательного центра математики, Университет ИТМО, Санкт-Петербург, Россия; e-mail: [email protected].

Багмутов Александр Сергеевич, аспирант образовательного центра математики, Университет ИТМО, Санкт-Петербург, Россия; e-mail: [email protected]. Катасонов Владислав Геннадиевич, студент образовательного центра математики, Университет ИТМО, Санкт-Петербург, Россия; e-mail: [email protected]. Попов Игорь Юрьевич, доктор физико-математических наук, профессор образовательного центра математики, Университет ИТМО, Санкт-Петербург, Россия; e-mail: [email protected].

Работа выполнена при поддержке гранта РНФ (проект 22-11-00046).