The Laguerre pseudospectral method for the reflection ... - Springer Link

Journal of Mathematical Chemistry, Vol. 41, No. 4, May 2007 (© 2006) DOI: 10.1007/s10910-006-9083-z

The Laguerre pseudospectral method for the reflection symmetric Hamiltonians on the real line H. Tas¸eli∗ and H. Alıcı Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey E-mail: [email protected]

Received 8 February 2006

Hermite–Weber functions provide a natural expansion basis for the numerical treat¨ ment of the Schrodinger equation on the whole real line. For the reflection symmetric Hamiltonians, however, it is shown here that the transformation of the problem over the half line and use of a Laguerre basis is computationally much more efficient in a pseudospectral scheme. ¨ KEY WORDS: Schrodinger operator, quantum mechanical oscillators, singular Sturm– Liouville problems on the real line, spectral and pseudospectral methods, Laguerre polynomials, Laguerre collocation points AMS subject classification: 65L60, 81Q05, 65L15, 34L40, 42C10

1.

Introduction

This is the third in a series of papers dealing with the numerical solutions of the singular Sturm–Liouville eigenvalue problems, described by the Hamiltonian H=−

d2 + V (x), dx 2

x ∈ (−∞, ∞)

(1)

by means of a spectral method. In recent articles [1] and [2] (hereafter referred to as PI and PII, respectively), Tas¸eli and Ersec¸en and Tas¸eli and Alıcı presented the continuous (spectral, Rayleigh–Ritz) and discrete (pseudospectral, collocation) variable representations of the scaled Hermite–Weber basis, respectively, applied to the problem being considered with several non-singular quantum mechanical potentials V (x). In PI and PII, we consequently deduced that the convergence rates of the Rayleigh–Ritz and pseudospectral methods based on the normalized Hermite– Weber functions were almost the same except for the reflection symmetric potentials, where V (x) = V (−x). For a reflection symmetric Hamiltonian, the ∗ Corresponding author.

407 0259-9791/07/0500-0407/0 © 2006 Springer Science+Business Media, Inc.

408

H. Tas¸eli and H. Alıcı / The Laguerre pseudospectral method

potential may be regarded as a function of x 2 , i.e., v(x 2 ) := V (x), and hence the spectrum can be decomposed into two disjoint subsets containing solely the even and odd eigen-levels, respectively. In general, however, the nature of the Lagrange interpolant in a pseudospectral scheme, particularly in the Hermite pseudospectral method (HPM) suggested by PII, is not suitable for an independent treatment of the even and odd parity wavefunctions. For this reason, we reported a doubling in the truncated matrix sizes in PII compared with those of PI, when the calculation of the eigenvalues of symmetric potentials within the same accuracy is required. Therefore, the main aim of this paper is to introduce a pseudospectral method for potentials of the form V (x) = v(x 2 ) over x ∈ (−∞, ∞) in which trial solutions to the even and odd eigenfunctions can be proposed separately. First of all the symmetry of the potential function suggests the use of a quadratic transformation ξ = (αx)2 ,

α>0

¨ which converts the Schrodinger equation HΨ = E Ψ to the form 1 1 d E d2 2 − 2 v(ξ/α ) Ψ(ξ ) = − 2 Ψ(ξ ), ξ ∈ (0, ∞), ξ 2+ 2 dξ dξ 4α 4α

(2)

(3)

where α is an optimization parameter. Clearly, the Hermite–Weber functions in PI and PII provide an orthonormal basis over the whole real line whereas the transformed equation (3) is defined on the half line. Thus the HPM of PII is not preferable anymore. Alternatively, it seems that an appropriate Laguerre basis γ γ {L n (ξ )}n=0,1,... , in which the L n denote the Laguerre polynomials (sometimes the γ L n are known more precisely as the generalized or associated Laguerre polynomials of degree n and order γ ), may be used by recalling the fact that it is an orthogonal set on (0, ∞) with respect to the weighting function ξ γ e−ξ for γ > −1. Hence, section 2 is concerned with the pseudospectral formulation of the ¨ computational problem corresponding to the transformed Schrodinger equation (3) by means of a Laguerre basis. In the last section we present a number of computer experiments and discuss the results. 2.

The Laguerre pseudospectral method (LPM) Let us start with a solution of the type Ψ(ξ ) = ξ p e−ξ/2 y(ξ ),

p∈R

(4)


409

satisfying the asymptotic boundary condition at infinity, where the factor ξ p has been introduced to cope with the artificial singularity of (3) at ξ = 0. Note that the original Hamiltonian (1) is in fact regular everywhere except the “point at infinity”, and the additional singularity of (3) at the origin has been resulted from using the quadratic transformation (2). Substitution of (4) into (3) leads to the equation that the new dependent variable y(ξ ) must satisfy 2 1 α 2 2 1 y ξ y + (2 p + 2 − ξ )y + 2 α ξ − v(ξ/α ) + 2 p(2 p − 1) ξ 4α 1 2 (4 p + 1)α − E y (5) = 4α 2 implying that the unwelcome singularity located at the origin can be removed if p is either 0 or 1/2. Therefore, setting 2p +

1 2

=γ +1

(6)

we then express (5) in the more neatly form ξ y + (γ + 1 − ξ )y + q(ξ ; α)y = E(α, γ )y,

ξ ∈ (0, ∞)

(7)

with γ = ∓1/2, where q(ξ ; α) :=

1 2 α ξ − v(ξ/α 2 ) 2 4α

(8)

and 1 2(γ + 1)α 2 − E (9) 2 4α denote the modified potential and eigenvalue parameter, respectively. Now it is clear that the solutions in (4) with p = 0 (γ = −1/2) and p = 1/2 (γ = 1/2) yield even and odd eigenfunctions, respectively, on returning back to the original variable x via (2). Accordingly, as we expected, the new setup of the problem in (7) allows us to determine symmetric and antisymmetric states separately, for the two specific parameter values of γ = −1/2 and γ = 1/2. Furthermore, (7) is much akin to the differential equation E(α, γ ) :=

ξ y + (γ + 1 − ξ )y = −ny,

n = 0, 1, . . .

(10)

γ

of the Laguerre polynomials L n (ξ ), especially, when q(ξ ; α) is viewed as a perturbation over the zero potential. In the traditional Rayleigh–Ritz method, one may propose a trial solution of the form yT (ξ ) =

N n=0

cn ψn (ξ )

(11)

410


i.e., a truncated eigenfunction expansion in terms of the normalized Laguerre polynomials γ (12) ψn (ξ ) = n!/Γ (γ + n + 1) L n (ξ ) substitute into (7), and multiply both sides by ψm (ξ ) in order to arrive at the algebraic system Ac = E c with the matrix elements ∞ q(ξ )ξ γ e−ξ ψm (ξ )ψn (ξ )dξ (13) Amn = −nδmn + Q mn , Q mn = 0

upon performing inner products relative to the weight ξ γ e−ξ over ξ ∈ (0, ∞), where δmn is the Kronecker’s delta. Obviously, the constants cn represent the Fourier coefficients in the continuous expansion of y(ξ ), which is actually a polynomial of degree N . On the other hand, a pseudospectral method is based on choosing N + 1 nodes or collocation points, say ξ0 , ξ1 , . . . , ξ N , and then using the actual values of y(ξ ), that is, y0 = y(ξ0 ), y1 = y(ξ1 ), . . . , y N = y(ξ N ), at the specified nodes to construct an interpolation polynomial of degree N yT (ξ ) =

N

yn n (ξ )

(14)

n=0

by means of the Lagrange formula [3], which suggests another polynomial approximation to a typical solution of (7). Such a pseudospectral scheme in which the N th degree Lagrange polynomials n (ξ ) are defined by the normalized or standard Laguerre polynomials ψ N +1 (ξ ) = n (ξ ) = (ξ − ξn )ψ N +1 (ξn )

γ

L N +1 (ξ ) , d

γ L N +1 (ξ ) (ξ − ξn ) ξ =ξn dξ

n = 0, 1, . . . , N (15)

is called a LPM, where the nodes ξn are the real, distinct and positive roots γ of L N +1 (ξ ). Here, recall the key property of the Lagrange polynomials that n (ξm ) = δmn for all m and n. In this article, the discretization procedure of (7) by the LPM in its full generality, for any arbitrary γ parameter, is presented keeping in mind that we will eventually take γ = ∓1/2 to approximate the even and odd solutions of the original reflection symmetric system. First we need to fix the nodal points ξn accurately. Employing the wellknown relation γ

γ

γ

(n + 1)L n+1 (ξ ) = (2n + γ + 1 − ξ )L n (ξ ) − (n + γ )L n−1 (ξ )


411

we find the three-term recursion n(n + γ )ψn−1 (ξ ) − (2n + γ + 1 − ξ )ψn (ξ ) + (n + 1)(n + γ + 1)ψn+1 (ξ ) = 0 (16) for the normalized Laguerre polynomials and consider its first N + 1 equations, provided by n = 0, 1, . . . , N , to write the inhomogeneous linear system (ξ I − R)x = b,

(17)

where R is the tridiagonal symmetric matrix ⎡ ⎤ γ +1 − γ +1 0 ··· 0 ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ .. .. . ⎢− γ + 1 ⎥ γ +3 − 2(γ + 2) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . . ⎢ ⎥ (18) . . R=⎢ . . 0 − 2(γ + 2) 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ √ .. .. .. ⎢ ⎥ . . N (γ + N ) . γ + 2N − 1 − ⎢ ⎥ ⎣ ⎦ √ 0 ··· 0 − N (γ + N ) γ + 2N + 1 of dimension (N + 1) × (N + 1), b the (N + 1)−vector T b = 0, 0, . . . , 0, (N + 1)(N + γ + 1) ψ N +1 (ξ )

(19)

with only one non-zero component, and I the identity matrix. To determine the nodes, we require that ψ N +1 (ξ ) = 0 and notice that the system (17) reduces to the standard matrix eigenvalue problem Rx = ξ x. Hence the first N + 1 roots ξn of the Laguerre polynomials appear to be the eigenvalues of the matrix R. Then we put the interpolant (14) into (7) N

N ξ n (ξ ) + (γ + 1 − ξ )n (ξ ) + q(ξ ; α)n (ξ ) yn = E(α, γ ) n (ξ )yn (20)

n=0

n=0

and demand its satisfaction at the grid points ξm for m = 0, 1, . . . , N , leading to the discrete representation y = E(α, γ ) y B

(21)

of our differential equation for the computation of an eigenvector y = [y0 , y1 , . . . , y N ]T corresponding to a modified eigenvalue E. Here, the general has the form entry Bmn of B mn + qm δmn , Bmn = K

m, n = 0, 1, . . . , N

(22)

412


mn the in which qm := q(ξm ; α) stands for the diagonal potential matrix and K kinetic energy term that is expressible as (2) (1) mn = ξm dmn + (γ + 1 − ξm )dmn K

(23)

(1) (2) = n (ξm ) and dmn = n (ξm ) in terms of the so-called differentiation matrices dmn (see equations 16 and 19 of PII for details). More specifically, making use of the differential-difference relation d γ γ γ ξ L n (ξ ) = n L n (ξ )−(n + γ )L n−1 (ξ ) ⇒ ξ ψn (ξ ) = nψn (ξ )− n(n + γ )ψn−1 (ξ ) dξ (24)

and the differential equation (10), we obtain ⎧ ξn ψ N (ξm ) 2 ⎪ ⎪ ⎨ 1 ξm − ξn ξm ψ N (ξn ) (1) = dmn 2⎪ 1 ⎪ ⎩ (ξn − γ − 1) ξn and (2)

dmn

⎧ ⎪ ⎪ ⎨

if m = n (25) if m = n

1 2 ξn ψ N (ξm ) 3 (ξm − γ − 1) − ξm − ξn ξm ψ N (ξn ) 1 ξm − ξn ξm =

3⎪ 1 1 ⎪ ⎩ (ξn − γ − 1)(ξn − γ − 2) − N ξn ξn

if m = n (26) if m = n.

explicitly. Nevertheless, as stated in PII, it is again more amenable to consider the total kinetic energy matrix given by (23), the derivation of which can be accomplished after some algebra, ⎧ ψ N (ξm ) 12ξn ⎪ ⎪ if m = n ⎪ ⎨ 2 (ξm − ξn ) ψ N (ξn ) 1 mn = − K (27) 6⎪ 1 ⎪ 2 ⎪ 2N + (γ − ξn ) − 1 if m = n ⎩ ξn by exploiting the functional identities such as (10), (16) and (24) available for the nm , however, its simple structure suggests mn = K Laguerre polynomials. Clearly K immediately introducing a similarity transformation P B = P −1B

(28)

to replace the coefficient matrix of (21) with a symmetric one. Actually, if P is chosen as a diagonal matrix of the form ψ ψ N ,m ψ N ,N

N ,0 ψ N ,1 P = diag √ , √ , . . . , √ , . . . , √ (29) , ψ N ,m = ψ N (ξm ) ξ0 ξ1 ξm ξN


413

then the matrix B can be defined by Bmn = K mn + qm δmn , which is symmetric, since K mn reduces to ⎧ √ 12 ξm ξn ⎪ ⎪ ⎪ 1 ⎨ (ξm − ξn )2 K mn = − 6⎪ 1 ⎪ ⎪ ⎩ 2N + (γ − ξn )2 − 1 ξn

(30)

if m = n (31) if m = n

with K mn = K nm . Besides their simplicity, the symmetric matrix elements now depend only on the grid points ξk and do not require any evaluation of the Laguerre polynomials. Hence, instead of the system (21), we may consider the symmetric matrix eigenvalue problem B u = E(α, γ )u

(32)

because the similar matrices share the same eigenvalue spectrum. Also, the connection formula y = Pu

(33)

serves to return back to an eigenvector y of the non-symmetric system (21), by the help of a calculated eigenvector u of the symmetric matrix. 3.

Numerical experiments and conclusion

The spectral and pseudospectral formulations in section 2 are applied to a class of symmetric potentials by making use of the Laguerre bases with γ = −1/2 and γ = 1/2. Specifically, we consider the test problems provided by the generalized anharmonic oscillators V (x) = x 2 + v2k x 2k ,

v2k > 0

for k = 2 and 3, the symmetric double well potential (SDWP) 2 V (x) = v4 x 2 − 21 v4−1 , v4 > 0

(34)

(35)

and the non-polynomial Gaussian potential (GP) V (x) = −e−βx , 2

β>0

(36)

whose very well-known spectral properties can be found in the literature [4–7]. Typical computations for the ground state eigenvalues of the quartic and sextic oscillators as a function of the coupling constants v4 and v6 are displayed in tables 1 and 2, respectively. In all tables, NRR , NHPM and NLPM

414

H. Tas¸eli and H. Alıcı / The Laguerre pseudospectral method Table 1 Ground state energies of the quartic oscillator V (x) = x 2 + v4 x 4 , as a function of v4 .

v4 10−4 10−2 1 10 103 104 105

E0 1.000 1.007 1.392 2.449 10.639 22.861 49.225

074 373 351 174 788 608 447

986 672 641 072 711 870 584

880 081 530 118 328 272 229

200 382 291 386 046 468 625

111 460 855 918 063 891 157

122 533 657 268 622 759 076

834 843 507 793 042 867 387

155 905 876 906 669 963 001

30 98 61 19 4 5 1

NRR

NHPM

NLPM

αopt

8 17 25 26 30 30 30

15 32 51 53 56 56 56

8 17 25 27 27 28 28

1 1 2.1 3.1 6.5 10 14

Table 2 Ground state energies of the sextic oscillator V (x) = x 2 + v6 x 6 , as a function of v6 . v6 10−4 10−2 1 10 103 104 105

E0 1.000 1.016 1.435 2.205 6.492 11.478 20.375

187 741 624 723 350 798 098

228 363 619 269 132 042 656

153 754 003 595 329 264 309

680 732 392 632 671 543 660

768 031 315 351 550 961 844

286 671 761 009 549 289 567

355 817 272 973 557 816 287

665 981 220 387 845 038 513

62 51 54 17 34 6 5

NRR

NHPM

NLPM

αopt

16 32 39 40 40 39 41

30 70 78 78 80 78 81

16 34 39 40 42 40 41

1 1.8 3.2 4.2 7.0 9.5 12

denote matrix sizes of the Rayleigh–Ritz, Hermite and Laguerre pseudospectral schemes, respectively, at which the desired accuracy is obtained. Table 3 exhibits the nearly degenerate states of a SDWP for v4 = 0.01, in which the two wells located symmetrically about the origin are sufficiently separated. As a last example, a few even-indexed eigenvalues of the GP (36) have been presented in table 4. The GP has both continuous and a finite number of discrete eigenvalues lying between −1 < E < 0. It is known that there exists a threshold value βthr for the parameter β beyond which the particular bound state being considered can no longer survive. It can be seen from table 4 that, as in the other methods, a remarkable slowing down of convergence occurs for very weakly bound states with E just below zero, when β approaches βthr . It should be noted that the continuous Rayleigh–Ritz treatment of the problem via Laguerre bases is equivalent to that of PI, where the sets of Hermite polynomials {H2n } and {H2n+1 } were used separately for the approximations of symmetric and antisymmetric state wavefunctions. In fact, if we recall the interrelations [8] −1/2

H2n (x) = (−1)n 22n n!L n

(x 2 )

(37)


415

Table 3 2 Nearly degenerate states of the SDWP, V (x) = v4 x 2 − 21 v4−1 for v4 = 0.01. n

En

0 1 2 3 4 5 6 7 8 9 10 11

1.404 1.404 4.170 4.170 6.870 6.870 9.489 9.498 12.049 12.049 14.514 14.514

048 048 193 193 088 088 578 578 309 309 205 205

605 605 605 605 833 833 387 387 486 486 022 048

297 297 999 999 714 714 187 191 334 673 981 121

706 706 310 310 024 046 870 178 092 006 239 017

882 882 127 219 612 802 055 212 592 847 103 338

425 602 833 613 172 425 194 320 332 573 429 991

707 566 875 291 315 995 418 856 880 312 421 612

570 280 071 198 168 681 356 961 171 477 443 415

82 56 30 73 49 89 55 14 6 9 9 8

NRR

NHPM

NLPM

αopt

52 52 54 54 56 56 58 58 59 59 60 60

104 104 108 108 112 112 116 116 118 118 120 120

52 52 54 54 56 56 58 58 59 59 60 60

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1

Table 4 2 Eigenvalues of the GP V (x) = −e−βx , as a function of β. β

αopt

NRR

10−3

0.2

30 35 45 55

10−2

0.3

10−1

0.2

NHPM

NLPM

n

En

56 70 85 103

28 35 43 52

0 2 4 6

−0.968 −0.846 −0.731 −0.621

752 820 125 888

703 196 549 650

41 70 70 70

81 128 128 128

41 69 69 69

0 2 4 6

−0.903 −0.550 −0.267 −0.068

763 801 463 692

987 980 773 054 539 687 567 95 670 798 557 886 254 842 935 81 693 629 351 027 251

70 70

148 148

74 74

0 2

−0.721 530 628 487 107 638 685 036 884 81 −0.007 90

034 725 125 443

398 804 734 182

668 118 739 657

606 603 132 155

599 225 375 148

656 951 767 987

91 44 29 66

and 1/2

H2n+1 (x) = (−1)n 22n+1 n!x L n (x 2 )

(38)

between the Hermite and Laguerre polynomials then it is not difficult to verify that the variational matrices in (13) are nothing but those in PI. However, the HPM of PII and the LPM of the present article stand for alternative numerical procedures of the problem. Evidently, we deduce from the numerical tables that NHPM ≈ 2NLPM . Therefore, the most efficient pseudospectral discretization ¨ of the Schrodinger equation over (−∞, ∞) having a symmetric potential is not the HPM, and it is suggested by the LPM, which supports the main argument

416


of this paper. On the other hand, we see that NLPM ≈ NRR because both Rayleigh–Ritz and LPM take care of the symmetry of the potential. Finally, the LPM which is considered here only for specific values of γ = ±1/2 might be much more generally applicable, for instance, to radial ¨ Schrodinger operators as well, where the radial variable is already defined on the half line (0, ∞) naturally. Acknowledgment ˙ ¨ ITAK, This research was supported by a grant from TUB the Scientific and Technical Research Council of Turkey. References [1] [2] [3] [4] [5] [6] [7] [8]

H. Tas¸eli and M.B. Ersec¸en, J. Math. Chem. 34 (2003) 177. H. Tas¸eli and H. Alıcı, J. Math. Chem. 38 (2005) 367. D. Funaro, Polynomial Approximations of Differential Equations (Springer-Verlag, Berlin, 1992). K. Banerjee and S.P. Bhatnagar, Phys. Rev. D 18 (1978) 4767. N. Bessis, G. Bessis and B. Joulakian, J. Phys. A 15 (1982) 3679. C.S. Lai, J. Phys. A 16 (1983) L181. M. Cohen, J. Phys. A 17 (1984) L101. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970) p. 779.