Electromagnetic space-time crystals. II. Fractal computational approach
aa r X i v : . [ qu a n t - ph ] O c t Electromagnetic space-time crystals. II. Fractal computational approach
G. N. Borzdov ∗ Department of Theoretical Physics and Astrophysics,Belarus State University, Nezavisimosti avenue 4, 220030 Minsk, Belarus
A fractal approach to numerical analysis of electromagnetic space-time crystals, created by threestanding plane harmonic waves with mutually orthogonal phase planes and the same frequency, ispresented. Finite models of electromagnetic crystals are introduced, which make possible to obtainvarious approximate solutions of the Dirac equation. A criterion for evaluating accuracy of theseapproximate solutions is suggested.
PACS numbers: 03.65.-w, 12.20.-m, 02.60.-x, 02.70.-c
I. INTRODUCTION
Fractal approach makes possible to obtain and/or re-search objects with any level of complexity by usingsimple algorithms [1]. It provides useful tools to de-sign noval devices, such as fractal antennas, filters, dif-fusers, absorbes, microwave invisibility cloacks, and frac-tal metamaterials [2–4], as well as effective algorithmsfor computer qraphics and fractal compression of digitalimages [5].In this paper we present a fractal computational ap-proach to calculating the fundamental solution of theDirac equation describing the motion of an electron in anelectromagnetic field with four-dimensional (4d) period-icity (electromagnetic space-time crystal, or ESTC) [6].The electromagnetic field is composed of three standingplane harmonic waves with mutually orthogonal phaseplanes and the same frequency. In this case, the Diracequation reduces to an infinite system of linear matrixequations. Each equation of the system relates 13 Fourieramplitudes [bispinors c ( n + s )], where the multi-index n = ( n , n , n , n ) is a point of the integer lattice L with even values of the sum n + n + n + n , and theshift s = ( s , s , s , s ) ∈ L takes all 13 values satisfyingthe condition g d ( s ) = 0 ,
1, by definition (see appendix) g d ( s , s , s , s ) = max {| s | + | s | + | s | , | s |} .In other words, each amplitude c ( n ) enters in 13 differ-ent matrix equations of the infinite system. The funda-mental solution of this system is obtained in the previouspaper [6] by a recurrent process. It is expressed in termsof an infinite series of projection operators. This processbegins with the selection of an infinite subsystem con-sisting from independent equations and the calculationof the projection operators ρ ( n ) = P ( n ) , n ∈ F ⊂ L ,which uniquely define the fundamental solutions of theseequations [6]. At each new step of the recurrent pro-cess, we add another infinite set of mutually independentequations (MIE) which, however, are related with someof the equations introduces at the previous steps. Con-sequently, we obtain an infinite set of independent finite ∗ [email protected] systems of interrelated equations [fractal clusters of equa-tions (FCE)]. It can be described as a 4d lattice of suchclusters. Each step of the recurrent procedure expandsFCE for which it provides the exact fundamental solu-tions. The presented in Sec. II fractal algorithm of thisexpansion is devised to minimize volumes of computa-tions and data files. Some MIE (aggregative MIE, orMIE1) just add one equation to each cluster of the pre-vious FCE lattice so that these enlarged clusters remainindependent. Other MIE (connective MIE, or MIE2), byadding each equation, interrelate a pair of neighboringclusters into a joint cluster, and a quite different FCElattice arises. Each fractal period includes connectionsin directions of n , n , n , and n axes, respectively. Thesmaller is FCE, the smaller are volumes of computationsand data files, which are necessary to find and to writedown the fundamental solution for this FCE. To simplifycalculations, we add a maximal possible number of MIE1before adding the next MIE2.In Sec. III, we discus the interrelations between thefundamental solution and approximate partial solutionswhich can be obtained in the framework of finite modelsof ESTCs. In Sec. IV, a criterion for evaluating accu-racy of approximate solutions is suggested, which plays agreat role in numerical analysis of ESTCs. Results of thisanalysis will be presented in the subsequent paper. Theintroduced in appendix sequential numbering of points n ∈ L drastically simplifies numerical implementation ofthe presented technique and analysis of solutions. II. FRACTAL SPLITTING OF THEFUNDAMENTAL SOLUTION
Due to the specific Fourier spectrum [6] of the 4d peri-odic electromagnetic field of ESTC, in the current seriesof papers we use indexing of Fourier components of thewave function and many other mathematical objects bypoints n = ( n , n , n , n ) of the integer lattice L witheven values of the sum n + n + n + n . The funda-mental solution S and the projection operator P of theinfinite system of equations under study are defined asfollows [6] S = U − P , P = + ∞ X k =0 X n ∈F k ρ k ( n ) , (1) + ∞ [ k =0 F k = L , F j \ F k = ∅ j = k, (2)where U is the unit operator and ρ k ( n ) are Hermitianprojection operators with the trace tr [ ρ k ( n )] = 4. Tospecify the lattices F k , we split the lattice L into fractalsubsets and designate stages of calculation by ( s t , p h ),where s t = 0 , , . . . , and p h = 1 , s t .Let L set [( a , a , a , a ) , ( b , b , b , b )] be the subset of L given by a i ≤ n i ≤ b i , i = 1 , , ,
4, where a i and b i are some integers. At the initial stage s t = 0, we split L into subsets L ss [(0 , , s ] ≡ L ss [(0 , , s ] which can beobtained by periodic translation of the central subset L cs (0 , ≡ L cs (0 ,
2) = L set [( − , − , − , − , (2 , , , p st (0) = { , , , } , i.e., byshifts s = 4( k , k , k , k ), where k i are integers. Here, L ss [( s t , p h ) , s ] signifies the subset obtained by a shift s of the central subset L cs ( s t , p h ) of stage ( s t , p h ). Eachsubset L ss [(0 , , s ] contains 128 points of L .The lattice L can be composed of point lattices num-bered u = 1 , , . . . , and specified by the center c ( u ) andthe list of periods p ( u ). At stage (0 , F of 8 sublattices ( u = 1 , . . . ,
8) with equal peri-ods p ( u ) = p st (0) = { , , , } and the following list ofcenters c L (0 , ≡{ c ( u ) , u = 1 , . . . , } (3)= { (0 , , , , ( − , − , − , − , (1 , , − , − , (1 , − , , − , ( − , , , − , (0 , , , , (2 , , , , (2 , , , } ⊂ L cs (0 , . It is easy to verify that, any two points n, n ′ ∈ F satisfythe condition g d ( s ) >
2, where s = ( s , s , s , s ) = n ′ − n . In this case, ρ ( n ) ρ ( n ′ ) = 0 [6].At stage (0 , F , . . . , F which have the same periods p ( u ) = p st (0) = { , , , } , numbers u = 8 + k = 9 , . . . ,
14, and the list ofcenters c L (0 , ≡{ c ( u ) , u = 9 , . . . , } (4)= { (0 , , − , − , (0 , − , , − , ( − , , , − , (0 , , , , (1 , , , , (1 , , , } ⊂ L cs (0 , . For any points n and n ′ of lattices with numbers u and u ′ (1 ≤ u, u ′ ≤ n ′ − n = c ( n ′ ) − c ( n )+4 { k , k , k , k } ,where k i are some integers. It is easy to check that g d ( n ′ − n ) > k i is not zero. Becauseof this, one can calculate ρ k ( m ) at k = 1 , . . . , ρ j ( n ) for which j = 0 , . . . , k − n belongs to the subset L ss [(0 , , s ] containing m .This conclusion follows immediately from the recurrentrelations presented in [6].We define lattices F k in such a way that, at any stage( s t , p h ), calculations of ρ k ( m ) and ρ k ( m ′ ) can be carriedout independently at different subsets: L ss [( s t , p h ) , s ]and L ss [( s t , p h ) , s ′ ]. To fulfil this condition, one can addonly a finite number of point lattices at each stage, inparticular 8 and 6 at stages (0 ,
1) and (0 , k > F k comprises only point lattice with u = 8 + k . At stages s t = 1 , , ,
4, which constitute thefirst cycle of fractal expansion, we have the following listsof periods: p ( u ) = p st (1) = { , , , } for u = 15 , . . . , , = p st (2) = { , , , } for u = 43 , . . . , , = p st (3) = { , , , } for u = 103 , . . . , , = p st (4) = { , , , } for u = 403 , . . . , , and the central subsets L cs ( s t , p h ): L cs (1 ,
1) = L set [( − , − , − , − , (2 , , , , L cs (1 ,
2) = L set [( − , − , − , − , (2 , , , , L cs (2 ,
1) = L set [( − , − , − , − , (2 , , , , L cs (2 ,
2) = L set [( − , − , − , − , (6 , , , , L cs (3 ,
1) = L set [( − , − , − , − , (6 , , , , L cs (3 ,
2) = L set [( − , − , − , − , (6 , , , , L cs (4 ,
1) = L set [( − , − , − , − , (6 , , , , L cs (4 ,
2) = L set [( − , − , − , − , (6 , , , . At the phases p h = 1 and p h = 2 of any stage s t , equalnumbers of point lattices are added, namely, 14, 30, 150,910 for s t = 1 , , ,
4, respectively. At s t = 1, the centersof lattices ( u = 8 + k = 15 , . . . ,
42) are defined as c ( u + 14 p h ) = c ( u ) + ( − p h (0 , , , , (5)where u = 1 , . . . , , p h = 1 ,
2. At s t = 2 ,
3, and 4, thecenters of the lattices added at p h = 1 and p h = 2 arerelated by shifts as follows: c ( u + 30) = c ( u ) + (4 , , , , (6) s t = 2 , u = 8 + k = 43 , . . . , c ( u + 150) = c ( u ) + (0 , , , , (7) s t = 3 , u = 8 + k = 103 , . . . , c ( u + 910) = c ( u ) + (0 , , , , (8) s t = 4 , u = 8 + k = 403 , . . . , . To define c ( u ) at p h = 1, we use the lists: c ′ L (4 ,
1) = { (3 , , , (5 , , , (3 , , − , (5 , , − , (6 , , − , (4 , , − , (4 , , − , (5 , , − , (5 , , − , (4 , , − , (5 , , − , (4 , , − , (3 , , − , (3 , , − , (4 , , − , (4 , , − , (3 , , − , ( − , , , (1 , , , ( − , , − , (1 , , − , (2 , , − , (0 , , − , (0 , , − , (1 , , − , (1 , , − , (0 , , − , (1 , , − , (2 , , − , (0 , , − , (2 , , − , ( − , , − , ( − , , − , (2 , , − , (0 , , − , (2 , , , (0 , , − , ( − , , − , (1 , , − , ( − , , , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , , , ( − , , − , (3 , , , (5 , − , , (3 , , − , (5 , , − , (6 , , − , (4 , − , − , (4 , , − , (5 , , − , (5 , , − , (3 , , − , (4 , , − , (5 , , − , (4 , , − , (3 , , − , (3 , − , − , (3 , , − , (4 , − , − , (4 , , − , (4 , , , (4 , , − , (3 , − , − , (3 , , − , ( − , , , (1 , − , , ( − , , − , (1 , , − , (2 , , − , (0 , − , − , (0 , , − , (1 , , − , (1 , , − , ( − , , − , (0 , , − , (1 , , − , (2 , − , − , (0 , , − , (2 , − , − , ( − , , − , (2 , , − , ( − , − , − , ( − , , − , (2 , , − , (2 , , − , (0 , − , − , (1 , , − , (0 , , − , (2 , , − , (2 , , , (0 , , , (0 , , − , ( − , − , − , (1 , , − , ( − , , − , (1 , − , − , ( − , − , , ( − , , − , ( − , , − , ( − , − , − , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , , − , ( − , − , − , ( − , , − , ( − , − , − , ( − , , − , ( − , , − , ( − , , − , ( − , − , − , ( − , , − , ( − , , − , ( − , , − , ( − , , , ( − , , − , ( − , − , − , (3 , − , , (3 , − , − , (6 , − , − , (4 , − , − , (5 , − , − , (5 , − , − , (3 , − , − , (4 , − , − , (5 , − , − , (4 , − , − , (3 , − , − , (3 , − , − , (4 , − , − , (4 , − , , (3 , − , − , ( − , − , , ( − , − , − , (2 , − , − , (0 , − , − , (1 , − , − , (1 , − , − , ( − , − , − , (0 , − , − , (1 , − , − , (2 , − , − , (0 , − , − , ( − , − , − , (2 , − , − , ( − , − , − , (2 , − , − , (2 , − , − , (1 , − , − , (0 , − , − , (2 , − , − , (0 , − , , (1 , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − , ( − , − , − } , c ′ L (3 ,
1) = { (6 , − , , (3 , − , , (3 , , , (5 , − , , (5 , − , , (4 , − , , (4 , − , , (4 , − , , (5 , − , , (3 , − , , (2 , − , , ( − , − , , ( − , , , (1 , − , , (1 , − , , (0 , − , , (0 , − , , (0 , − , , (1 , − , , (2 , − , − , ( − , − , , (2 , − , , ( − , − , , ( − , − , , ( − , − , , ( − , − , , ( − , − , , ( − , − , , ( − , − , − , ( − , − , } ,c ′ L (2 ,
1) = { ( − , , , ( − , , , ( − , , , ( − , , , ( − , , , ( − , , } . Each five consecutive centers n = ( n , n , n , n ) = c ( u )in any of Eqs. (6)–(8) have the same projection ontothe three-dimensional (3 d ) space consisting of points( n , n , n ). They differ only by values of n , namely, n = 4 , , − , , − | n | + | n | + | n | is even, and n =3 , − , − , , − | n | + | n | + | n | is odd. In particular, c (43) = ( − , , , , c (44) = ( − , , , , . . . , c (47) =( − , , , − , c (48) = ( − , , ,
4) and so on. Because ofthis, the presented above lists of 3d projections c ′ L ( s t , c ( u ) at s t = 2 , ,
4, and u = 43 , . . . , , , (5 , , . . . , (8 , u = 2223with c (2223) = (4 , , , − , p st (5) = { , , , } , L cs (5 ,
1) = L set [( − , − , − , − , (6 , , , u = 108526 with c (108526) = ( − , − , , − , p st (8) = { , , , } , L cs (8 ,
2) = L set [( − , − , − , − , (18 , , , . III. APPROXIMATE SOLUTIONS
Numerical implementation of the obtained solution im-plies the replacement of the projection operator P (1) ofthe infinite system of equations [6] P ( n ) C = 0 , n ∈ L (9)by the projection operator P ′ = X k ∈ k L X n ∈ n L ( k ) ρ k ( n ) (10)of its finite subsystem P ( n ) C = 0 , n ∈ L ′ = [ k ∈ k L n L ( k ) ⊂ L , (11)where k L is an ordered finite list of integers, and n L ( k )is a finite list of points n ∈ F k , taking into account.Here, C is the so-called multispinor [6] defined as the set C = { c ( n ) , n ∈ L} of the bispinor Fourier amplitudes c ( n ) of the wave function in the Dirac equation, treatedas an element of an infinite dimensional linear space V C .The projection operator S ′ = U − P ′ (12)defines the exact fundamental solution of Eq. (11) andan approximate solution of Eq. (9). In particular, usingthe described above fractal lattices F k , we can set k L = { , , . . . , } , n L ( k ) = F k \ L cs (4 , . (13)In this case, the system (11) contains 5150 equa-tions, including all 2048 equations with n ∈L set [( − , − , − , − , (4 , , , ⊂ L . In some specialapplications, it may be advantageous to restrict both therecurrent relations [6] and the system (11) to a subset of k L (13) and subsets of n L ( k ) (13), i.e., to a more simplefinite model of the infinite electromagnetic crystal.In the subsequent paper, we will illustrate the pre-sented technique by some results of its computer sim-ulation. To this end, we restrict our consideration to thecase when the amplitude C specifying a partial solu-tion [6] is given by C = a j e j ( n o ) , n o = (0 , , , , (14)and L ′ ⊂ L cs (4 , e j ( n ) is the basis in V C [6], andsummation over repeated indices is carried out from 1 to4. In this case, the relation C = { c ( n ) , n ∈ S d } = S ′ C = C − P ′ C (15)describes the four-dimensional subspace of exact solu-tions of Eq. (11), i.e., for any given bispinor a = a a a a , (16)it specifies a partial solution, where S d ⊂ L is the subsetof L with nonzero bispinors c ( n ), for brevity sake, it willbe referred as the solution domain.Bisbinors c ( n ) and a are linearly related as c ( n ) = S ( n ) a , (17)where S ( n ) is the 4 × (cid:10) θ i ( n ) , S ′ e j ( n o ) (cid:11) of the operator I ( n ) S ′ I ( n o ) = S ij ( n ) e i ( n ) ⊗ θ j ( n o ) . (18)Here, θ j ( n ) = e † j ( n ) is the dual basis in the space of one-forms V ∗ C , I ( n ) = e j ( n ) ⊗ θ j ( n ) is the projection operatorrelated with point n ∈ L . From Eqs. (10), (12), and (18)it follows S ( n ) = U δ ( n − n o ) − X k ∈ k L X m ∈ n L [ k ] R k ( n, m, n o ) , (19) where U is the unit 4 × δ ( n − n o ) is the Kroneckerdelta, matrices R k ( n, m, n o ) are defined in [6]. Substitut-ing of c ( n ) into the Fourier series, specifying the bispinorwave function Ψ [6], givesΨ( x ) = X n ∈ S d c ( n ) e iϕ n ( x ) ≡ E ( x ) a , (20)where x = ( r , ict ), and E ( x ) = X n ∈ S d e iϕ n ( x ) S ( n ) (21)is the evolution operator. In terms of the dimensionlesscoordinates r ′ = r /λ = X e + X e + X e , X = ct/λ , and the dimensionless parameters q = ~ k m e c , q = ~ ωm e c , Ω = ~ ω m e c , (22)the phase function ϕ n ( x ) can be written as ϕ n ( x ) = ( k + k n ) · r − ( ω + ω n ) t = 2 π [( n + q / Ω) · r ′ − ( n + q / Ω) X ] , (23)where n = n e + n e + n e , Ω = ~ ω / ( m e c ), ω isthe frequency of the electromagnetic field, k = ω /c =2 π/λ is the wave number, ~ is the Planck constant, m e isthe electron rest mass, c is the speed of light in vacuum.The evolution operator E ( x ) is the major character-istic of the whole family of partial solutions Ψ( x ) (20).In particular, it provides a convenient way to calculatemean value h A i of an operator A with respect to functionΨ( x ) h A i = a † A E a a † U E a , (24)where A E = Z dX Z dX Z dX Z dX E † ( x ) AE ( x ) , (25) U E = Z dX Z dX Z dX Z dX E † ( x ) E ( x )= X n ∈ Sd S † ( n ) S ( n ) . (26)In the subsequent paper, we will use four finite modelsof ESTC, designated p -models with p = 0 , , ,
3. Theydiffer in level of accuracy, volume of calculations, andfield of application. The most simple 0-model with k L = { } and L ′ = { n o } is sufficient to obtain the free spacesolution from Eq. (20) as the limiting case at vanishingfield. In p -models with p >
0, the list k L begins with zeroand contains in order increasing numbers k of all lattices F k satisfying the condition g d [ c (8 + k )] ≤ p . The set L ′ comprises all points n of these lattices, complying withthe restriction n ∈ L cs (4 , F (see Sec. II). For example, in 1-model we use the list k L = { , , , , , , , , , , , , } and the system (11) containing 998 equations. In p -models with p = 2 and p = 3, k L has 69 and 210 mem-bers, and the system (11) consists of 1520 and 2199 equa-tions, respectively. IV. EVALUATING ACCURACY OFSOLUTIONS
The distinguishing feature of the presented techniqueis that each step of the recurrent procedure expands thesubsystem of equations for which it provides the exactfundamental solution. One can check the calculation foraccuracy by using relations [6] ρ † k ( n ) = ρ k ( n ) = ρ k ( n ) , tr [ ρ k ( n )] = 4 , n ∈ L , (27) ρ k ( m ) ρ l ( n ) = 0 if k = l or (and) m = n. (28)In terms of matrices R k ( m ′ , m, n ′ ), they can can writtenas: X n ∈ F d ( k,m ) tr [ R k ( n, m, n )] = 4 , (29) X p ∈ F d ( k,m ) R k ( m ′ , m, p ) R k ( p, m, n ′ ) = R k ( m ′ , m, n ′ ) ,m ′ , n ′ ∈ F d ( k, m ) , (30) X p ∈ F d ( k,m ) ∩ F d ( l,n ) R k ( m ′ , m, p ) R l ( p, n, n ′ ) = 0 (31)if k = l or (and) m = n, and m ′ ∈ F d ( k, m ) , n ′ ∈ F d ( l, n ) , where F d ( k, m ) is the subset of L containing n ′ withnonzero matrices Φ k ( m, n ′ ) ( F -domain, see [6]). For theproblem under study, the Dirac equation reduces to aninfinite system of homogeneous linear equations with ma-trix coefficients V ( n, s ) [6]. Substitution of Eq. (17) intothe left side of these equations reduces it to the form V S ( n ) a , where V S ( n ) = X s ∈ S V ( n, s ) S ( n + s ) , (32)and S is set of shifts s with g d ( s ) ≤ S (A.28)]. At n ∈ L ′ , theequation V S ( n ) a = 0 is satisfied at any a , because inthis domain V S ( n ) ≡
0. This provides means for finalnumerical checking of the fundamental solution S ′ of thesystem (11) and the evolution operator E ( x ) (21) foraccuracy. Let D be a differential operator in a space V Ψ of scalar,vector, spinor, or bispinor functions, and k Ψ k be thenorm of Ψ on V Ψ . The functional R : Ψ
7→ R [Ψ] = k Ψ D kk Ψ k (33)where Ψ D = D Ψ, evaluates the relative residual at thesubstitution of Ψ into the differential equation D Ψ = 0.It provides a fitness criterion to compare in accuracy var-ious approximate solutions of this equation. For an exactsolution Ψ, the residual Ψ D vanishes, i.e., R [Ψ] = 0. IfΨ D = 0, but R [Ψ] ≪
1, the function Ψ may be treatedas a reasonable approximation to the exact solution, andthe smaller is R [Ψ], the more accurate is the approxima-tion. In terms of distances d = k Ψ k and d D = k Ψ D k ofΨ and Ψ D to the origin of V Ψ (the zero function), onecan graphically describe R [Ψ] as shrinkage in distance R [Ψ] = d D /d . The functional R , as applied to a familyof functions Ψ( x , λ ) with members specified by a param-eter λ , results in function R [Ψ( x , λ )] of λ , denoted below R ( λ ) for short.To introduce this criterion in the problem under con-sideration, we first transform the Dirac equation in [6] tothe equivalent equation D Ψ = 0 with the dimensionlessoperator D = X k =1 α k (cid:18) − i ~ m e c ∂∂x k − A ′ k (cid:19) − i ~ m e c ∂∂t + α , (34)where α j are Dirac matrices, and A ′ k is defined in [6].From Eqs. (20) and (34) followsΨ D ( x ) = D Ψ( x ) = D ( x ) a , (35)where D ( x ) = D E ( x ) = X n ∈ Sd [ D n − D A ( x )] e iϕ n ( x ) S ( n ) (36)is the evolution operator describing the family of remain-der functions Ψ D , and D n = X k =1 α k ( q k + n k Ω) − U ( q + n Ω) + α , (37) D A ( x ) = X k =1 α k A ′ k ( x ) . (38)The norm of Ψ D (35) can be written as k Ψ D k = q a † U D a , (39)where U D = Z dX Z dX Z dX Z dX D † ( x ) D ( x )= X m,n ∈ S d S † ( m ) [ D m D n δ ( n − m ) (40) −A ( m, n ) D n − D m A ( m, n ) + A ( m, n ) U ] S ( n ) , A ( m, n ) = X k =1 α k X j =1 [ A jk δ ( n − m + s j )+ A ∗ jk δ ( n − m − s j ) (cid:3) , (41) A ( m, n ) = X j,l =1 ( A j · A l δ ( n − m + s j + s l )+ A j · A ∗ l δ ( n − m + s j − s l )+ A ∗ j · A l δ ( n − m − s j + s l )+ A ∗ j · A ∗ l δ ( n − m − s j − s l ) (cid:1) , (42) s = (1 , , , , s = (0 , , , ,s = (0 , , , , s = ( − , , , ,s = (0 , − , , , s = (0 , , − , , vectors A j and their components A jk are specified in [6].Thus, for the function Ψ (20), from the definition (33)follows R = s a † U D a a † U E a , (43)where U E and U D given by Eqs. (26) and (40), respec-tively. V. CONCLUSION
The projection operator S ′ (12) defines the exact fun-damental solution of the finite subsystem (11) which ex-pands with each new step of the recurrent process. Therelations presented above form the complete set which issufficient for the fractal expansion of this subsystem toa finite model of ESTC of any desired size. A criterionfor evaluating accuracy of the approximate solutions, ob-tained by the use of such model, is suggested. It plays aleading role in search for best approximate solutions inthe framework of the selected model. The correspondingexamples will be presented in the subsequent paper. Appendix
In this series of papers, we intensively use index-ing of various mathematical objects by points n =( n , n , n , n ) of the integer lattice L with even valuesof the sum n + n + n + n . The introduced below se-quential numbering of these points drastically simplifiesboth numerical implementation of the presented fractaltechnique and analysis of solutions, because it takes intoaccount the specific Fourier spectra of the electromag-netic field of ESTC and the wave function, as well thestructure of the finite models of ESTCs described above.It is of particular assistance in the analysis of partial so-lutions with the localized amplitude C (14). Let us define functions g d ( n ) and g d ( n ) of n =( n , n , n , n ) ∈ L as follows g d ( n ) ≡ g d ( n , n , n , n ) = | n | + | n | + | n | , (A.1) g d ( n ) ≡ g d ( n , n , n , n )= max {| n | + | n | + | n | , | n |} . (A.2)For any n ∈ L , integers g d ( n ) and g d ( n ) have the sameparity. First we split L into the infinite sequence of finitesubsets G { p } ( p –generations) composed of all n ∈ L with g d ( n ) = p, p = 0 , , , . . . . Next we split G { p } into sub-sets G { p,r } composed of members n with g d ( n ) = r ≤ p .Then we split G { p,r } into subsets G { p,r,n } of members n = ( n , n , n , n ) with equal values of n . Finally, wesplit G { p,r,n } into subsets G { p,r,n ,n } of members withequal values of n . These inclusion relations can be writ-ten as n ∈ G { p,r,n ,n } ⊂ G { p,r,n } ⊂ G { p,r } ⊂ G { p } ⊂ L . To introduce a sequential numbering i = 0 , , , . . . ofpoints n ∈ L in the direction of increasing p = g d ( n ), wefirst assign the global number i = 0 to the single member n o = (0 , , ,
0) of G { } ≡ { n o } and local numbers i =1 , . . . to members of G { p,r,n ,n } , which differ from oneanother only by values of n and n , as follows i = 1 for n = 0 and n ≤ , = 2( R + n ) for n < , = 2( R + n ) + 1 for n > , = 4 n for n = 0 and n > , (A.3)where R = | n | + | n | = r − | n | , see also Fig. 1. Thetotal number of G { p,r,n ,n } members depend on R as N ( R ) = 1 for R = 0 , = 4 R for R > , (A.4)and i = 1 , . . . , N ( R ).Next we introduce local numbers i of G { p,r,n } mem-bers as i = M ( r, n −
1) + i , (A.5)where M ( r, n ) = n X n ′ = − r N ( r − | n ′ | ) (A.6)= 0 for n < − r, = 1 + 2( n + r )( n + r + 1) for − r ≤ n ≤ , = 1 + 2 r ( r + 1) − n ( n + 1 − r ) for 0 < n < r, = 2 + 4 r for n = r > , and i is defined by Eq. (A.3). The total number of G { p,r,n } members is N ( r ) = M ( r, r ) = 1 for r = 0 , = 2 + 4 r for r > , (A.7)
112 3412 34 56 78 - - n - - n FIG. 1. Numbers i (A.3) at R = 0 (the central point), R = 1(4 points connected by the dash lines) and R = 2 (8 pointsconnected by the solid lines. and i = 1 , . . . , N ( r ). One can visualize G { p,r,n } asa set of elements n = ( n , n , n , n ) with projections( n , n , n ) onto the three–dimensional space, lying inthe eight faces of the regular octahedron with the sixcorner points ( ± r, , , (0 , ± r, , (0 , , ± r ).To enumerate elements of G { p,r } , we specify thenumeration order of its subsets G { p,r,n } by j =1 , . . . , j max ,where j = 1 for r < p = | n | and n < , = 2 for r < p = | n | and n > , = − n for r = p and n < , = 1 + n for r = p and n ≥ , (A.8) j max = 2 for r < p = 1 + p for r = p. (A.9)All these subsets have the same total number of members N ( r ), so that N ( p, r ) = 2 N ( r ) for r < p, = (1 + p ) N ( p ) for r = p (A.10)is the total number of G { p,r } members enumerated by i = ( j − N ( r ) + i , (A.11)where i is given by Eq. (A.5).At any given p , p –generation G { p } consists of subsets G { p,r } , where r has the same parity as p and takes k max = p − p r min = [1 − ( − p ] / r max = p .The total number of elements of all subsets G { p,r ′ } ⊂ G { p } with r ′ ≤ r ≤ p is given by M ( p, r ) = k X j =1 N [ p, p − k max − j )] (A.13)= 0 for r < r min , = 23 ( r + 1)(2 r + 4 r + 3) for r < p, = 1 for r = p = 0 , = 43 p (4 p + 5) for r = p > , where k = r/ − p ] /
4. The total number of p -generation members is N ( p ) ≡ M ( p, p ), and thesemembers are enumerated by i = M ( p, r −
2) + i , (A.14)where i is given by Eq. (A.11).Finally, we introduce the global numbering [ n =( n , n , n , n ) ∈ L 7→ i ] of the lattice L points as i = M ( p −
1) + i = M ( p −
1) + M ( p, r − j − N ( r ) + M ( r, n −
1) + i , (A.15)where M ( p ) = p X k =1 N ( p ) = 23 p ( p + 1)(2 p + 2 p + 5) (A.16)is the global number of the last element of p -generationordered as described above.With this numeration, L becomes the ordered infite setand the inverse mapping s : i s h ( i ) = ( n , n , n , n )is defined as follows. The number i = 0 defines n o =(0 , , , i >
0, we first find the generation number p from the condition M ( p − < i ≤ M ( p ) (A.17)and calculate the local number i = i − M ( p − . (A.18)Next we determine r and i from the relations M ( p, r − < i ≤ M ( p, r ) (A.19) i = i − M ( p, r − . (A.20)The relations (A.8), (A.9) and (A.11) make it possiblefirst to find j from the condition j − < i N ( r ) ≤ j (A.21)and then n = ( − j p for r < p, (A.22)= ( − p + j +1 j + (cid:2) ( − p + j − (cid:3) / r = p, i = i − ( j − N ( r ) . (A.23)Thereafter we find n and i : M ( r, n − < i ≤ M ( r, n ) , (A.24) i = i − M ( r, n − . (A.25)Finally, we obtain the last two components n = − R + i / (cid:2) ( − i − (cid:3) / , (A.26) n = ( − i ( | n | − R ) , (A.27)where R = r − | n | .As an illustration let us consider the values of thefunction s h ( i ) at i = 0 , . . . ,
68, which define all mem-bers of p -generations with p = 0 , ,
2. The total num-ber N ( r ) (A.7) of the members of the set G { p,r,n } de-pends only on r , in particular, N (0) = 1 , N (1) = 6,and N (2) = 18. The first value s h (0) = n o = (0 , , , G { , , } ≡ G { } ≡ { n o } .The following six values [ s h ( i ) , i = 1 , . . . ,
6] are the mem-bers of the set G { , , − } , whereas the next six mem-bers [ s h ( i ) , i = 7 , . . . ,
12] are members of G { , , } . Eachof the sets G { , , − } and G { , , } has only one member: s (13) = (0 , , , − s (14) = (0 , , , i = 15 , . . . , i = 33 , . . . ,
50; and i = 51 , . . . ,
68, thefunction s h ( i ) gives the members of G { , , } , G { , , − } ,and G { , , } , respectively. Consequently, the list S ofthe first 69 values of the function s h ( i ), which containsmembers of p -generations with p = 0 , , S = { (0 , , , , (A.28)(0 , , − , − , (0 , − , , − , ( − , , , − , (1 , , , − , (0 , , , − , (0 , , , − , (0 , , − , , (0 , − , , , ( − , , , , (1 , , , , (0 , , , , (0 , , , , (0 , , , − , (0 , , , , (0 , , − , , (0 , − , − , , ( − , , − , , (1 , , − , , (0 , , − , , (0 , − , , , ( − , − , , , (1 , − , , , ( − , , , , (2 , , , , ( − , , , , (1 , , , , (0 , , , , (0 , − , , , ( − , , , , (1 , , , , (0 , , , , (0 , , , , (0 , , − , − , (0 , − , − , − , ( − , , − , − , (1 , , − , − , (0 , , − , − , (0 , − , , − , ( − , − , , − , (1 , − , , − , ( − , , , − , (2 , , , − , ( − , , , − , (1 , , , − , (0 , , , − , (0 , − , , − , ( − , , , − , (1 , , , − , (0 , , , − , (0 , , , − , (0 , , − , , (0 , − , − , , ( − , , − , , (1 , , − , , (0 , , − , , (0 , − , , , ( − , − , , , (1 , − , , , ( − , , , , (2 , , , , ( − , , , , (1 , , , , (0 , , , , (0 , − , , , ( − , , , , (1 , , , , (0 , , , , (0 , , , } . [1] K. Falconer, Fractal Geometry: Mathematical Foundationsand Applications (John Wiley, 2003).[2] M. W. Takeda, S. Kirihara, Y. Miyamoto, K. Sakoda, andK. Honda, Phys. Rev. Lett. , 093902 (2004).[3] B. Hou, H. Xie, W. Wen, and P. Sheng, Phys. Rev. B ,125113 (2008). [4] W. J. Krzysztofik, Microwave Review , 3 (2013).[5] R. E. Chaudhari and S. B. Dhok, Int. J. Comput. Applic.57