RESONANCE ASYMPTOTICS FOR QUANTUM WAVEGUIDES WITH SEMITRANSPARENT MULTI-PERFORATED WALL Текст научной статьи по специальности «Математика»

Resonance asymptotics for quantum waveguides with semitransparent

multi-perforated wall

A. M. Vorobiev

ITMO University, Kronverkskiy, 49, St. Petersburg, 197101, Russia lenden31@yandex.ru

DOI 10.17586/2220-8054-2021-12-4-462-471

A pair of coupled quantum waveguides with a common semitransparent wall is considered. The wall has a finite number of small windows. We consider resonance states localized near each window. The presence of several windows forces one to describe their common influence differently from that of the single-window case. Using the "matching of asymptotic expansions" method, we derive formulas for resonances and resonance states.

Keywords: resonance, resonance state, waveguide, asymptotics, semitransparency. Received: 5 June 2021 Revised: 12 August 2021

1. Introduction

There is a large class of mathematical and physical problems concerning to "systems with small coupling windows", i.e., systems of connected quantum waveguides, Helmholtz resonators and other structures with perturbation caused by small windows on different boundaries. Such systems have long been of interest to physicists and mathematicians. Transport properties of waveguides and other additional phenomena caused by resonators are widely used in electrodynamics and theories of acoustic and electromagnetic waves.

Currently, scientists study such systems in the context of nanoelectronics. The development in this field has led to the creation of a new class of objects used in electronic devices - quantum threads (nanowhiskers), quantum dots, antidots and so on. Of course, "quantum" is a keyword here because studying mesoscopic systems (systems where the coherence of the phases of the electrons is preserved on a scale much larger than atomic) is absolutely impossible if we ignore quantum properties of the electron. Obviously, taking into account the quantum behavior of an electron requires the development of fundamentally new physical, and, most importantly, mathematical approaches.

When we consider mesoscopic systems, we actually mean studying wave propagation in waveguides and other structures. Properties of this propagation are described as spectral properties of corresponding mathematical operator. This is usually a Schrodinger operator. Laplace operator is also applicable for problems considering the ballistic mode. Resonance phenomena are widely used for development of new nanoelectronic devices.

Systems with small coupling windows were studied since the beginning of the 20th century [1,2]. Currently, we use term "asymptotic analysis" for large class of problems including resonance phenomena. Great variety of electronic device caused great variety of systems in resonance problems. Different combinations of waveguides and resonators were considered [3-6], in addition to different geometry characteristics [7,8], windows amount [9-11], and boundary conditions [12,13].

The core of the mathematical approach in this work is "Matching asymptotic expansions of solutions of boundary problems". This method was previously described in [14]. This method is intended for boundary problems of equations containing naturally occurring small parameter. It is very typical for mathematical physics problems, in particular for problems we described previously.

Semitransparency was studied by Ikebe, Shimada, Exner, Kreicirik, Popov in [15-20] in terms of spectral and scattering properties. Resonance asymptotics for such systems were studies in [21,22].

2. Preliminaries

Let us consider the pair of quantum waveguides in two-dimensional Cartesian coordinates with widths d_ and d+.

The common wall of both waveguides is semitransparent and is coupled through small windows. Semitransparency was studied in [21] for the same system but with single window. In this work, we consider finite number of windows. We will rely on the results of that work but we won't describe in detail the preliminary calculations. So let's take a quick look at used formulas and assumptions.

/ d+

aa>„ -a<»m+1 aw,„_

-d.

Fig. 1. Waveguides with common semitransparent wall (windows abscissas are relative to each center)

The wall semitransparency is described by the parameter a. Generally, a e (0; where zero value means no barrier and infinity means absolutely nontransparent barrier. Zero value is not included because actually it is not the case of the considered problem.

Boundary conditions on the walls are the key point: we choose the Dirichlet conditions on non-common walls but semitransparency causes special requirements for common wall. When a wave passes through the barrier, a jump occurs in the derivative of the considered function u(x1, x2), so, we will consider specific boundary conditions:

(u+=u_ (1)

I u+ — u_ = au,

where u+__are vertical derivatives at the top and bottom of the wall. The conditions of such type appear if one

considers the singular potential supported on hypersurface. These potentials have been intensively investigated during last two decades (see, e.g., [23-25]).

Problems with several windows may be very different. The number of windows can be finite or infinite; the "order of smallness" can vary for each window. We consider somewhat simple case - finite number of windows with the same order of smallness.

Let us have n coupling windows with the centers at the points (xq; 0), where q = 1..n. Each window has width

2aq = 2a--umum_1um+1. Parameter a is exactly common order of smallness. We will construct asymptotics in

a. This also means that wq, q = 1..n, describes the relative sizes of all the windows.

Unperturbed eigenfunctions of the transversal problem for the Laplace operator with semitransparency can be written as follows:

x (x) = j —Cn sin(d_v„)sin((x2 — d+)v„),X2 > 0, (2)

n y Cn sin(d+v„) sin((x2 + d_)vn), x2 < 0.

One can see that it satisfies conditions (1). The corresponding eigenvalues are denoted as vn and can be found from the equation —vctg(d+v) — vctg(d_v) = a. For investigating resonances, we need to choose "threshold" value. Threshold is such value of vn, that there are no summands with imaginary part exponent in Green's function series which are lower than vn. Imaginary exponent corresponds to periodic summands which mean propagating waves. We will seek terms of asymptotic expansions close to the second threshold v2.

"Perturbed" eigenvalue, also known as quasi-eigenfrequency, will be denoted as ka. Difference between ka and v2 is actually small value. We will use convenient expansion for its asymptotics:

_ rc [ j / 2] 1

k2 =£ E kjiaj W -. (3)

an

j=2 i=o 0

The Green function for waveguide is well-known [25] and appears as follows in our case:

G±(x, y, k) = g Xn(x2) • Xn(y2) • e_Pn|xi_yi|, wherep± = pn = v7^^—^.

n=i

2pn

2

2

Finally, we can write down eigenfunctions for perturbed case near each window:

a

œ n / a \

^a(x) ^y^f-kä ■ E Pj+1 Dy, ln - G+(x,y,k)

j=0 q=1 V a0/

œ [(j-1)/2]

x G sqo(a/a0)i/2,

y=(xq ;0) ?=1

^a(x) = E E 'vjil aj lni —,x G U # .. )i/2 , (4)

ji V a/ an' 2ao(a/ao)1/

j = 1 i=0 \a' a0 q=1

n

^a(x) = -VVf—kî ■ E E aj Pj+1 ( Dy, ln - ) G-(x, y, k)

j=0 q=1 V a0 '

,x G Il Sq

a

y=(xq ;0) q=1

a0(a/a0)i/2

Here, S^ )1/2 is as sphere with a center at q-th window. Last important thing is differential operator Pj. It is described as follows:

an / \ m-1 [(j-1)/2] / \i

Pi(D„, ln £) = a™D' D'y = In, = £ E o(r) ln ¿)Dr\ m > 2,

y v 7 j=1 i=0

Pm(Dy, ln — ) = aqPm(Dy, ln — ).

a0 a0

Here, aq are very important coefficients which describe operator action in distribution between windows. 3. Calculating

Boundary problems for vji Ç - j from (4) can be obtained in the following manner. We substitute the series (4)

x

and (3) into the Helmholtz equation (for k = ka) and then change variables £ = —. The coefficients in the terms with

a

the same powers of a and ln — should be equal. Hence, we obtain the following problems:

«0

j-3 [p/2]

AçVji = - E E Apqvj-p-2,i-q, £ G R2\Y, vji = 0, £ G Y, (5)

p=0 q=0

where y = {£|£2 = 0 A £1 g (-to; -1] U [1; and Apq are the coefficients of the series:

k2 = EE Apq ap lnq

— — a0

p q

-

As next step we need to introduce operator Mpq(U) - it changes variables in expressions U (£ = —, ln r = ln p+ln a) and filters summand with ap lnq —<£>(£). Also, Mp = E Mpq, it is used to obtain all summands with ap.

a0 q

3.1. Calculating of k20

Considering matching at q-th window we can obtain the following a-degree selections:

a 1mA ±VVf-kl ■ E P?G±(x,y,ka)|y=(xq;0)

-1M. I -J-. /,,2 _ i.2 . T^?'

q=1

a(110)C^vl sin2(dTV2)cos2(d±V2) " . 1 (1) _1 . „

± —-2T-■ E apPsin 6 T -a?k2oa(0)p 1 sin 6.

2 p=1 n

(6)

An extremely important point to note - k-singularity has its place in all window summands but x-singularity exists

n

only in current window, so we have E ®P for positive p degree and just aq for negative degree being in context of

p=I

q-th window.

In accordance with the lemma concerning the vj representation in the form of a linear combination of harmonic functions, we need to choose those functions as a summary of all windows. Function /1(z) = - (z + %/z2 - l) was

used for single window so we choose summary function as follows:

f1(z) = E aq"qZl

9=1

W,

9

™ Z - Xq ™ ™ "^q

q=1 "q q=1 4z q=1 4z q=1 \ 4 16 / z

- ± J + ^ 1 - ,z ^ +„,

E«,7T + £

W^Xq

1 q 4z 1 q 4z2 1 q

q=1 4Z q=1 4Z q=1

n I w2X2 \ 1

+ E aq "V1 + WH T. +

4 16 / z

For different coordinates, we will use notation /¿(z) = Xj(z) + ¿ii(z).To match terms increasing on p ^ to in accordance with (6), we shall select v10(£) in such a way:

te-, a10)C22V22 sin2(d_V2)cos2(d+V2) «wC2^2 sin2(d+V2) cos2(d-V2) vio(U = -ö- • yi(S) +--ö- • yi(£ ).

2

Hence, we can match terms of order p 1 sin 9 for each window and get the system of equations:

1

T~aq k2oa10) = ±

a^Cfvl (sin2(d_v2) cos2(d+v2) + sin2(d+v2) cos2(d_v2))

n W2 "p

£ '"V "

£ a ^ = ea e =_-2ko_

p q' nC2v2 [sin2(d_v2) cos2 (d+v2) + sin2(d+v2) cos2(d_v2)]

p=1

-4k2o

nC2v2 [1 - cos(2d_v2) cos(2d+v2)]

This system has nontrivial solution if:

det

^ 1 ß 1

CW2 ß •••

••• c^n - ß J

q2 where is actually harmonic capacity of the corresponding segment in R2, so we will use sometimes a symbol:

= -. Determinant of this matrix is (—ß)

e = E

1

= > еш_ =Ф

E J - в J. Hence:

-4k2o

q=1

q ' nC2 v2 [1 - cos(2d_v2)cos(2d+v2)]

= E ^

q=1

С22

k20 =--¿p2 [1 - cos(2d_V2)cos(2d+V2)]^ c^ •

q=1

If you compare it with k20 for the single window case, you will see that the only difference is sum J2 cw . It is

q=1

1

actually something like "common size" of the windows, characterized by sum of capacities. For n =1 it equals -.

3.2. Calculating of k30

Procedure for k30 is absolutely the same. Selecting of a2 gives us:

±а-2мЛ v^^ ■ E PqG±(x, y, ka)|y=K.0)

-a10^C| sin2(dTv)v3 cos(d±v2) sin(d±v2)

1

^ app2 cos 20 T — aqкзоа^Р 1 sin(0), p=1 n

2

2

.•• c

W1

С

0

c

c

2

2

2

±a-2M^v7^!-^ ■ E a ■ (x,y,k„)|y=(xq;a)J =

a(i0}C| sin2(dTV2)vf ■ sin(d±V2)cos(d±V2) " • a 1 , (2) -2 oa

—2-^^——-------1 E sino a9k3aa1ap 2 cos20.

Harmonic function of the second order uses the same formula except one more multiplier -9 and uses f2(z) 1

z ■ fi(z) = 1 (z2 + ^v/z2-!)

h(z) = E «9

9=1 V -9

n ~2 n O™ ~ n —2 n 2r—2x

v^ 2 z v^ 2 2x9z v^ 2 —9 v^ 2 9X9

E«g--J - £ a9- ^ - ^ - ^ " ...,Z ^

q=1 —9 9=1 —9 9=1 16z 9=1 16z

2 ■ - ^ + ...,z ^-TO.

E2 9 I V^ 2---9 9 ,

9=1 a9^ + 9=1 a9+

Unfortunately, it contains not only positive p degree, so we need to subtract corresponding additional summand using

X1(e):

-a11a)C| sin2(d_v)vf cos(d+V2) sin(d+V2) ^ v2a(5) =-g-a2(5)

-a1o)C2 sin2(d+v)v2 cos(d_V2) sin(d-V2) ^ , .

-X2(f )

8

E

x*,

99

—2a1a)C| sin2(d_v)v3 cos(d+V2) sin(d+V2) 9=1

--n-X1(S)

8

E

9

9=1

X9 x9

-2a11)C2 sin2(d+v )v2 cos(d_V2) sin(d-V2) 9= 9 9 ^ -8--n-X1(S )

E a9 9=1

a^Ci, sin2(d_v2)vf ■ sin(d+V2) cos(d+ ^2)^7-

-2-y1(e)

a121)C2 sin2(d+v2)vf ■ sin(d-V2) cos(d-V2) . -o-y1(5 ).

Equations system is similar to previous:

2

1 , (1) 1 a1o)C| sin2(d_v2)vf ■ sin(d+V2)cos(d+V2) / " ~v

T a9k3aa1a = ±-2- ^

n 2 \p=1 4

a12a)C22 sin2(d+V2)v2 ■ sin(d_V2)cos(d_V2^ A -A T 2 Lt14)

En O O -8k3a

= ^«9 =

_1 p 9' nClvf [sin(d_v2) sin(d+v2) sin((d_ - d+)v2)]

Determinant of this matrix is (-a)n-1 ^ ^ E - ^ . Hence:

a = ^^ _-8fc3a_= ^^

^ = ¿t ^ nC22v23 (sin(d_V2)sin(d+V2)sin((d_ - d+)v2)) = ^^

-nC22v2 (sin(d_v2) sin(d+v2) sin((d_ - d+)v2)) k3a =-8-^ .

9=1

3.3. Calculating of k40

As we previously noticed, equation (5) is homogeneous for vi0(£), v20(£), but becomes more complicated for the next step. So we need to solve the Poisson equation:

n2

A«V30 = — dp vio, vso(0 = 0, £ e y, y = =0 A £i e (—to; —1] U [1;+to)}.

The solution of this boundary problem can be presented as:

V30(£) = «30(C) + «30(C),

where «30(£) is solution of homogeneous Laplace equation satisfying the boundary conditions (as we seek for previous steps) and «30(£) is particular solution of inhomogeneous Laplace equation satisfying the boundary conditions. M-operator expressions are not much different with single-window ones:

r^,-n n 1 m A _L n a(i0)k20C2 sin2(dTvi)v^ cos2(d±vi) .

a 3M30 (±Vv| — k^ • PiG±(x,y,ka)|y=^ = ± E ap —-^-^-^psin9

±E

aio^Clv2 sin2(dTv) cos(d±v)

'H\fv2[— vi

p=i

p 2

(i)i + • n (i) 1

1 iß t aioaq

' n

p=i n

± E • ai0)k2og±p sin 0 T a^aq — &4op - i sin 0

— cos(d±v2)v3p3 sin 30 ± sin(d±v)k20pcos 0

(8)

tE

3M3o (±^V2r—k2 • a • P2G±(x,y,ka)|y=o) a^C*! sin2(dTv2)v2 • sin2(d±v2)

p2 cos 20 t _aqk4Oai20)p 2 cos 20,

p=i

-3M3o fc2 • a2 • a(i0)D3G±(x,y,ko)| J

±

p=i

—ai30)C2 sin2(dTV2)v4cos2(d±V2^ (3) V22 _i (3) 2 -3 ■ oa

-—-^- • psin0 ± aio aq — k20p i sin0 ± aV aq —k40p 3 sin30,

2 2n n

M30 (±yV2—fc2 • a2 • a20)D2G±(x,y,ko)| J

a JM3o

n

= E ap p=i

22OC2 sin2(dTV2)vf sin(d±V2)cos(d±V2^ 1 (3) _2 oa

—-1-psin 0 T — aqk30a2o P 2 cos 20.

(9)

(10)

(11)

The harmonic summary for f3(z) = 1 ^z3 + 3 Vz2 + (Vz2 — 1)3^ can be presented as follows:

i"3(z) = E aq^3f3

q=i

3xq z

.3 q

*qWq 3

E aq-3 - E

q=i — q=i

n 3 3 - q

-E aq-316Z - ...,z

+ E

3x2 z

.3_q

'q-q -3

3 3-q

+ E aq-3^ + ..., z ^

q q=

i * -3 q=i

q q 16z

q=i

Here, we also obtain a set of positive degrees instead of one so we will need the same idea as for previous order. Separating summands from (8)-(11) with positive p degrees, we can obtain the following representation for «30(£):

«30 (o = &i/à(o + ^2iX2(e*)+£11X1 (e)+£iiXï(É*)+ _ ^32?3(e) + /^(e*) + /22*2(0+feîMO) + /^Yùo+/^Yte*) '

where coefficients can be found:

-a(i20)C| sin2(d-V2)v| • sin2(d+V2) ~ aiO'C sin2(d+v2)v| • sin2(d-V2) /2i = -0-, /2i = -0-,

ßii

ai0)k2OC2 v2 sin2(d_v) cos(d+v) sin(d+v)

/A \

aqxq

+ 2

q=i

E aq \ q=i /

ß2i

a

P

8

a

a

P

zx

q

q

3

~ a(11)k2aC!v2 sin2(d+v) cos(d-v) sin(d-v) P11 =-o-

+ 2

E <

9=1

9=1

/21,

a1a)C2v2 sin2(d_v2) cos2(d+v2) ~ aH/clvl sin2(d+v2) cos2(d_v2) /32 =-T--, />32 =-

48

48

/A \

/ > a,x9

/22 =3

9=1

E a9 I

9=1

/A \

/ , a,x9

/32, />22 = 3

9=1

E a9 I

9=1

/32,

/12 =

a1H)k2aC2 sin2(d-v1)v2 cos2(d+v1) 2«^/-

+ a(1l)k2ag+ - 3

^ E a9xg ^ 9=1

E a9 /

9=1

/32 +2

E <

9=1

99

iE«,,

\ 9=1 /

/22

^a'Cl sin2(d-v2 )v| cos2 (d+v2) a23i) sin2(d-v2)v3 sin(d+v2) cos(d+v2)

+

> a1a)k2aC2 sin2(d+v1)v2 cos2(d-V1) (1) _ /512 =-/ „ 1--+ a1Q k2ag- - 3

^ E a9 xg ^ q=1

2iyV2 - v 1

iE«,|

q=1

/32 +2

E

q=1

f^C2 sin2(d+v2)v| cos2(d_v2) a2'H)C| sin2(d+v2)v| sin(d_v2) cos(d-v2)

E a9 /

q=1

/22

Particular solution of inhomogeneous equation can be obtained by integrating:

a(o)C!v! sin2(d_v2) cos2(d+v2)

(£) = -na c2 v2 sin (d-v2) cos (d+v2) ^ (£) + _1H

(1) 2 2 2

sin2(d+v2) cos2(d_v2)

^n,

>3a(0 =

2 aH^C2 v2 sin2(d_v) cos2(d+v)

x

^=1

E a, ) 8 P3 sin 0 + ( E a,^ ) p ln p sin 0 - 8 ^E a, ( ^ + 79 ) ) p- sin 30 - ..., £2 > 0,

2 2 2

—2x2 —2

q=1

8 W=1 U 4

16

n —2

1n

- E a^-q plnpsin0 + - E a

2 2 2

J, x9 ^ — 9 J!

, 9=1 9 8 /------------- ' 8 vq=r"M 4 ■ 16

+ 79 p-1 sin30 + ..., £2 < 0,

ad/clv2 sin2(d+v2) cos2(d_v2)

2 a1a c2 v2

2

(£ a,

1 / n / —2x2 —2 qIA j plnpsin0 - ^ ( E a, (^p + 16 ) )p-1 sin30 - ..., £2 > 0,

2

—2 1 n —2 x2 —2

- ( EaJ ^p3 sin 0 -( ¿a, p ln p sin 0 E a J + ) p-1 sin30 + ..., £2 < 0.

9=1

1

8

,9=1

8

,9=1

16

a

2

2

99

a

2

2

2

2

2

Resonance asymptotics for quantum waveguides with semitransparent multi-perforated wall Finally, we obtain full solution for £2 > 0:

V3ü(£) = ß2ip2 cos 20 + ßiipcos 0 + ß32P3 sin 30 + ß22P2 sin 20 + ßi2psin 0

E aq4)

vq=i ,4\

2 aio Civ! sin2(d—v2) cos2(d+v2)

2 aio'c!v2 sin2(d+v2) cos2(d—v2)

E aq I 8 P3 sin 0 +

vq=i 1

- 8 15 aq

q2

aq — I plnpsin0 8

^ + 3)1 p2 si"30-■••

w,

2

E a^^ plnpsin0

k q=i

- 8 IqS aq

22 w2x2 wq

4 + -SJJ p^ sin30 - -

+ E (ßi2b+ + ßi2b-j + ß22b+j + ß22b-j + feb- + ß326-j)p2 sin j0

j = i

TO

y2j

2j

j=i

For £2 < 0, one has:

+ E (ßii alj — ßiiaij + ß2ia2j + /?2ia2j)p j cos j0.

V3ü(£) = ß2ip2 cos 20 + /?iipcos 0 — ,032p3 sin 30 — ,022p2 sin 20 — ßnpsin 0

2 aiü^C!v2 sin2(d—v2) cos2(d+v2)

aiÜ^Cfv2 sin2(d+v2) cos2(d—v2)

2 aio C2 v2 v2

— E a^—p plnpsin0+

^q=i

1 / ™ / w?x? wf . . .

+ 8 qEEi M W4Xq + if) ^ sin30 + -

2

I n \ l / n w2

H EaJ 8 p3 sin 0 — ( E^a^-^q ) p ln p sin 0+

1 / ™ /w2x2 w4 . . _i .

+— > aq +--- p i sin30 + ...

8 Lti M 4 16

+ E ( —ßi2b+j — 0i2b-j — ß22b+j — /022b—j — ß32b+j — ß32b-j)p-j sin j0

j = i

"ij

2j

2j

+ E (ßiia+j — Aia- + ßna2,- + Z^ia2,-)p j cos j0.

j=i

Matching terms of order p i sin 0, one obtains:

Ta(i0)aq -k4ü ± a(i3ü)aq x2-kn , n 2n

±v.

±

2 a(i1ü)C22v22 [sin2(d_v2) cos2(d+v2) + sin2 (d+v2) cos2(d_v2)]

+ Z^H b — i + ß22b+i + ,022 b—i + ß32b+i + /?32b

8 (ps an ^

wpxp + wp4

l6

ii

i 3a( i) Co Vo S+

2 2 4 w2xp w4

(i) , (3) v2 7 2 3aiü C2 v2 S+ f wpxp , wp v^ 3wp aiü e2 v2 S +

— «iü aq-k4ü + «iü aq 22 k2ü = V2 -^--E ap[ -r- + 16 E ap^ ■ -48-

+ E ap i.

ano)k2ü elvi 2^Y/v2 — v2

,p=i

S+ + ano)(g+ + g- ) ■ k2ü +

(11ü) C22 v24

3w4 a^CVS,

p=i

l6

Si

+

(3)C 2V3 '2ü e2 v2 / • 2

(sin2(d_v2) sin(d+v2) cos(d+v2) — sin2(d+v2) sin(d—v2) cos(d_v2))

^ y ] aqx, ^

3

q=i

qq

aq

\ q=i /

a nlü)C22v24 S+ +2

48

E

q=i

qq

\

aq

\ q= i /

E

q= i

qq

\

aq

\ q= i /

aiü)C2 v2 48

S2

where S± = sin2(d—v2) cos2(d+v2) ± sin2(d+v2) cos2(d—v2).

2

2

2

x

2

X

2

2

3

As we can see this system of equations is not linear because of expressions like will consider only imaginary part of both sides and it will make system actually linear:

^ E apxp^

p=I

v E ap i

\ p=i /

That's why we

Z10 aq

1

E "p ^

Vp=i

4

(1) no

k2oC?

1 vi

(sin2(d_v1) cos2(d+v1) + sin2(d+v1) cos2(d_v1))

=>

ap —, p =

2\J v2 — v2Imk40

p=1

E'

q=1

nk20C2v2 (sin2(d_v1) cos2(d+v1) + sin2(d+ v1) cos2(d_v1))

=>

2\J v| — v2Imk,

40

nk20C^v2 (sin2(d_v1) cos2(d+v1) + sin2(d+v1) cos2(d_v1))

=>

Imk

40 =

—n2C2v2C|(1 — cos(2d-v1) cos(2d+v1)) (1 — cos(2d_v2) cos(2d+v2))

We can compare this formula with result from single-window case:

E

Vq=1

Imk

—n2C2v2C2v2

40

128^ —

(1 — cos(2d-v1) cos(2d+v1)) (1 — cos(2d_v2) cos(2d+v2))

and verify it is absolutely consistent, the difference is just in multiplier ( J2 characterizing common size of

q=l

windows. As usual, it is the square of the coefficient appearing in the first order term. 4. Conclusion

The results for the imaginary part of the resonance allow us to pose the problem of minimizing/maximizing the lifetime of the resonance by changing the configuration of the windows, e.g., fixing the summary size of windows. The results pertaining to the real part of the resonance, give one an estimation of the shift of resonance with respect to the threshold. These results can be useful for the description of "quantum waveguide - quantum dot - quantum waveguide" systems. One can find such systems in different nanotechnology applications.

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