Pergamon Computers Math. Applic. Vol. 34, No. 5/6, pp. 613-625, 1997

Copyright©1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved

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H e l m h o l t z P r o b l e m for P l a n e Per iodica l S truc tures

P. G. AKISHIN Laboratory of Computing Technique and Automation

Joint Institute for Nuclear Research, 141980 Dubna, Russia akishin@ict a28. dubna, j int. su

F. B o s c o International Solvay Institute for Physics and Chemistry

Boulevard du Triomphe, B-1050 Brussels, Belgium

S. I. VINITSKY Laboratory of Theoretical Physics

Joint Institute for Nuclear Research, 141980 Dubna, Russia

Abs t r ac t - -The plane Helmholtz problem of the periodical disc structures, with the phase shifts conditions of the solutions along the basis lattice vectors and the Dirichlet conditions on the disc boundaries, is considered. The Green function satisfying the quasi-periodical conditions on the lattice is constructed. The Helmholtz problem is reduced to the boundary integral equations for the simple layer potentials of this Green function. The methods of the discretization of the arising integral equations are proposed. The procedures of calculation of the matrix elements are discussed. The reality of the spectral parameter of the nonlinear continuous and discretized problems is shown. Some numerical results are presented.

K e y w o r d s - - H e l m h o l t z equation, Plane periodical structure.

1. I N T R O D U C T I O N

The Helmholtz problem for the plane periodical structures is arising in study of the quantum motion of a free particle on the periodical system of reflecting discs [1,2]. This problem, treated as the quantum Sinai billiard [2] or the quantum Lorentz gas, attracts attention of many inves- tigators, for example, see [3,4]. However, a solution of the general problem is far from complete. So, solving the problem with nonorthogonal basis lattice vectors and quasi-periodical boundary conditions is an open question. Moreover, even for the known version of "desymmetrized" Berry's billiard [2] with the periodical conditions, the evaluation of the energy levels near the so-called close packing limit, when the distance between discs tends to zero, is an open question, too. In the general case, which has been discussed in the recent paper [4], two additional parameters treated as quasimomenta are introduced. That is why the question about quasicrossing of the energy levels with respect to all free parameters should be investigated. In the paper [4], the Helmholtz problem has been reduced to solving the integral equations for the potentials of simple and double layers of the Hankel function on the boundary of the complex domain consisting of the forth arcs of the discs and the forth pieces of the sides of the hexagonal cell.

In the present paper, we propose the approach of reducing the Helmholtz problem to the boundary integral equations for the simple layer potentials of the special Green function of the

We thank I. Antoniou and Yu. Kuperin for fruitful discussions. We thank the Commission of the European Communities for financial support in the frame of EC-Russia Collaboration Contract ESPRIT P 9282 ACTCS.

Typeset by .A~TEX

613

u(x) = O, x e OS.

periodical lattice on the boundary of one disc only. The methods of discretization of the boundary integral equations are suggested. The first numerical results obtained in the framework of this approach are presented.

Now we give the mathematical formulation of the solving problem. Let {ej, j = 1, 2} be a system of the noncolinear basis vectors in the plane R 2, which defines the fundamental domain f t ( f t : x = a e l +/~e2, 0 < a < 1, 0 _</~ _< 1), see Figure 1. We define the cell ~ of the reflection structure by means of arrangement of the disc S of the radius R in the fundamental domain n : ~ = f~ \ S, see Figure 2. The wave function u ( x ) in the domain f/satisfies the Helmholtz equation:

Au + Au = 0, (1.1)

with Dirichlet condition on the boundary 0 S of the disc S

u(x + ej) = e~PJu(x),

Vu(x + ej) = e~PJVu(x),

The wave function is expanded on all the cells of the periodical structure with the help of the following conditions:

¼

j = 1, 2, (1.2)

j = 1, 2. (1.3)

The problem is to find the spectral parameter A and the wave function u ( x ) with respect to the parameters Pl,P2 and the radius R.

Q

e2

Figure 1.

A

Q

L2

Figure 2.

614 P .G. AKISHIN et al.

In Section 2, we construct the special Green function of the periodical lattice. In Section 3, the Helmholtz problem is reduced to the boundary integral equations for the simple layer potentials of this Green function. In Sections 4 and 5, two methods of discretization of arising the boundary integral equations are discussed. The former is based on the piecewise approximation of the density. The latter uses the Fourier series of the density, and the explicit analytical form of matrix elements of the secular equation is obtained. The reality of the nonlinear spectral parameter of the boundary equations is proved. In Section 6, the results of numerical simulation for the low ladled part of the spectrum for the "desymmetrized" Berry billiard with the periodical boundary conditions and the rectangle quantum billiard with nonzero quasimomenta are presented.

2. G R E E N F U N C T I O N O F T H E P E R I O D I C A L L A T T I C E

Let {ej, j = 1,2} be the system of the noncolinear basis vectors in the plane R 2, determined early. We define the Green function G(z, z0) -- G(z, x0, pl, 1o2, ~) as the solution of the Helmholtz equation in the fundamental domain N:

AG + ,~a = 6(z - xo), (2.1)

Helmholtz Problem 615

where A does not belong to the spectrum of the homogeneous problem. The function G is expanded on all the plane R 2 by means of the quasi-periodical conditions:

G(x + ej) = e-~P~G(x), (2.2)

with the parameters {pj, j = 1,2}. Using the continuity of the function G and its derivatives, one can link the corresponding values on the opposite sides Lj and Lj+2 of the boundary 0f~ of the fundamental domain ~ (Figure 1):

G(X)IL~+ 2 = e-iPJa(X)lL~ , o £

On= G(X)lL¢+2 = -e-iP¢ G(X)ILj'

(2.3)

(2.4)

we have coefficients C~l being defined as

Ckl :

where

I A [ - I ~ (A - II~k ÷ v~l12) ' (2.12)

uk : (Pl + 2k~r)fl, vl -- (P2 + 2br)f2. (2.13)

Applying the inverse transformation, we obtain that G(x, z0) is defined by the following relation:

lffi~ e-i(Pa+2klr)(=-z°'Ia) e-i(p~+2br)(x-zo,f2) G(z, xo) : Ia1-1 ( ' A - ] ( ' ~ I ~ ~ ( ' ~ ~ " (2.14) k, -oo

where ~ is the normal derivative of the function G in the point x. Let {/j , j -- 1,2} be the attendant system of vectors connected with {ej, j = 1,2} by the

following relation: i f , , e~) = ~ , j . (2.5)

We introduce the new variables {yj, j = 1, 2} defined as

Yl = (X, fl) , Y2 = (x, f2). (2.6)

After this change of the variables, the domain fl is transformed to the unit square [0, 1] x [0, 1], and the boundary conditions may be written in the form

G(1,y2) = e-imG(O, y2), G(yl, 1) = e-~mG(yl ,0) ,

Oy, G(X, Y2) = e-iPl C~y, G(O, Y2), (2.7)

Ov2 G(yl , 1) = e-~P2 0~2 G(yl , 0).

In the new variables the Laplacian A has the form

0~2 2 02 02 /x = 11/1112 Yl + 11/211 b~y22 + 2(fl , /2)c 9yly2, (2.8)

and the 6-function is defined by the relation [5]

6(z - z0) = IAI-16(y - y0),

where IAI is the Jacobian of the transformation from y to x. The set of functions {e -i(p+2k~r)u, - c o < k < co} forms the basis in the space of one-dimensional functions {¢(y)} which satisfy the quasi-periodic relation

¢(y + I) = e-iP¢(y), (2.9)

We find the Green function in the form o0

G(yl,y2) = E Gkle-i(Px+2kTr)In e-i(P2+211r)It2" (2.10) kJ---c~

Using the orthogonality of functions Ckt (Y):

¢~i(Y) : e -i(pa+2k~r)ln e -i(p2+21~)v2, (2.11)

6 1 6 P . G . AKISSIN et al.

3. B O U N D A R Y I N T E G R A L E Q U A T I O N S

Now, we multiply the equation (2.1) by the solution of the initial problem u(x), subtract the equation (1.1) multiplying by G, and integrate over the domain ~ \ S containing the disc S (see Figure 2), and obtain

( AGu - AuG)ds:~ = u( xo ). (3.1) ks

From equation (3.1), it follows that

L( /0 u(xo) = oa ou ( a - n Onz Onz ] dl~+ \Onz nx - - u - b-h-2n, u d t z . ( 3 . 2 )

S

Here oa and o,, ~ are the limits of the normal derivative when point x tends to the boundaries of domains ~ and S, respectively, for the first and second integrals in equation (3.2). We show that the first integral with respect to OFt equals zero. Indeed, this integral is the sum of four integrals

n .__ i

(3.3)

Using the property of the Green function G and solution u(x) from (1.2), we obtain that

j j + 2

As a result, the equation (3.2) reduces to the following integral equation with respect to the boundary OS only:

u(xo) = s ~n~ G - ~ u dlx. (3.4)

Taking into account that on the boundary OS of the disc S the solution u(x) = O, we have

fo OU(x) U(Xo) = a(x , Xo, A) dlz. (3.5) S On=

Thus, one can find the solution u(x) in the form of the potential of single layer for the Green function G(x, xo) from the previous section. For determining A, the following integral equation on the boundary OS of the disc S takes place:

~os a(x)G(x, A) dlz = O, 6 0 S , (3.6) XO, XO

where a(x) = Onx *

It should be noted that only real values of the spectral parameter A correspond to the solution of this problem. Multiplying the equation (3.6) by a(x), integrating over the boundary OS, and substituting the explicit representation of the Green function (2.14), we have

fos fos exp(-i(pl + 2k~r)(x - xo, f l ) - i(p2 + 2/~r)(x - x0,/2)) × a(x)a(xo) dlx dlxo k,, (~ - INk +viii 2) -- 0.

(3.7) Then the following equation takes place:

k,l (A - ]luk + vl[]2) = 0. (3.8)

Helmholtz Problem 617

Taking the complex conjugate of the equation (3.8), we obtain the same equation

[fo$ exp(-i(pl -}- 2kzf)(x, f l ) - - ~(J92 "]- 217~)(X, f2))O'(X) dlz[ 2 = O. (3.9)

k,l (~ -- Ilu~ + viii 2)

After subtraction of the equation (3.8) from equation (3.9), we have

I A _ iluk + Vlll2l 2 = o. (3.10)

Taking into account that the expression in the square brackets is greater than zero, we have = A. This means that the spectral parameter A satisfying nonlinear equation (3.6) is real. It should be noted that the equation (3.6) has additional solutions which are not solutions of

initial problem (1.1)-(1.3). Let u*(x) be the solution of the Helmholtz equation (1.1) inside the disc S with the Dirichlet boundary condition:

u* (x) = O, x • OS.

We multiply the equation (1.1) by the Green function G, subtract the equation (2.1) multiplying by u*(x), and integrate over the domain S. After this we obtain

s(AU*G - AGu*)dsx = u*(xo).

From this equation, it follows that

u*(x0)= s k, on= G - ~ u* dl=.

Taking into account that u*(x) = O, x e OS, we obtain that A and u*(x) satisfy equation (3.6). The eigenfunctions u* i x) and eigenvalues A* of the internal Helmholtz problem for the disc S

may be expressed through the Bessel functions and these nodes and found analytically.

4. D I S C R E T I Z A T I O N O F T H E B O U N D A R Y I N T E G R A L E Q U A T I O N S

In the papers [6,7], the procedure of discretization of the boundary integral equations and algorithms of the determining eigenvalues of the arising nonlinear problems based on the Newton method with the extraction on founded roots was described. For discretization of the boundary integral equations, the collocation method with the piecewise approximation of the density a(x) was applied. The main difficulty of using the collocation method for integral equations (3.6) consists of the fact that the Green function (2.14) is a complex valued function. It means that solutions of the problem A may go out from the real axis to the complex plane as a result of the discretization errors of the boundary integral equations. That is why we use the momentum method for discretization of the problem. The boundary aS is approximated by right N-polygons. Let {S~} be sides of the inscribed N-polygon and ai be the approximation of a(x) on the sides Si. Then the discretized system of equations has the form

N

This system may be written in the matrix form:

i = 1, N. (4.1)

A(A)b = 0, (4.2)

618 P.G. AKISHIN et al.

where b = ( o ' 1 , o ' 2 , . • • , O'N) T and elements of the matrix A(A) are corresponding double integrals from equation (4.1). The equation (4.2) has a nontrivial solution when the determinant of the matrix equals zero:

det[A(A)] = 0. (4.3)

As a consequence of the chosen method of the discretization (the moment method), the elements of matrix A(A) satisfy the following relations:

aij = a;i. (4.4)

This means that this matrix is complex Hermitian. The reality of A may be proved by using the method from Section 3.

Thus, we constructed the approximated equations (4.2),(4.3) for determining eigenvalue A of the problem (1.1)-(1.3).

5. E X P L I C I T F O R M O F T H E S E C U L A R E Q U A T I O N

In the previous section, the discretization of the boundary integral equations based on piece- wise-polynomial approximation of the density a(x) has been proposed. The matrix elements were expressed by means of double integrals of the Green function. In this section, we improve the method of reduction of the boundary integral equations to an algebraic system using the Fourier series expansion of the density a(x).

The vectors x (x E OS) belonging to the circle of the radius R with center in the origin of coordinates may be written in the parametric form

x(¢) = (R cos ¢, R sin ¢)T, (5.1)

where 0 < ¢ <_ 21r. Let the function a(x) be expressed in the Fourier series

o o

o ( x ) = ~ ~me -'m~. (5.2)

From the equations (3.6),(5.2) it follows that

Crm G(x(¢),xo(¢l),A)e-im¢Rd¢ = 0. (5.3)

Substituting the Green function from equation (2.14) in the equation (5.3), we obtain

oo am E°° Jo"2~r exp{-iR((uk +vt)'e(dp)-e(¢'))} +vtll 2 (5.4) m ~ - - o o k~l=--oo

where e(¢) = (cos ¢, sine) T. The expression ((uk + vt), e(¢)) may be transformed to the form

((uk + v t ) ,e (¢) ) = Iluk + viii cos(~ -- Ck0, (5.5)

where (Uk + Vt, al)

~kt = aretan (uk + vl, a2)'

al and a2 are the orthogonal system of unit vectors in the Cartesian system of coordinates. Then equation (5.4) may be written in the form

o o o o

om ~ exp{i[ll~k + v, llneos(¢' - ¢~,)]} A~,m = 0, (5.6) ,,,=-~¢ ~ , l=-~ ,X -I lu~ + vtll 2

Helmholtz Problem 619

where

ff Aklm = exp{ - i ( l luk + v~llRcos(¢ - eke) + m¢)}Rd¢. (5 .7 )

Using the integral representation of the Bessel functions [8]

ff Jm(Z) = (21rim) -1 exp(iz cos u) cos(mu) du,

we obtain the analytical expression for the coefficients Aktm

Aktm = ( 2 7 r i m R ) J m ( - R l l u k + vzll) exp(-imCkl). (5.8)

Multiplying equation (5.6) by exp(in¢') and integrating over the interval [0, 2r], we have

oo oo A k l m ~

"" ~ (~, - I1,~,+ ~zll 2) = o, (5.9)

where n varies from - c¢ to ~ . It should be noted that the coefficients A~zm decrease fast as k, l, m tend to infinity. Taking a finite number of members in (5.9), we obtain the approximated system of equations for determining am and A:

M K,L A k , , , A k l , = 0. (5.10)

rn---.- M k, l f f i - K ,L

Let B be the complex matrix with elements bnra:

K,L Ak lmAkln ( 5 . 1 1 )

b,,,-,,= ~ (A -- I-F~k¥~II")" k,lffi-K,L

Then, instead of the system of equations (5.9), we consider the reduced system of equations

M a,nbnrn = O, n = - M , . . . , M . (5.12)

m ffi - M

ILl

R Figure 3. Berry's billiard for N -- 33: k -- - 1 6 , - 1 2 , - 8 , - 4 , 4 , 8 , 12, 16 (k is ordinal number of harmonics); pl -- 0, p2 -- 0.

620 P. G. AKISXlN et al.

R - 0 . 1

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. . . . . • . ; | = . . . . . . . . . . . . . . . . : : : : . - " . . : : h : i •

a " . , - ' " : : : ; : ' : i • " . . . . . ; i : i i ; | ' h , . • , , . . . | . . - , , . . , , ' t | ; ; h ; ; ' : " . , i . :~ : . . . . . : : , , . , . • I ' , ; , : : : : : * . . . . . o • : . . . , * , * * , * * * . • * , | * * ' ' ' * • * ' ' * " " , ~ I , " 11 . l . I * ' * " I * I . . . . . | i •

, tnn ~ ; . : l . , . . . . . . . . . . . . . . . . . : . : : : " . . . . . . . . . . . . . . . . . . . . . . . . . : : : : : " . . . . . . . . . . . . . . . : , . , - : i

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. . . . : h i : . . . . . . . . . . . . . . . . . . : I . : . : . . . . .

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| . . o " ° . * * * * , . " : | .

2 . 0 0 " : ' : . . . . . . . I . . . . ' . . . . . I , . . . . . . : : : : . . . . . . . . . . . : : . . . . . . . . . . . . . , , • . . . . . . . . . . . . . . . . . . . , , : . . . . . . . . . . . . : . . . . . . . • , * * * i I l l l * ° e l * ml I * l O l l , i o*

I l l O I I I I ! 0111111 .1111 I I • 1 • 1 = * o o l * * * l . i ! 11111 i l I I I * 1 I I I I i l l l l l I I i i i I l t l l i I I I I I I I t l l l l i l i I t I I I I I I I I I I

i i i

• . . . . . . . . : ; : : : : : ! 1 = I l l * " l i = i :11::::: ' . '" . . . . . . . . . . . . . . "*" . . . . . * , I I ' l l l m l l l l " " 1 ° i l l l lO lO* l i I I I I I ! 1 I I I I I l l l l I l i l l l l l l I . I I " 1 I I l l l l I l l t l I J i l l I I l l l l i l l l i I I I I I I l l i l l l l l l l l l I I I I I I . . . . . ' . ' " : ' : : : : : : : : : : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : , , : = : ,

1.UU . : : . ' U : : . . . . . . . . . . ' . . . . . . .

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * • * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ' * ° ' ' , - o * * , * . ° , * * . , * o * * * . . * . .

0.00 . . . . . . . . 1'.(~6 . . . . . . . 2' ()0 . . . . . . . 3' 0() . . . . . . . 4~ 6() . . . . . . . 5' (~0 . . . . . . . 6' I ~ ' 0 . 0 0 . . . . . pl

Figure 4. R - 0 . 2

W

5.00:1"=: . . . . ": o:* .." "-. :: l :; . . . . :==" " ' : . ; ~ . . . . . ; ; ~ : : . . . . . . . . . . . . ! :~1=; ' • " ~ , . : :

:!=:: .... :!,,!: .... :::::: ' : :::::::: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ::::::::: ' ::::: ..... :I,,F ..... :~: : [ " " : = =: . . . . . : := i . . . . . : : : : = o . , = . : : : : : : = = . . == == . . = : : : : : : : . l . . O : : : : : . . . . . i i :: . . . . . := : : * "

** °** ~le.$**.=o*'* *'*..o*.i.e| ~ ,=" *o * 0 = * * • I | • * o * * ' ° * * * * ° m * * ° * * I * l * * * l I * 4 . 0 0 - t * . . . . . =, . . . . . . . . . . . . . =. " . . . . . . . • . . . . . . . . . . . . . *= . . . . . ! " * * * = * " * ' * = * * , ' ' . ' ' °

: [ . . . : = . . . . . . . . . : := l ,= . . . . . . . . : := . . . . . . . . . . . =, , ; ; ; , ,= . . . . . . . . = , .= : : . . . . . . . . =, i : : : . . . . . . . . . : : . . . * * * * - . ,o *o o =* ,o

. • . * . . . . : o , : " " : , ° : . . . . * . . * " . . " . : . , : : " - . • . . . . : : , , : : " . . . " " ' . . . . . . . . . . ~ . . . . . . . : I i , ,=: . . . . . . . . . . . . . . . . ." . : , = ' " . . . " " . . . " : , , : • .

3 . 0 0 • , l= : . . . . . . . . . . . . . . . . . : : , . , I ' * * " . . . . : . , . ° : : . . . . . . . . . . . . . . . . . . I i ° • | * * I * , =* * i ' l =

1 i l l l l I l l i l l l l l l l l l l l l l l l l l i i ~ l l

. . . . . . . . . . . . . : , . l : : . . . . . . . . . : l : : : . : : = i : . . . . . . . . . . . : : , . , : : . . . . . . . . . . . . . . . . . . . . . . . . . . : : : : , . . I : : . . . . . : l * :1 : , =: . . . . . . : i , . , : : : : : . . . . . . . . . . . .

2.00 , : : " . . . . . . . . . . . . . . . . . . . . . " . . . . . . . . . : : l . . . . : : : : , . , : : . . . . . . . . . . . . . . . . . . . " . - . : : , . , : : : : . . . .

" * ' : l . l : : : :o .=: . . . . 1 l * . . , . * * : : S * , * * * * * * * * * * * * * * * * * * * * * * * * * * * . o : : : , . * * ° , : | o , .

• , e e i o l l l | | * l , | , * l . * . o , * , * . , , • • l . • O i l ' . . . . . . . . . . . : l i e * . l , * , * , , , * , , , • , * , . , . I | ! I l l l | e l | * . . . - : : : : l . . . n : : : . • . . . .

* ° * - 0 m | : • • * * * * * , , * ' • ° • * * * * * ° * * * ° , • * * , . : ! e ' * * * 1.00 I I | : . . . . . : : |ao **=1! . . . . . . . . . . . . . . . . . : :1o.* *=:: . . . . . . : : l l

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * | l l I | , , • , , l = . e l , . * * * | I I I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

° ° ° o . od . . . . . . . . lOO' . . . . . . . . i . . . . . 6a . . . . . . . 3. . . . . . . . N . . . . . . . i 6d . . . . . . . pl

Figure 5.

The system (5.12) has a nontrivial solution when the determinant of the matrix B equals zero. Note tha t the error of reduction in (5.10) may be majorized by the value of the order

c (5.13) 5: 5: 5: ÷v, tl31ml J [m[>Mlkl>K[ll>L

Here C is the maximum of module of the j t h derivative of the density a(¢) with respect to ¢. I t means tha t the error decreases fast if K, L, N tend to infinity.

Thus, for the fixed values of the parameters Pl, P2 of the periodical structure, we obtained the approximated equation for determining A and or(C). Note tha t truncated matrix B is complex

Helmholtz Problem

R-0.3

621

5.00

ILl

4 . 0 0

3.00

2.00

1.00

0.00

"', 'III IIII|•| ' ~ !7 . . . : : . . . . . : :::::: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : :::::::: . . . . . . . : : , , : . . . .

. . * * . . . • : : ~ | 1 1 1 * e l l l l l l l l l l l l l e * 1 1 1 ~ . . • . = . . . . . 0 . . * l l * " * * • { 1 0 1 1 ' * ' * * ' ' * * " * * * * * * • * l l l l i • * ' * ' : l l l l l l I •

'=I* el|l *'*" *=* ,****10 |e,**,,i, ****" . .**'" : I " i . . . : l , . , i . . . . . . . . , . . . . . . * . . . l i . * ' l . . . . . . . . . . ~,::-" . .... • . . . . . . . . . . . . .... ....

........ ::, ~ .............. ~:::, ,:...:;:, , .......... ~, *:;:...~, ,::: .............. :: ,:: ........

".. . - ' * : , , : :~ : : , , : ' " . . . ."

"::s;';: . . . . . . . . . . . . . : : ' ; , : : "

~:~ .................... ~ .~1: ............. :x,= , ...............................

..'u~ '::'m ,- ............ :' ": ................... : ......... ;" 'l: ..... : : : ' " i ' ~ - : ~ : : ~ P " " • • ' * i l l e O l l l l * ' l * • " I I l | ' l l * * * * * * * * * * * * * * * * * * * * * * * * * * ' * ' 0 ' * • * ' ' • ' ' ' e i . : i , . . • | 1 . " " "

........ : ' " ' : : : ! ! I ; : : : ; IH : : : . ' " ' ........ 'lli: ................................................. ::::lla

. . . . . . . . . . . . . . : : | l l o . , I ; . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : | l O . . 1 1 : . . . . . . . . . . . . . . . .

~d ....... I'.6d ....... 2'.6d ....... 3'.6d ....... ~.~d ....... s'.dd ....... ~.dd'

pl Figure 6.

R - O . ~

5 . 0 0

4.00

3.00

ILl

2.00

1.00

0.00

** ** i•• i.• ° ~ : , , * . . . . . . . . . . i , . l l ' . o l / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | h I | ! , I | : ' ~ . . . . . . | , ' : ~ .

• . . . . I . I | . , h . , . , , • . . . ° , . , , , . , . , , * , * , * - . * * * * * , * . * , . * , • , . * , . , * * * . , - . , * , . * * • . l * .s I . I , . . , •

..:..:. ...: ................................................. .. . . ~' ......... ~..:..:..

~h*l|**,•,,,.**,, ''''',•,,, .,,,•,''* ,,,,,,,,,,*,,*~I ''I''

''*., ''*'*''' .... ' .... $~III*,,III~:: .... ' ..... ''''' ,.''

**lllllll~,,e,*,*,,*,*,*,,,*,,h,~llllllll'' lllli,**o* • OlllUlllllllllll,,,.,,,,I,,,,,,,,••,••,,,,,,,,,,l,,•,,,*,llllll " "

*° '*•• • ° * ' " * , , , • ° ° ° . . , * * , , , . * * 0 , , , . , , . . . , * * • . - , ° , , e el*am °, I l l " , , ,

• . ° , • , ""la * * ' * * ' • ' * ° * * . . l • . . o , I , . , 0 , , , . , , , * I ° l j • ° j l l " * ' * . . . = , I .... ."t, ,i:: ...................... Ill,,*. .~1111:I. *,USll|l ................. .... :i, ,I: 1 .....

'II:::: ........................................................ :;::!I: • , l : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ::::, "

. . . : : : u , . , u : : : . . . .

d ....... 1'.dd ....... ~.6d ....... 3'.6d ....... ~.dd ....... ~.6d ....... ~.~d' pl

Figure 7.

Hermitian and using the techniques developed earlier, one can show that the roots A* of the

secular equation

det IIB(A*)IJ = 0 (5.14)

are real. Using the integral representation for u(z) from (3.5), the found values of the spectral parameter A, and the density ~(~), one can recalculate the values of the wave function u(z) in an arbitrary point of the plane R 2.

6. N U M E R I C A L SIMULATION

On the basis of proposed methods and algorithms the code for solving the plane Helmholtz problem has been created. The discretization from Section 5 has been used. The numerical

622 P.G. AKISHIN et al. R=0.5

5.00

4.00

3.00

I L l

2.00

1.00

0 . 0 0 0

°1. °" = . I . ° . . . . .= . • • • . . ° • ¢ ° ° • ° . . . . . ' " ° . • ° • . °= •

.°iI.,.l.,° ,°,°.•,.,,, .°°°,•°°,I" ..,l,,~Ill I '°°'' ..... ,,,,~IIllle • II°°'.oli • llllI]~,., .... '' .... °...'°,°,,,. ,,,°'°'''°*,o

. . . . . . . . . . . . . . : . . . : . . . . . . : . . . . . . . . . . . . . . . . . . . : . . . . . . : . . . : . . . . . . . . . . . . . . " ' ' ' ' ' . . ° . . . ° ° . . . . ° . . . . . • ° ° ° ' ' ' ° ° "

" • ° ' ' ° ° ' 4 ° ° . ° . , . °

° ' ' ' ' O a ' H ° ° . O O . O . • ° ° ° ° ° . ° ° ° ° H . . ° , , . ° • • " o ' ' ~ . - O.°. . . . . ° , , . ° . ° ° ° . ° . , ° . ° . ° ° ° ° . ° ° ° . ° . . g . ° . . ° , ' ° ° . ° ° . ° ' , . . ' ° ° ° ° ° ° ' ° ' ' ° g ' ° ' ° ° ° ° ' ° ° ' ° ' ' 0 " " . . . . . . " ° ' ' '

. . . . . . :e = : : . : : n t : o . . - ° "

s . . . . , . . , o . - ' ' ' " ' - , , , , , , , . . . , ' ' ' ' =

"•''''' le ..°.,•.,t. ¢l,,........ "''°'',,°

. . . . . . . . . . . . . . . . . . := • , l : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : : : ' ¢ " = = . . . . . . . . . . . . . . . . . . .

" : : : : = , , . . , : : : : :

1 . = . . . . . • . . . • ' ° ° ' ' ' " " ' * ° ° ' ' ' . . . H • . = . ! . l l | l l l l l ! i i i 1 | 1 1 1 1 i i i i i i i i 1 | 1 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1 1 1 1 | 1 1 ~o ~.~o 2.60 ~.oo ~.oo 5.oo ~.oo

p]

2.00

1.50

I L l 1 .00

0.50

Figure 8.

R = O . I O N = 9

* - - - N = 1 7 . . . . . . ° . . , e 16 . • . . . . = , . . . . . .

" . . . " ' . , . , : : . . , . . . . , . . . . , . . . . , . . . . , . . : : , . . * ' " . . . " " . . . " ' . , . . . , . : : : : ' , . . . . , " . . . . "

' ' , , , ° l * , i o ~l °° ' ' * , • °° ..*.o o *°o .. . °°*.° °,°. o°. I .°,.. °°,.. .°°° .°,°.

=,.°. =..°. .I,. " ..,.. ..° °° °.,°° ..°°*o'" .° °o.° °.*....,o°..*." ....° .. "°*°°°.

. ' " . , . . . . , . . . . , . . . . , . . : . ~ . . . . . . . F : . . , . . . . , . . . . , . . . . , . . " - . . . : . . , . . . . , . - . . , . . . . * . . . . . : . . . : : . " - . * . . . . , . . . . "* . . . . , . . . : ¢ ,

,i~. °,....,.. .o... . ° . . . .~°..°,° ~ , I " , ° . , * i . . ' * ' ' ' ° * ° ° " , ° . . " ° ' ° " " ° ' ' * . , . . " ' ° * ' ° ° ' * ° ' ° ' * -~°o° |

L' .~: : . . . . . . . . . . . . . . . . . . . . . . ::~-:,: • ". ' ' 4 . . •* '* .°

" . , . ° . . , - " • "4.• . •*•" • . * . . • . . * - "

" * ' , ° ° . • • . , , , . . ° , • , • ' * °

, .oQ~°'°'$1'''*''il*'°*°~'~..* ,.q.'*'°''~'''" " "'''*'*''*''..*.°

....,....* .... *.°'' ''*...°,....~ ....

o.oo ......... ~6 ........ ' ......... ' ......... ' ........ ~ 6~ ....... ~.66 ' 0.00 1. 2.00 3.00 4.00 .

pl Figure 9.

results of computation of the spectrum for orthogonal basis lattice vectors and zero phase shifts conditions with the known ones [2] have been performed. Results of calculations of energy levels 5 < En(R) <_ 100 of the "desymmetrized" Berry billiard (Figure 3) are reproduced till the radius R = 0.4 and new parts of the energy levels E~(R) over interval R = [0.4,0.5] are produced by means of the proposed approach using the N = 33 harmonics of the corresponding discretized nonlinear problem (i.e., k = - 1 6 , - 12 , - 8 , - 4 , 4, 6, 8, 12, 18, where k is ordinal number of har- monics). The solid lines correspond to solutions of the internal plane Helmholtz problem. The picture of spectrum with respect to radius of discs are in good agreement with Berry's picture [2]. The numerical study for the general case billiard for orthogonal basis lattice vectors and nonzero quasimomenta Pl,P2 is carried out. In Figures 4-8, the numerical energy levels E are presented

H e l m h o l t z P r o b l e m 623

ILl 1.00

R = 0 . 2 0 N = 9

ILl 1.00

2.00 * N = 1 7

1111:::: " " ' " , . ' "

o~O° ,o°° • o ° ~ , . . s ' " . * " " * . . . . ' " "

o°l ~•°.°$•°o°~ oO "o

t . 5 0 ~ . . ~ . . . , . . . . , : : . * : ' " * . . . . * . . . . * . . . . * . . . . * ' " 1 " . : : , . . . . , . . . ; & . . . ,

• , , . . + . . . . ~ . . . . . . . . . . . . . . : . . . . . . . . . . . : : . . . . . . . . . . . . . . . * - . + . . . . l" • • ooO.. °°

. . . . . """ • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "i : : , . . . . "

. . " . : : , . . . . . . . . . • . . . . . . . . . . . . . : : ; ~ . . . . " . .

.~ . . . . . . . . . . . , . . . . , . " . . . . ~. o° ° °$ . • o.,O" . . °

" . ° ° . , ° ° °$ . " °. • ° ° • . , . . = & . - . * ' ° . . "

° o . . " * - ° 0 . , . . . ° , . ° ° ° , . . . . • . ° . .

" ° ° . o .°" °°

'"""::::.:=... • ........ @@@ @0@

0.50 • . . . . * ° ° ' ° * ' ' ' ' * . . . . • 0 8 . ~ . . . . , . . . . ~ . . . . * . . . . * ' ' ' ' ~ . . . . • . . . . *

°'°°0.06 . . . . . . . . , . 6 6 . . . . . . . 1.66 . . . . . . . 3'.66 . . . . . . . i.&~ . . . . . . . s'.66 . . . . . . . d6~' p l

Figure 1 0 .

R = O . 3 0 N = 9

2.00 * N = 1 7 " . , . . , . . . . . . . . i : . . . * . . . . * . . . . * . . . : i . . . . . . . . , . . , . . ' "

i I j o o , , o o l • I I • • °° ' Q ° ' * * . e e" • . . . . * . . . . . " * . o . . ~ • °,oO" ~. o° °o o " ' . , 0

" . ~ . . . . , . o ~ . . . . , . o . . , . . . . , . 0 . , . . o o , o . o o $ o . . o , ° . . ° , ° . . . $ . . . o , . * . , . . . . * " ° ' * ' " ' T ° ' * . . . . $ ' : * °oe~e °° °e el °° °•el° . . . , . I , , ] L . . . . e . . . . , l . . . , . . a$ . . . . $ . . . . $ . . . . , . . . . $ . . . . , . . . . , . . . . , . . . • $ n " * ' " : $ ' ' " * ' " ' 1 j " : ' $ . . . .

" . . 0 . . " . . . . " ' ; . . . " . . , . . " 1.50 "'*" "" "'" " ' " " ' * " • . o ~ . . o°°° . . . ° ..~..o •

. . . . . , . . . ; ~ ~ : . . . * " . . . • • ° . o , . . : , . . . . * . . . . $~,*, $ . ° " .

°o °•Oo° ooO°

°•*eo • • ° " °Oo. °°oO

. . . . , . . . . , . . . . , ° ° ~ ' ~ ° . . e . . . . * . . . . • . . . . * . . . . $ . . . . * . * . 0 e . . . . , . . . . , . . o ° , . . . . * . . . t . ' ° . . , . . . . , . . . . , . . . .

0 . 5 0 ° ° • * ° e ° . ° O°°Oo°.eOOaoo. . . . . • • o ° ° o . ° O O o* ° ° ' ° * °°

"~:=~,:~'

0 . 0 0 , , , , , . , , . , , . , . , , , , . , . , , , , , , . , . . , , , . , , , , , , , , ,

6 1.&i i.66 ~'.66 ~.66 s'.66 6',~ p l

Figure 1 1 .

for different values of radius R with respect to quasimomenta Pl (P = pl). In Figures 9-13, the comparmon of numerical energy levels E is presented for total number of harmonics N = 9 and N = 17. From these figures, one can see that the proposed algorithms provide the stable calculations of levels of the spectrum for small radius, but for radius of discs near R = 0.5 the dependence of the dimension of discretized problem holds. For evaluating the spectrum in the general case and especially near the close packing limit, additional investigations should be done.

7 . C O N C L U S I O N

We have reduced the Helmholtz problem (1.1)-(1.3) to the boundary integral equation for the simple layer potentials of the constructed Green function on the boundary of only one disc. This

624 P. G. AKISHIN et aL

R=O.4-O N = 9

. . ; ;g. * - - - N = 1 7 ... 2.00 . . . . P ' . . .. .. . . " ; "

~.. . . . . ; ; . . . . . " ,

• " " " . o ' $ ' " g '$ ' . . - ' " ' * • " , , , , ' I Q , . , o

1.50

I , I 1 . 0 0

0.50

0.00 0

, • O - , o . . ' Q • , . ." ;,... . . . ; ".. . . . ' * " ; . . . .~. . . .~ $ .... ~ . . . . ;" *...g

. , • * . .

. . . . • . . . : 8 " . . , . . . . • . . . . • . . . . • ' - . o * - . . . • . 1 0 . . . . ~ . . . . , j . * . . . . • . . ' . * . . . . • . . . . * . . . . , . . ' 8 . . . . • . . . . . . " . e . . . ' i " " e ' . . . , " ' . .

le I , a o o , e J o ' ~ eml " " ° ' . ~ e e e Q o e , " , ) ,:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :.

" '" 'g '" '"" '" '"" ..... " ...... :, .s::'"" ...... " '° '""'g'"'" '" '""

,d ....... I'.6d ....... i.6d ....... d.6d ....... ,;.6d ....... s'.dd ....... £6d' pl

Figure 12.

R = 0 . 5 0 - - - N = 9

2 0 0 . . . . . . . * N = 1 7 . . . . • ..~ " * * * l . " I , , I o . l o g o l • "ae°a i . ° " . . * o" *

j I ° l l e l t l 4 1 1 o l ° 4 ~ ~ J l l e . t l . l l p l l l ' e j

• ~ * • • • • •

1.50 .-"" " *"" . . . . . . . . . . . . . . . • " ' - . i ' ' • • a ' ' ' e * , , , , o , o * ' Q ' ' * • ' o

• ;" • • " ; , . . * • • 'Q

• " , , f r e e | ° , • .4 . . ' • . . . . . . . .;. ; . * " . . . • ' . j ,o e , , ' ~ ' * • * I ' ' ° ' . - , ge l 4 , " , . . . . . . . . ; * ;, . . . . . . . . . , . . . . . . . . . . . . . . . . . ; . . . . ;" • , ; . . . . ; . . . . . . . . . . . . .

LLJ 1.00 ~ . . . . * * • * .. .

" S . . ° o • t

. . . . ~ . . . . • . . . . • . . . . ~: , .~ . . . . • . . . . • . . . . • . . . . • . . . . • . . . . , . . . . , . . . . , . . . . • . . . . ~ . . : , . . . . , . . . . • . . . . t . . . .

0.50 ~ " ' " . . . . . . . . . " "

1 "::s.s:~ "g

0 . 0 0 I l e l l i e I l J l I I I I I I l l | I S l l l i l l l j l l I I I l l i l l l l l l I I i i i j l l I l l l i s i j l l 0.00 1.00 2.00 3.00 4-.00 5.00 6.00

pl Figure 13.

a p p r o a c h allows one to eva lua te the spec t rum of the p lane Helmhol tz p rob lem for nonor thogona l

basis l a t t i ce vec tors and nonzero quas imomen ta Pz, P2. The a lgor i thm of solving nonl inear dis-

c re t ized p rob lem is improved. Two me thods of d i sc re t iza t ion of b o u n d a r y in tegra l equa t ions have

been developed. T h e first one is more universal . I t m a y be used for a wide t ype of domains . Bu t

th i s m e t h o d needs c rea t ion of fast a lgor i thms of ca lcula t ion of the Green funct ion and i ts deriva-

t ives wi th respec t to the spec t ra l pa ramete r . The second approach allows one to eva lua te the

e lements of the d iscre t ized m a t r i x analyt ical ly , bu t i t m a y be used only for discs. Bo th me thods

of d i sc re t i za t ion of t he b o u n d a r y integral equa t ions preserve the Hermi t i c i t y of the d iscre t ized

ma t r i ce s and the rea l i ty of t he s p e c t r u m of the a p p r o x i m a t e d problems. As one can see, the above

Helmholtz Problem 625

methods provide new construct ive ways in the s tudy of the quan tum mot ion of a free particle on

the periodical sys tem of reflecting discs or some plane periodical s t ructures with quasi-periodical

b o u n d a r y conditions. The generalization of the above approach to the three-dimensional case can

be made in the framework of this considerat ion and will be useful for s tudy of the hard sphere

gas.

R E F E R E N C E S

1. Ya.G. Sinai, Dynamical systems with elastic reflections. Ergodic properties of scattering billiards, Russ. Math. Surv. 25, 137-189, (1970); Introduction to Ergodic Theory, Princeton University Pres8, Princeton, NJ (1976).

2. M.V. Berry, Quantizing a classically ergodic system: Sinai's billiard and the KKR method, Ann. Phys. 131, 163-216 (1981).

3. P.G. Gaspard and G. Nicolis, Transport properties, Lyapunov exponents, and entropy per unit time, Phys. Rev. Left. 65, 1693--1696 (1990).

4. Yu.A. Kuperin, S.B. Levin, Yu.B. Melnikov and B.S. Pavlov, Quantum Lorentz gas: Effective equations and spectral analysis, Preprint IPRT, Nr. 19-94, St. Petersburg, (1994).

5. L. Hermander, The Analysis of Linear Partial Differential Operators, Vol. 3, Springer-Verlag, New York, (1983).

6. P.G. Akishin and I.V. Puzynin, Implementation of Newton's method in Sturm-Liouville difference problems, Preprint JINR, 5-10922, Dubna, (1997).

7. P.G. Akishin, V.P. Akopian and E.P. Zhidkov, Solution of the two-dimensional Helmholtz equation by the boundary integral equation method, Preprint JINR, Pll-87-738, Dubna, (1987).

8. P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part 2, McGraw-Hill, New York, (1953).