Reccurent sequenses in solving the Schrödinger equation

Transcription

1 It. Jl. of Multiphysics Volume 9 Numbe 5 57 eccuet sequeses i solvig the Schödige equatio Alexade F. Polupaov Kotel iov Istitute of adio-egieeig ad Electoics of the ussia Academy of Scieces, Mohovaya st. /7, 59, Moscow, ussia ABSTACT A explicit umeical-aalytical method is demostated fo accuate solvig the Schödige equatio i those cases whe this equatio educible to a system of coupled odiay diffeetial equatios with sigula poits. Fudametal system of solutios is costucted as algebaic combiatios of powe seies, powe fuctios ad logaithmic fuctio i the eighbouhood of the egula sigula poit ad as asymptotic expasios of solutios i the eighbouhood of the iegula sigula poit. The method is based o the calculatio of ecuet sequeces of costat matices of coefficiets i powe seies ad i ivese powe seies i asymptotic expasios usig deived ecuet elatios, that maes possible to calculate solutios at ay give poit usig oly algebaic computatios ad elemetay fuctios. I tu it maes possible to solve accuately the eigevalue poblem ad scatteig poblem ad to deive aalytical expessios fo the wavefuctios. The method is applied to calculatios of eegies ad wavefuctios of the discete spectum ad wave fuctios of the cotiuous spectum of the hydoge-lie atoms ad of acceptos i semicoductos. Keywods: umeical methods, Coulomb potetial, odiay diffeetial equatios, sigula poits, ecuet sequeces, eigevalue poblem, scatteig poblem. INTODUCTION We coside the Schӧdige equatio HΨ () z = EΨ(), z () whee the Hamiltoia H is a quadatic fom of the mometum, i those cases whe Eq. () is educible to a system of coupled odiay diffeetial equatios with sigula poits. Hamiltoia H ca be epeseted i the followig fom: () H = Ap + B P + V(), m= m m () whee p = -i is a mometum opeato ( = ), ae the compoets of the ieducible spheical teso opeato of the secod a [] composed of the compoets of the P m () *Coespodig Autho: sashap@cplie.u; sashap55@ mail.u

2 58 eccuet sequeses i solvig the schödige equatio symmetic teso P = p p δ p, i i i ad A, B m ae costat matices. We suppose that 3 V() is the Coulomb potetial. Usig the expasio of the wavefuctio Ψ(z) i the complete basis fuctios of agula vaiables o if ecessay the fuctios of agula ad spi vaiables, such as fuctios i the -S couplig scheme (see [], ad the efeeces cited theei) it is possible to educe () to a system of coupled adial equatios: w d + p d + ( q + q + q ) =, (3) d d whee w, p ad q i ae costat matices, = ( τ ) is a -dimesioal columfuctio, is a quatum umbe of the obital agula mometum opeato = p, τ is a set of some quatum umbes chose i accodace with a Hamiltoia symmety. Hemitia chaacte of the Hamiltoia imposes the followig coditios o the matices of coefficiets: w = w* > (this matix is popotioal to the ivese masses matix), * * * * * * p = 4w p, q = q + w p, q = q.,, Thee ae two sigula poits of Eq. (3): the egula sigula poit = ad iegula sigula poit =. Eq. (3) descibes Coulomb states of a paticle (o of two attactive paticles, i.e. of excito) i vaious systems. I case = these ae states of a hydoge-lie atom, if > (3) descibes states of, e.g., a shallow accepto impuity o excito i semicoductos i diffeet appoximatios. Usig the substitutio φ = we educe the system of secod-ode + equatios (3) to a system of fist ode equatios: φ d, = α() φ α() = α + α + α. (4) d It is obvious ow that = is the egula sigula poit (by defiitio). Hee α = Q ( P ), α =, Q α = Q ae matices, Q = w ( q p ) +, = Q w q,, P = w p. The methods of costuctio of the fudametal system of solutios of Eq. (4) i the eighbouhood of the egula sigulaity = ad solvig the sigula eigevalue poblem ad the scatteig poblem, that ae based o the ecuet sequeces pocedue wee developed i [3, ] i geeal case, i.e. whe a() i (4) is a abitay holomophic at the poit = N N matix fuctio. I the peset pape we deive solutios of (3) i the eighbouhood of the iegula sigulaity = i geeal case of abitay usig a method specifically petiet i case of Eq. (3) (sectio ), ad study paticula poblems ( =, i (4)), i.e. deive solutios of (3) ad solve the eigevalue poblem ad elated poblems both i the case whe exact aalytical solutios ae ow, that maes possible to estimate the accuacy of the method, ad i cases whe oly umeical methods ae applicable (sectios 3, 4). Note that the method based o the use of ecuet sequeces of costat coefficiets allows us to compute the solutio at ay give poit, usig oly simple algebaic computatios ad elemetay fuctios, without the use of ay step-by-step pocedues.

3 It. Jl. of Multiphysics Volume 9 Numbe SOUTIONS IN THE NEIGHBOUHOOD OF THE IEGUA SINGUAITY = I ode to costuct the fudametal system of solutios () of Eq. (3) i the eighbouhood of the iegula sigulaity = we deive asymptotic expasios of the fuctios f() = () at. We educe the system of equatios (3) to a system of fist f ode equatios usig the substitutio Χ () = df : d dχ() = β() Χ () = β + β + β Χ(), d (5) whee β = Q, β =, (6) Q P β = Q, To aalyse the stuctue of the matix β whose fom detemies the behaviou of the solutios of (5) at we coside the followig eigevalue poblem β ξ λ η = ξ η, whee ξ, η ae -dimesioal vectos, ad we obtai fom (6) η = λξ; -Q ξ = λ ξ. Note that the matix Q is simila to a Hemitia matix, i.e. its eigevalues ae eal, ad eigevectos fom complete system i the -dimesioal space. We suppose hecefoth that thee ae + positive ad - egative eigevalues of the matix Q ( = ), ad thee ae o multiple oes amog them. Thus the matix Q is simila to the diagoal matix diag( ε,..., ε, ε,..., ε ). Hece β is a plai matix, its eigevalues ae ulie umbes: + ± i ε,, ± i ε, ± ε,, ± ε, ad eigevectos ae of the fom + ξ ( ) ( ε ) ξ / ( ), ξ ( ) ε ( ) ξ / ( ), =,,. Obviously these eigevectos fom complete system i the -dimesioal space. We pefom the followig substitutio: whee Χ = T, T β T = Β, =,., (7) Ζ / / / / / / / / Β = diag[( ε ),,( ε ), iε,, iε, ( ε ),, ( ε ), iε,, iε ]. + +

4 6 eccuet sequeses i solvig the schödige equatio The Eq. (5) assumes the fom dζ Β Β = Β + + Ζ. d (8) The asymptotic expasio of solutios of Eq. (8) Z() at is of the fom (see [4]): Ζ + + D D D p () = exp l = p C. p (9) C i Eq. (9) is a abitay -dimesioal vecto, ad D ae matices, ad the followig elatios ae valid: ) = ˆ ; ) D ; 3) ; = Β ( ),, α =,, αα 4) D = diag,. Costat matices, D, (ecuet sequeces) satisfy the followig ecuet elatios: ( ) Β Β = D + D Β Β + ( ). l l l l l= () ( ) αβ It is easy to compute all equied compoets, α β, ad ( D ) αα usig the fact that B o is a diagoal matix with ulie eigevalues. 3. THE CASE OF A HYDOGEN-IKE ATOM The followig adial equatio (Eq. (3), = ) is cosideed, d d d ( + ) α E = d () () whee is the quatum umbe of the obital agula mometum, a is a itege (α = coespods to the Coulomb attactio i a hydoge atom), () is a Coulomb adial wavefuctio, E is the eegy. Geeally speaig, distaces ad the eegy ae measued i the uits of a= κ /( me 4 ) (effective Boh adius) ad κ y = me /( ) (effective ydbeg costat) espectively, whee e electo chage, m electo effective mass o educed mass i case of two attactive paticles (excito), κ static dielectic costat. Note that this equatio descibes the states of a hydoge atom o hydoge-lie doo impuities i the diect-gap semicoductos. 3.. FUNDAMENTA SYSTEM OF SOUTIONS The eighbouhood of the egula sigula poit =. The egula solutio of Eq. () is of the fom () = a, = whee coefficiets a satisfy the followig ecuet elatio a a + Ea = C, a = ( )( + + ), >. () (3)

5 It. Jl. of Multiphysics Volume 9 Numbe 5 6 a is a abitay costat C that is detemied by omalizatio. The secod, iegula solutio of Eq. () assumes the fom: I () = b + l (). = (4) Hee () is the solutio () with a coefficiet a : b + Eb = + Coefficiets b satisfy the ecuet elatio: a. b + Eb + (+ ) a b = D, b =, b =, ( )( + + ) (5) Coefficiet b is a abitay umbe ad we set b = that esults i the popotioality of a l ad D. The value D is detemied by omalizatio. As follows fom the esults of [3], powe seies i (), (4) coveges uifomly i the I whole iteval (, ) ad (), () ae liea-idepedet solutios of Eq. (3). It is clea that to calculate adial wavefuctios with ay give accuacy at a abitay poit ˆ,< ˆ <, it is sufficiet to tae ito accout oly a fiite umbe N i the powe seies. The value of N is limited oly by the compute esouces ad oud-off eos i the pocess of computatios. The eighbouhood of the iegula sigula sigula poit =. As it follows fom the esults of sectio 3, the asymptotic expasios of the fuctio f () = () ad its deivative at ae as follows F ~ + ( P ) = df P ωi, ~ λ ω, λ = E i (6) d = ( B ) i ω = C exp (( B ) + ( B ) l ), i =, (7) i i i i = ( ) Matix ecuet sequeces P ad B ae detemied by the followig ecuet elatios ( ) p AP PB = PB A P + PB ( ) P,. s s s s s= s= (8) Matices B ae diagoal ( B ) = ( B ) δ, B = A, ( P ) = δ, (P ) ii =,, ad i i i i i A = λ α, ( + ) A =, A =. λ λ 3.. THE MATCHING OF THE SOUTIONS. EIGENVAUE POBEM The covetioal eigevalue poblem fo Eq. () is i the followig: oe should fid such value of the eegy E: E < (eigevalue) that the solutio () is fiite at =

6 6 eccuet sequeses i solvig the schödige equatio ad () at, oe should fid also the wavefuctio coespodig to this value of E ad omalized by the coditio (eigefuctio): () d= (9) To solve the poblem it is ecessay to pefom matchig at some itemediate poit = ȓ, < ȓ < of a fuctio f () = () ( is the egula at = solutio ()) ad its deivative (the left solutio) with a ight solutio that teds to zeo as ad its deivative (defied by (6) (8) with i = ). We desigate by Φ ( E ) a colum composed of the values that tae the left solutio ad its deivative whe a = (3) at the poit = ȓ at some give E, ad by Ω ( E ) - a colum composed of the ight solutio ad its deivative whe С =. We defie a matix ( ) AE ( ) = Φ ( E), Ω ( E). To complete the matchig it is ecessay to esue compliace with the followig equatio: AE ( ) χ( E) =, χ( E) = a ( E) C ( E). () Thus the pocedue of calculatig some eigevalue E = E is educed to the umeical solutio of the equatio det AE ( ) = () The, afte fidig the eigevalue E ad usig (), (9) oe detemies values of costats C (E ) ad a l (E ) ad thus completely detemies eigefuctio ad its deivative o (, ). I ode to estimate the accuacy of the computatios the eegies ad wavefuctios of a few lowest boud states of a hydoge atom wee calculated. I this case exact aalytical solutios of the poblem ae well-ow (see e.g. [5], [6]). Eige eegies equal: E = = Eigefuctios (adial wavefuctios) of two lowest states with,, / = ae of the fom: = e, = e ( / ) (the fist idex is the picipal quatum umbe, the secod is ). Calculated values ae close to exact oes with high accuacy. I paticula, i case of above metioed states 4 sigificat digits i calculated eigevalues ad i the eigefuctios coicide with the exact oes. Eigevalue poblem i the case of a fiite iteval. I this case the eigevalue poblem fo Eq. () o a fiite iteval [, ] is i the followig: oe should fid such value of the eegy E that the solutio () is fiite

7 It. Jl. of Multiphysics Volume 9 Numbe 5 63 at = ad ( = ) =, oe should fid also the wavefuctio coespodig to this value of E ad omalized by the coditio: () d= () Note that the poblem (i the simplest model) about states of a hydoge-lie doo impuity, located i the cete of a semicoducto spheical quatum dot of the adius (see e.g. [7]) is educed to this oe. The poblem should be solved umeically sice aalytical solutios ca be foud oly i the cases whe the value coicides with the positio of a ode of some wavefuctio of the poblem at the semi-ifiite iteval [7]. I the peset case the poblem of fidig of eigevalues is educed to the umeical solvig of the equatio ( =, E ) =, whee (, E ) is the egula solutio (). Note that idex i this otatio coespods to the oe i the otatio of the solutio of the poblem at semi-ifiite iteval (i.e. whe ). Fo defiiteess we set a = i (). Afte fidig some eigevalue E = E ad implemetatio of itegatio () oe has a expessio () with a = / Q fo a omalized solutio, whee aa m + m+ 3 Q =. (4) = m= ( + m+ 3) Some esults of calculatios ae demostated i Table. Hee idices i the otatio of the eegy E = Е l coespod to those i the otatio l (). Table : The depedece of the eegies of the lowest thee Coulomb states of the adius (3) E, E, E, , , , ,

8 64 eccuet sequeses i solvig the schödige equatio I paticula it ca be see fom the table that at fiite eegy level is o loge coespods to the umbe (picipal quatum umbe): accidetal (Coulomb) degeeacy of the states with =, =, is emoved. As it was metioed above the exact aalytical solutio ca be foud i the case whe the value of coicides with the positio of a ode of some wavefuctio of the poblem at the semi-ifiite iteval. It ca be see fom the table that at = the computed lowest eigevalue (goud state eegy) equals E =.5 (with high accuacy, i.e. zeoes afte 5). To this value coespods the solutio = / () Ce ( /) of the poblem at the semiifiite iteval, which has a ode at =. It meas that this fuctio is the eigefuctio of the poblem o the fiite iteval which is omalized accodig to () if we set C = / Q, whee Q= 4e. Oe ca estimate the accuacy of the method i paticula compaig the value of Q computed usig Eq. (4): Q = , ad usig exact fomula: Q= 4e = sigificat digits i computed values of the eigefuctio coicide with the exact oes at all < <. Eigevalue poblem i the case of the had coe potetial with a Coulomb tail. Models that use the had coe potetial ae cosideed i vaious applicatios, ad methods of solvig poblems of quatum mechaics i the pesece of this potetial ae of special iteest (see e.g. [8]). A model of the had coe potetial with a Coulomb tail may be useful to descibe the shallow states of the boud multiexcito complexes i semicoductos. The followig eigevalue poblem fo Eq. () is cosideed: oe should fid such value of the eegy E: E < that the solutio () = at HC ( HC is a adius of the had coe potetial) ad () at, oe should fid also the wavefuctio coespodig to this eigevalue ad omalized by the coditio: () d =. HC (5) To solve this poblem it is ecessay to fid the coefficiet M = M(E) i a liea combiatio of egula () ad iegula (4) solutios fo which the followig coditio is satisfied: I ( = ) = ( = ) + M( E) ( = ) =. HC HC The it is ecessay to pefom matchig of a fuctio f whee is the () = (), () left solutio, ad its deivative with the ight solutio ad its deivative (see ()) at some itemediate poit = ȓ, < ˆ <, the calculate a eigevalue by the umeical HC solutio of Eq. () ad fid the wavefuctio, give the coditio (5). Some esults of the calculatios ae peseted i Figues ad. 4. COUOMB HOE STATES Coulomb hole states, such as the states of shallow acceptos o excitos i semicoductos with degeeate valece bads, ae descibed by the uttige Hamiltoia [9]. Withi the so-called spheical appoximatio [] uttige Hamiltoia ca be witte as [, ] HC (6) () () H = p μ ( P J ) +. (7)

9 It. Jl. of Multiphysics Volume 9 Numbe 5 65 Figue : The wavefuctio of the goud state i the case of the adius of the had coe potetial HC =.4 ( =, E = -.53) Figue : Depedece of the goud state eegy of the adius of the had coe potetial Hee p is the mometum opeato; P ad J ae ieducible spheical teso opeatos of the secod a [] deived fom the compoets of p ad vecto J epesetig the 3 pseudospi agula mometum with J = ; μ = (6γ + 4 γ )/5 γ, whee γ i ae empiical 3 costats uttige paametes of the valece bad [9]; the eegy ad distaces ae 4 measued i uits of = m e / κγ ad a= κγ / m e espectively, m is the mass a

10 66 eccuet sequeses i solvig the schödige equatio of a fee electo, κ is the static dielectic costat. Hamiltoia (8) is spheically symmetic i the coupled obital ad spi spaces ad the total agula mometum F = + J is a costat of motio. Wavefuctios ca be witte as []: { h l z h l z } Ψ = ( β f + f ) JFF + ( f β f ) +, JFF, whee JFF Z ae fuctios i the -J couplig scheme [, ], β = 3 F+ F + 3/ F / /. The fuctios f h (), f l () ae expessed usig the compoets of a solutio of Eq. (3) = : f = ( + β ) ( β + ), f = ( + β ) ( β ). Eq. (3) is educed h + l + + f h to a set of systems of equatios of the same fom ( = ), with f = istead of, with f l diagoal matices w, q, q : w = diag ( - μ, + μ), q =, q = E ad with a ati-diagoal. matix p. Explicit expessios fo the matices p ad q i (3) ae peseted i []. Each system of equatios (3) ad each state coespods to a cetai value of the total agula mometum F (half-itege) ad paity P = (-). Note that i the peset case Eq. (3) descibes a couplig of states of two paticles with diffeet masses, i.e. /( - μ) (heavy hole) ad /( μ) (light hole), by the Coulomb potetial. Two egula at = exact solutios of (3) ( left solutios) ae of the fom () () f f, = ρ = = () ρ () () f = f + Kf l, (8) whee ρ = + 3, ρ = + ad ecuet sequeces the ecuet elatios: f i ad a costat K ae foud fom () Γ ( ρ ) f = () () () Γ ( ρ ) f + q f + q f =, =,, ( f =, < ) Γ ( ρ ) f = () Γ ( ρ ) f + q f = () () () () Γ ( ρ ) f + q f + q f + K[( ρ + ) + p ] f = ρ+ ρ, =, 3, (9) Hee Γ (ρ) is a sequece of the matices: Γ ( ρ) = ( + ρ)( + ρ ) + p ( + ρ) + q, =,, Fo completeess, we peset also the expessio fo the two emaiig iegula solutios of the fudametal system of solutios: (3) ρ3 (3) f = f + f l Kf = () () l,

11 It. Jl. of Multiphysics Volume 9 Numbe 5 67 (4) ρ l 4 (4) (3) () f = f + f l f + Kf = l. 6 3 () To deive all ight solutios, i.e. asymptotic expasios of solutios at we use the method of Sectio ad educe (3) to a system of 4 fist-ode equatios (8) usig the substitutio f df d = S Z, whee S = λ λ h h λ l λ l, λ hl, E = μ / (3) B i (8) is a diagoal matix: B = diag (λ h, - λ h, λ l, -λ l ), ad the asymptotic expasio of Z is Z ()~ expb Dl D C. + + = = (3) Hee C is a abitay colum of 4 costat elemets. ecuet sequeces ad D ae give by the ecuet elatios (). Deived expessios (8 3) detemie left solutios of Eq. (3) ad all fou ight solutios. As follows fom the esults of [3], powe seies i the expessios fo f (), f (), f (3) ad f (4) coveges uifomly i the whole iteval (, ) ad these solutios fom a fudametal matix of solutios of Eq. (3). I ode to solve the eigevalue poblem it is ecessay to pefom matchig at some itemediate poit = ȓ, < ȓ < of a liea combiatio of two egula left solutios (8) ad thei deivatives with a liea combiatio of two ight solutios that ted to zeo as ad thei deivatives, i.e. those solutios (3), (3) that coespod to the eigevalues -λ h ad -λ l of the matix B. Usig the coditio of o-tivial cosistecy of the esultig system of liea homogeeous algebaic equatios we fid the eegy levels (see ()). The we compute the omalized wavefuctios as i subsectio 3.. I this case the omalizatio coditio has the fom: ( β + ) ( f + f ) d =. = I ode to calculate wavefuctios of the cotiuous spectum of the Hamiltoia (7) it is ecessay to match a liea combiatio of two egula left solutios (8) (ad the deivatives) with a liea combiatio of fou ight solutios (3), (3) (E > i this case) at some itemediate poit = ȓ, < ȓ <. We choose two liea combiatios of ight solutios that give the heavy-hole ad the light-hole adial i-solutios, whose asymptotic behaviou at ae h l + ih ( ) e + Shhe i f h ()~ μ i ( ) h h Slhe ( + μ) l ih il + μ ( ) l ih i, Shle f ()~ μ (3) l i ( ) h l + il il ( ) e + Slle

12 68 eccuet sequeses i solvig the schödige equatio (we have omitted the logaithmic phase i expoets fo bevity). Hee / hl, = ( E/( μ)), S αβ, ae elemets of the patial S-matix coespodig to a give value F of the total agula mometum ad paity P. This matix is symmetic ad uitay. I the peset study the method is used to calculate eegy ad wavefuctio of the goud state (F = 3/, P = ) ad the wavefuctios of states of the cotiuous spectum (with F = /, 3/, 5/ ad P = -) as fuctios of the eegy E > fo diffeet values of μ, i.e. fo a shallow accepto impuity i diffeet semicoductos. The dipole optical tasitios of a hole fom the goud state of a accepto ae allowed oly to these states of the cotiuous spectum ad thus it is possible to calculate the specta of the photoioizatio coss-sectio of shallow acceptos i semicoductos. The choice of the wavefuctios of the cotiuous spectum, asymptotic behaviou of which is descibed by (3) ad coespods to the scatteig poblem, maes it possible to calculate the patial photoioizatio coss sectios that coespod to ceatio of sepaately a heavy ad light hole i the valece bad. Usig the calculated values of eegies ad wave fuctios, as well as the explicit expessio fo the photoioizatio coss sectio of a shallow accepto, see [], we calculated the specta of the photoioizatio coss sectio fo shallow acceptos i vaious semicoductos (i.e. fo diffeet values of μ). esults of calculatios ae peseted i Figues 3 ad 4. I the captio of Figue 3 E GS is the calculated values of the eegy of the goud state (values of mateial paametes see i []). The value μ =.36 coespods to the diect excito i GaAs. Note that as is evidet fom Figues 3 ad 4 fo values of μ close to, i.e., fo lage values of the effective mass of the heavy hole, spectum of photoioizatio coss sectio diffes geatly fom this spectum fo hydoge-lie atom (μ = ). Figue 3: Specta of the photoioizatio coss sectio of shallow acceptos i e semicoductos. σ is i uits of σ = 4π a. - μ =.97 (IAs, E GS = -5.98), / 3 c - μ =.766 (Ge, E GS = -.64), 3 - μ =.6 (ZTe, E GS = -.53), 4 - μ =.36 (E GS = -.53).

13 It. Jl. of Multiphysics Volume 9 Numbe 5 69 Figue 4: Patial specta of the photoioizatio coss sectio. - σ, - σ h (μ =.97); 3 - σ, 4 - σ h (μ =.36). Uits ae the same as i Figue CONCUSIONS A umeical-aalytical method with some applicatios wee demostated fo accuate solvig the Schӧdige equatio i those cases whe it is educible to a system of coupled odiay diffeetial equatios with sigula poits. Fudametal system of solutios is costucted as algebaic combiatios of powe seies, powe fuctios ad logaithmic fuctio i the eighbouhood of the egula sigulaity ( left exact solutios), ad as asymptotic expasios of solutios i the eighbouhood of the iegula sigulaity ( ight solutios). I the famewo of the method, i ode to solve the eigevalue poblem ad the scatteig poblem it is ecessay to pefom matchig of a pope liea combiatio of left solutios ad thei deivatives with a pope liea combiatio of ight solutios ad thei deivatives at some itemediate poit = ȓ, < ȓ <. The method is based o the calculatio of ecuet sequeces of the costat matices of coefficiets i the powe seies ad i the ivese powe seies i the asymptotic expasios usig deived ecuet elatios, that maes possible to calculate solutios at ay poit, < < usig oly the simple algebaic computatios without usage of ay covetioal step-by-step o vaiatioal pocedues. I tu, it maes possible to solve accuately both the eigevalue poblem ad the scatteig poblem ad to deive aalytical expessios fo the wavefuctios. The method is used fo calculatios of states of the discete spectum of a hydoge-lie atom ad of a shallow accepto impuity i semicoductos, states of the cotiuous spectum of a accepto i the statemet coespodig to the scatteig poblem, ad the specta of the photoioizatio coss sectio of shallow acceptos i vaious semicoductos. I coclusio we ote that, as is obvious, use of ecuet sequeces of coefficiets i the powe seies maes it possible to efficietly ad accuately solve the Schödige equatio i cases whee it is educed to the ODE without egula sigula poit (see, e.g., [] [5]).

14 7 eccuet sequeses i solvig the schödige equatio EFEENCES [] Edmods A.. Agula mometum i quatum mechaics. Piceto Uivesity Pess, Piceto, NJ, 96. [] Galiev V. I., Polupaov A. F., Accuate solutios of coupled adial Schödige equatios. J. Phys. A: Math. Ge., 999, Vol. 3, [3] Galiev V. I., Polupaov A. F., Shpalisi I. E., O the costuctio of solutios of systems of liea odiay diffeetial equatios i the eighbouhood of a egula sigulaity. Joual of Computatioal ad Applied Mathematics, 99, Vol. 39, [4] Wasow W. Asymptotic expasios fo odiay diffeetial equatios. Itesciece, New Yo, 965. [5] adau. D., ifshitz E. M. Quatum mechaics. No-elativistic theoy. 3 ed. Pegamo Pess, 99. [6] Messiah A. M.. Quatum mechaics. Itesciece, New Yo, 958. [7] Polupaov A. F., Galiev V. I., Nova M. G. Effect of the spi-obit iteactio o the optical specta of a accepto i a semicoducto quatum dot. Semicoductos, 997, 3 (), [8] Yamasai Sh. A ew method fo teatig the had coe potetial. Pogess of Theoetical Physics. 6, 5 (), [9] uttige J. M. Quatum theoy of cycloto esoace i semicoductos: Geeal theoy, Phys. ev., 956,, 3 4. [] Baldeeschi A., ipai N.O. Phys. ev. 973, B8, [] Koga Sh. M., Polupaov A. F. Optical absoptio ad photoeffect specta of shallow accepto impuities i semicoductos. Sov. Phys. JETP, 98, 53 (), 9. [] Polupaov A. F. Eegy spectum ad wave fuctios of a electo i a suface eegy well i a semicoducto. Sov. Phys. Semicod., 985, 9 (9), 3 5. [3] Galiev V. I., Kuglov A. N., Polupaov A.F., Goldys E., Tasley T. Multichael caie scatteig at quatum-well heteostuctues. Semicoductos,, 36 (5), [4] Polupaov A. F., Galiev V.I., Kuglov A.N. The ove-baie esoat states ad multichael scatteig by a quatum well. It. Jl. of Multiphysics, 8, (), [5] Polupaov A. F., Evdocheo S. N. Accuate solutio of the eigevalue poblem fo some systems of ODE fo calculatig holes states i polyomial quatum wells. It. Jl. of Multiphysics, 3, 7 (3), 9 6.