Asymptotic iteration method for eigenvalue problems

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1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 6 ( PII: S ( Asymptotic iteration method for eigenvalue problems Hakan Ciftci 1,Richard L Hall and Nasser Saad 1 Gazi Universitesi, Fen-Edebiyat Fakültesi, Fizik Bölümü, Teknikokullar, Ankara, Turkey Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montréal, Québec HG 1M8, Canada Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, Charlottetown, PEI C1A 4P, Canada Received 8 August 00 Published 1 November 00 Online at stacks.iop.org/jphysa/6/11807 Abstract An asymptotic iteration method for solving second-order homogeneous linear differential equations of the form y = λ 0 (xy + s 0 (xy is introduced, where λ 0 (x 0 and s 0 (x are C functions. Applications to Schrödinger-type problems, including some with highly singular potentials, are presented. PACS number: 0.65.Ge 1. Introduction Second-order homogeneous linear differential equations arise naturally in many fields in mathematical physics. There are many techniques available in the literature that can be used to solve these types of differential equation with boundary conditions. The main task of the present work is to introduce a new technique, which we call the asymptotic iteration method, to solve second-order homogeneous linear differential equations of the form y = λ 0 (xy + s 0 (xy (1.1 where λ 0 (x and s 0 (x are defined in some interval, not necessarily bounded, and λ 0 (x and s 0 (x have sufficiently many continuous derivatives.. The asymptotic iteration method Consider the homogenous linear second-order differential equation y = λ 0 (xy + s 0 (xy (.1 where λ 0 (x and s 0 (x are functions in C (a, b. Inorder to find a general solution to this equation we rely on the symmetric structure of the right-hand side of (.1. Indeed, if we differentiate (.1 with respect to x, wefind that y = λ 1 (xy + s 1 (xy ( /0/ $ IOP Publishing Ltd Printed in the UK 11807

2 11808 HCiftci et al where λ 1 = λ 0 + s 0 + λ 0 and s 1 = s 0 + s 0λ 0. If we write the second derivative of equation (.1, we get y = λ (xy + s (xy (. where λ = λ 1 + s 1 + λ 0 λ 1 and s = s 1 + s 0λ 1. Thus, for (n +1th and (n +th derivatives, n = 1,,...,wehave y (n+1 = λ n 1 (xy + s n 1 (xy (.4 and y (n+ = λ n (xy + s n (xy (.5 respectively, where λ n = λ n 1 + s n 1 + λ 0 λ n 1 and s n = s n 1 + s 0λ n 1. (.6 From the ratio of the (n +th and (n +1th derivatives, we have d dx ln(y(n+1 = y(n+ y = λ ( n y + s n λ n y ( (n+1 λ n 1 y + s. n 1 λ n 1 y (.7 We now introduce the asymptotic aspect of the method. If we have, for sufficiently large n, s n = s n 1 := α λ n λ n 1 (.8 then (.7reduces to d dx ln(y(n+1 = λ n λ n 1 (.9 which yields ( x ( y (n+1 λn (t x (x = C 1 exp λ n 1 (t dt = C 1 λ n 1 exp (α + λ 0 dt (.10 where C 1 is the integration constant, and the right-hand equation follows from (.6 and the definition of α. Substituting (.10into(.4weobtain the first-order differential equation ( x y + αy = C 1 exp (α + λ 0 dt (.11 which, in turn, yields the general solution to (1.1as ( x y(x = exp α dt [ C + C 1 x exp ( t (λ 0 (τ +α(τdτ Consequently, we have proved the following theorem: Theorem. Given λ 0 and s 0 in C (a, b, then the differential equation y = λ 0 (xy + s 0 (xy ] dt. (.1 has a general solution (.1 ifforsomen>0 s n λ n = s n 1 λ n 1 α (.1

3 Asymptotic iteration method for eigenvalue problems where λ k = λ k 1 + s k 1 + λ 0 λ k 1 and s k = s k 1 + s 0λ k 1 (.14 for k = 1,,...,n.. Some illustrative examples.1. Differential equations with constant coefficients If s 0 (x and λ 0 (x are constant functions, for example, λ 0 = 4 and s 0 =, then, for the differential equation y = 4y y,thecomputation of λ n and s n by means of equation (.14 implies λ n = 1 (n+ 1 and s n = (n+1 1. s Condition (.1 implies lim n n λ n = 1which yields from (.1 the general solution y(x = C e x + C 1 e x, as expected by the application of elementary methods. Generally speaking, if we consider the differential equation (1.1 with λ 0 (x and s 0 (x constants, we have from (.14 λ n = s n 1 + λ 0 λ n 1 and s n = s 0 λ n 1. Consequently, the ratio s n /λ n becomes s n s 0 λ n 1 s 0 = = λ n s n 1 + λ 0 λ n 1 s n 1 /λ n 1 + λ 0 which yields, by means of (.1, that s n s 0 = λ n s n /λ n + λ 0 therefore ( sn s n + λ 0 s 0 = 0. (.1 λ n λ n This is a quadratic equation that can be used to find the ratio s n /λ n in terms of λ 0 and s 0. Therefore, the expected solutions for the differential equation (1.1 with constant coefficients follow directly by means of (.1... Hermite s differential equation Many differential equations which are important in applications, such as the equations of Hermite, Laguerre and Bessel, can be solved using the method discussed in section. Asan illustration we discuss here the exact solution of Hermite s equation by means of the iteration method; other differential equations can be solved similarly. Hermite s differential equation takes the form f = xf kf <x<. (. Here we have λ 0 = x and s 0 = k. Using(.14, we can easily show that n+1 δ = λ n+1 s n s n+1 λ n = n+ (k i n = 0, 1,,... (. i=0 Therefore, for the condition δ = 0 to hold, we must have k a non-negative integer, usually

4 11810 HCiftci et al known as the order of the Hermite equation. Consequently, for each k,theratio s n /λ n yields k = 0 s 0 = s 1 = =0 f 0 (x = 1 λ 0 λ 1 s 1 k = 1 = s = = 1 λ 1 λ x f 1(x = x s k = = s = = 4x λ λ x 1 f (x = x 1 s k = = s 4 = = 6x λ λ 4 x x f (x = x x s 4 k = 4 = s 5 4x 16x = = λ 4 λ 5 1x +4x f 4(x = 1x +4x 4 4 and so on. Clearly, the expressions for the exact solutions f k (x generate the well-known Hermite polynomials. We can easily verify that the general form of f k (x, k = 0, 1,,...,is given in terms of the confluent hypergeometric functions [] by f k (x = ( 1 k k( ( 1 k 1 F 1 k; 1 ; x (.4 and f k+1 (x = ( 1 k k( k x ( 1F 1 k; ; x (.5 where the Pochhammer symbol (a k is defined by (a 0 = 1 and (a k = a(a +1(a +...(a+ k 1 for k = 1,,,..., and may be expressed in terms of the Gamma function by (a k = Ɣ(a + k/ɣ(a, when a is not a negative integer m, and, in these exceptional cases, ( m k = 0ifk>mand otherwise ( m k = ( 1 k m!/(m k!... Harmonic oscillator potential in one dimension Although the iteration method discussed in section can be applied to any second-order homogeneous linear differential equations of the form (1.1with λ 0 0, we shall concentrate in the rest of the paper on the eigenvalue problems of Schrödinger type. We shall show that equation (.1 with conditions (.1 and (.14givesacomplete solution for many important Schrödinger-type problems. Through a concrete example we explore the exact solutions of Schrödinger s equation for the harmonic oscillator potentials, namely ( d dx + x ψ = Eψ (.6 where ψ L (,. Inthelimit of large x,theasymptotic solutions of (.6 can be taken as any power of x times a decreasing Gaussian. With this in mind we write the unnormalized wavefunctions as ψ(x = exp ( x f(x (.7 where the functions f(x are to be found by means of the iteration procedure. Substituting (.7into(.6, one obtains d f df = x + (1 Ef (.8 dx dx which, by comparison with (., yields the exact eigenvalues E n = n +1 n = 0, 1,,... and the functions f n (x, n = 0, 1,,..., are the Hermite polynomials obtained above. Therefore, using (.4 and (.5, the unnormalized wavefunctions of the Schrödinger

5 Asymptotic iteration method for eigenvalue problems equation (.6are ψ n (x = ( 1 n n( ( 1 n exp x for n = 0,, 4,...and ψ n (x = ( 1 n n( ( n x exp x 1F 1 ( n; 1 ; x (.9 1F 1 ( n; ; x (.10 for n = 1,,...The normalization constant of ψ(xcan be computed by means of ψ =1, as we shall shortly show..4. Gol dman and Krivchenkov potential The Gol dman and Krivchenkov Hamiltonian is the generalization of the harmonic-oscillator Hamiltonian in three dimensions; namely ( d dr + γ(γ +1 r + ψ = Eψ (.11 r where ψ L (0, and satisfies the condition ψ(0 = 0 known as the Dirichlet boundary condition. The generalization lies in the parameter γ ranging over [0, instead of the angular momentum quantum number l = 0, 1,,...Forlarger,the exact solutions of (.11 are asymptotically equivalent to the exact solutions of the harmonic-oscillator problem with eigenfunctions vanishing at the origin. Therefore, we may assume that the unnormalized wavefunction ψ takes the form ψ(r = r γ +1 exp ( r f(r (.1 where, again, f(ris to be determined through the iteration procedure discussed in section. Substituting (.1 into(.11, we obtain d ( f dr = r γ +1 df + (γ + Ef (.1 r dr where λ 0 (r = (r (γ +1/r and s 0 (r = γ + E. Bymeans of equation (.14 we may compute λ n (r and s n (r. That result, combined with condition (.1, yields E 0 = +γ E 1 = 7+γ E = 11 + γ... respectively, that means E n = 4n +γ + for n = 0, 1,,... (.14 Furthermore, with the use of f(r = exp( s n /λ n dr, equation (.1, after some straightforward computations, yields n f n (r = ( 1 k n k Ɣ(n +1Ɣ(γ +n +Ɣ(γ + n k +1 Ɣ(k +1Ɣ(n k +1Ɣ(n + γ +1Ɣ(n +γ k + rn k. k=0 (.15 In order to show that (.15 together with (.1 yields the exact wavefunctions for the Gol dman and Krivchenkov Hamiltonian, we may proceed as follows. Using the Pochhammer identity (a k = Ɣ(a k/ɣ(a, wecan write (.15as n ( f n (r = n r n ( 1 k (γ + n +1 k 1 k k!(n +1 k (n +γ + k r k=0

6 1181 HCiftci et al Since (a k = ( 1 k /(1 a k we have ( n f n (r = n r n ( n k n γ 1 k! k=0 in which we have used Gauss s duplication formula ( ( a a +1 (a n = n n. n k ( 1 k (.16 The finite sum in (.16 isthe series representation of the hypergeometric function F 0. Therefore ( f n (r = n r n F 0 n, n γ 1 ; ; 1 r = ( 1 n n! L γ + 1 n (r = ( 1 n n( γ + ( n 1 F 1 n; γ + ; r. Consequently, the unnormalized wavefunctions take the form ψ(r = ( 1 n n( γ + n rγ +1 exp ( r ( 1F 1 n; γ + ; r for n = 0, 1,,...The normalization constant for ψ(r can be found using ψ =1which leads to the exact wavefunctions of the Gol dman and Krivchenkov potential, namely ψ(r = ( 1 n ( γ + n n!ɣ ( r γ +1 γ + exp ( r ( 1F 1 n; γ + ; r. (.17 Some remarks are in order: (1 The exact odd and even solutions of the harmonic oscillator potential in one dimension can be recovered from (.17 bysetting γ = 0 and γ = 1respectively. ( The exact solutions of the harmonic oscillator potential in three dimensions can be recovered from (.17 bysetting γ = l, where l = 0, 1,,...is the angular momentum quantum number. ( The exact solutions of the harmonic oscillator potential in N dimensions can be recovered from (.17 bysetting γ = l + 1 (N where N. 4. Singular potentials We discuss in this section the application of the iteration method discussed in section to investigate two important classes of singular potentials. The first class is characterized by the generalized spiked harmonic oscillator potentials which have a singularity at the origin, and the second class is characterized by the quartic anharmonic oscillator potentials where the perturbative term diverges strongly at infinity. Different approaches are usually applied to deal with each of these classes. The asymptotic iteration method, however, can be used to investigate the eigenvalues for both classes of potentials Generalized spiked harmonic oscillator potentials Since the interesting work of Harrell [7] on the ground-state energy of the singular Hamiltonian H = d dx + x + A x α x [0, A 0 α>0 (4.1 r

7 Asymptotic iteration method for eigenvalue problems 1181 known as the spiked harmonic oscillator Hamiltonian, the volume of research in this field has grown rapidly. A variety of techniques has been employed in the study of this interesting family of quantum Hamiltonians [4 14]. We shall investigate here the solutions of the spiked harmonic oscillator Hamiltonian (4.1inarbitrary dimensions by means of the iteration method. That is to say, we examine the eigenvalues of the Hamiltonian known as the generalized spiked harmonic oscillator Hamiltonian H = d dx + γ(γ +1 x + + A x [0, A 0 α>0 (4. x x α where γ = l + (N / forn. Clearly we may compute the eigenvalues of the spiked harmonic oscillator Hamiltonian (4.1directly by setting N = and l = 0in(4.. The wavefunctions (.1 ofthe Gol dman and Krivchenkov Hamiltonian suggest that the exponent in the power of x term of the exact solutions of (4. should depend, at least, on the parameters γ and A. Since the exact form of this term is unknown, we write the exact wavefunctions in the simpler form ψ(x = exp ( x f(x (4. where the functions f(x must satisfy the condition f(0 = 0 and remain to be determined by the iteration method. Substituting (4. into(4., we obtain d ( f df = x dx dx + 1 E + A γ(γ +1 + f. (4.4 xα x With λ 0 (x = x and s 0 (x = 1 E + A/x α + γ(γ +1/x,wemaycompute λ n (x and s n (x using (.14 and the eigenvalues may be calculated by means of the condition (.1. For each iteration, the expression δ = s n λ n 1 s n 1 λ n will depend on two variables E and x. The eigenvalues E computed by means of δ = 0should, however, be independent of the choice of x. Actually, this will be the case for most iteration sequences. The choice of x can be critical to the speed of the convergence to the eigenvalues, as well as for the stability of the process. Although, we do not have at the moment a specific method to determine the best initial value of x, wemay suggest the following approaches. For the spiked harmonic oscillator potential V(x= x + A/x α, one choice of x = x 0 could be the value of x 0 that minimizes the potential V, namely x 0 = ( αa 1/(α+.Another possible choice comes from noting that the ground-state energy of the harmonic oscillator or that of the Gol dman and Krivchenkov potential can be obtained by setting s 0 = 0. We may therefore start our iteration with x 0 obtained from s 0 = 0. For example, if α = 4, then s 0 = 1 E + A γ(γ +1 + = 0 x4 x implies A x 0 = p + p + (4.5 E 1 where p = γ(γ +1/(E 1. The results of our iteration method for the cases of α = 1.9 and α =.1 are reported in table 1 where we compute the eigenvalues E P nl by means of 1 iterations only. It should be clear that these results could be further improved by increasing the number of iterations. For the case of α = 4wereport our results in table wherein we used x 0 to start the iteration procedure, as given by (4.5.

8 11814 HCiftci et al Table 1. Acomparison between the exact eigenvalues E nl for dimension N = 10 of the Hamiltonian (4. computed by direct numerical integration of Schrödinger s equation and the eigenvalues Enl P computed by means of the present work. α = 1.9 α =.1 N E 00 E P 00 E 1 E P Table. Acomparison between the exact eigenvalues E 0 for one dimension of the Hamiltonian (4. with α = 4computed by direct numerical integration of Schrödinger s equation and the eigenvalues E0 P computed by means of the present work. A γ E P 0 E The quartic anharmonic oscillator potentials We investigate the Schrödinger equation Hψ = E(Aψ for the quartic anharmonic oscillators [15 19], where H = d dx + x + Ax 4. (4.6 In order to obtain the energy levels using the iteration method, we write the exact wavefunctions in the form ψ(x = exp ( x f(x. Consequently, after substituting in the Schrödinger equation, we obtain d f df = x dx dx + (1 E + Ax4 f (4.7

9 Asymptotic iteration method for eigenvalue problems Table. Acomparison between the eigenvalues E n,n = 0, 1,...,5, for the quartic anharmonic oscillator with A = 0.1 computed by direct numerical integration of Schrödinger s equation [19] and the eigenvalues E P computed by means of the present work. n E p E or λ 0 (x = x and s 0 (x = 1 E + Ax 4. We start the iteration in this case with x 0 = 0, the value of x at which the potential takes its minimum value. We report our computational results in table. 5. Conclusion One can find a Taylor polynomial approximation about x 0 for an initial value problem by differentiating the equation itself and back substituting to obtain successive values of y (k (x 0. This method is perhaps as old as the very notion of a differential equation. In this paper we develop a functional iteration method related to this general idea and specifically tailored for an important base-class of linear equations of the form L(y = y λy sy = 0. The iteration is assumed either to terminate by the condition s n /λ n = s n 1 /λ n 1 α, or this condition is imposed, as an approximation. After looking at some well-known problems which are exactly of this sort, our principal application is to Schrödinger eigen equations. These latter problems are converted to the base-type by first factoring their solutions in the form ψ(x = f(xy(x, where f(x is the large-x asymptotic form, and y satisfies L(y = 0. Some aspects of this approach, such as the iteration termination condition, the construction of asymptotic forms and the choice of x 0, still await more careful mathematical analysis. However, even in its present rudimentary state, the method offers an interesting approach to some important problems. This is especially so, as it turns out, for problems such as the spiked harmonic oscillator that are known to present some profound analytical and numerical difficulties. Acknowledgment Partial financial support of this work under grant nos GP48 and GP49507 from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by two of us (respectively RLH and NS. References [1] Saad N and Hall R L 00 J. Phys. A: Math. Gen [] Andrews G E, Askey R and Roy R 1999 Special Functions (Cambridge: Cambridge University Press [] Hall R L, Saad N and von Keviczky A 001 J. Phys. A: Math. Gen [4] Hall R L, Saad N and von Keviczky A 1998 J. Math. Phys [5] Hall R L and Saad N 000 J. Phys. A: Math. Gen. 569 [6] Hall R L and Saad N 000 J. Phys. A: Math. Gen. 551 [7] Harrell E M 1977 Ann. Phys., NY [8] Aguilera-Navarro V C, Estévez G A and Guardiola R 1990 J. Math. Phys. 1 99

10 11816 HCiftci et al [9] Aguilera-Navarro V C, Fernández F M, Guardiola R and Ros J 199 J. Phys. A: Math. Gen [10] Aguilera-Navarro V C, Coelho A L and Ullah Nazakat 1994 Phys. Rev. A [11] Znojil M 1991 Phys. Lett. A [1] Znojil M and Leach P G L 199 J. Math. Phys. 785 [1] Solano-Torres W, Estéves G A, Fernández F M and Groenenboom G C 199 J. Phys. A: Math. Gen [14] Hall R L, Saad N and von Keviczky A 00 J. Math. Phys [15] Simon B 1970 Ann. Phys., NY [16] Bender C M and Wu T T 1969 Phys. Rev [17] Bacus B, Meurice Y and Soemadi A 1995 J. Phys. A: Math. Gen. 8 L81 [18] Hall R L 1985 Canad. J. Phys [19] Mostafazadeh A 001 J. Math. Phys. 4 7