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Application of rod splines for solving the radial Schrödinger equation for coupled states
U.S.S.R. Cbmput.Maths.Math.Phys.VoL 22, No. 2, pp. 130-136, Printed in Great Britain
1982.
0041-5553/82/020130-07$07.50/O 01983. Pergamon Press Ltd.
APPLICATION OF ROD SPLINES FOR SOLVING THE RADIAL SCHRiiDINGER EQUATION FOR COUPLED STATES* L. ALEKSANDROV, M. DRENSKA and D. KARADZHOV Sofia, Bulgaria (Received 17 April 1980; revised 3 June 1981)
THE METHOD of rod splines is compared with other methods for solving the radial SchrGdinger equation,
and special features of the method are pointed out.
Introduction We described in [I ] a method of rod splines for solving eigenvalue problems for linear differential equations of high order, and we applied the method to radial Kadyshevskii equations (equations of infinite order). Here we shall apply the method to the radial Schr6dinger equation for coupled states a low-order equation); this equation can be solved by a variety of methods, see [2-61, etc. We shall compare our method with the finite difference method [5] and the method described in [4], taking the example of the Saxon-Woods potential (the optical potential) and the Coulomb potential.
1, Solution of Schriidinger’s equation for coupled states In the method of rod splines, the radial Schrijdinger
equation
(1.1) =x
s
[y(x,z)]*ds=l
(1.2)
XH
X= [z., s,] cXo= (0, m), where XH is taken fairly close to zero, while XK is fairly large, and in an interval of discrete energies Z = [ZH, ZK] , the choice of which depends on the behaviour of the potential v (x) as x + 00.
is considered in a finite interval
In Eq. (1.1) we use the following notation: 1 is the orbital momentum quantum number (1=0,1,. . . .), and fi is the relative mass, which in the system of units with fl = 1 has the form
R=m,m2/(m,+m,).
*Zh. vjkhisl. Mat. mat. Fiz., 22, 2, 375-381,
1982.
130
Solving the radial Schrbdinger equation for coupled states
131
To use the method of rod splines, we have to choose a rod function [l] ; to this end, we shall consider potentials v (x) E C (X0) with the following behaviour at the ends of the interval X0: as x+0,
u(x)+
=
UOO const
r<2,
(1.3) and asx+=, u(x)-+0
(1.4a)
or u(x) “VOX’,
uo, t=const>O.
(1.4b)
This class of functions u (x) defines the class of radial functions characterized by the asymptotic behaviour y
(x, 2) “0,x’+‘,
Y (x92) 42 exp [-k
Oj=const,
(z)4,
1/x+00,
&=const,
where, in case (I .4a), we have q=l, k(z) =(-~Ez)‘~, z&c have 4= (t+2) /2, k (z) = (2iiivo) ‘“Iq, ZEZC [0, =J).
(1.5)
x*00,
(-a~, 01,
(1.6)
and in case (1.4b) we
Relations (1.5), (1.6) show that, as the rod of the solution we can use the function p* (x, 2) =xl+l exp [-k(z)s’].
Using the identity a2 Z-,$+’
L(z+l) x2 -5+2
+ 2% [z-v (3) ]
>
(a++‘& (x, 2) )
1+x d --~+2iiiIz--u(r)l
We will solve the equivalent Schrijdinger equation x=X,
$+2
Z&,
(1.1’)
with the simpler rod
p2(x, z)=exp
[-k(z)sg].
(1.7)
L. Aleksandrov, M. Drensku and D. Karadzhov
132
In the case of problem (1 .l’), (1.2), (1.7), the main quantities [l] have the values ,jrr
Wzp:-Pp,,+D,(“),
P’&
of the method of rod splines
2,3,
d,,=qk (2) xy-’{gk(2)2,*-[2(1+1)+q--ll}exp[-k(z)qPl, dsj=O,
dz~=-2~k(z)xjp-‘exp[-Ic(z)qp], P*j’P2 (q,
where
(zj},
2),
2, . . . , N,
j=l,
1+1
Uj=2E[C_V(Zj)],
dij=2’-,
a:,=11 xj
nodes of the parabolic spline [ I] ,
is the mesh of interpolation
Let the pairs (I?, s(x, Z)), (z”, y”(5, 2) ) denote respectively the exact and approximate solutions (found by the method of rod splines) of problem (1 .l’), (1.2) and let the pair == considered on the semi-axis X0 (y, 2, Z”&), be the exact solution of problem (2, I/(x, 5, (1 .l’), (1.2). Also, let the conditions of Theorem 2 of [I] be satisfied. Then, as when solving equations of infinite order (see [ 1, Section 2]), we have
(1.8)
GM, (Jf,+M,M,) hZ+O (h’) + max I [q kc> -? (z> 1 sLp l;(5) {expL-k(Z)sql (OF,1
xe~p{-[k(T)-k(Z)]sq}l+
-
exp[-k(
Y)rq]}
(1.9)
I+
SUP
{exp[-k(Z)xQl-
I;(Z)
exp[-k(~)zqlI
1,
P$,-1
where h =
(XK -
is the mesh step of the parabolic spline,
XH)/N
Mi=2.5
max I q” (5) I,
Mp -
max Ip2(z, E) I, I
X
Ms= xx
XX
(1.10)
MI = marl sz(x) pzz’(X,z> 1, z
If lim max I q (5) -7 (5) I 4 XH4 x x,:-c= the second term on the right-hand side of (1.8), and the last three terms on the right-hand side of (1.9), can be made as small as desired by taking sufficiently small values of XH and h, and sufficiently large XK .
Solving the radial Schrbkiingerequation for coupled states
133
2. Numerical comparison of the method of rod splines with other methods
The method of rod splines was realized by means of the SPSOL subroutine in the type of construction of the basis spline described in [7]. The IBM 370/145 computer was used in the double accuracy mode.
The following two problems were solved. Roblem A. To find the coupled states in the case of the optical potential [4]
v(5)=where T-exp
[ (x-v~)/v~],
Vi
i+T
1
T
3
--;;(1ST) +
(2.1)
v,=-50, ~~-7,vs=0.6.
We wish to find the coupled states corresponding to I= 0 with Fii=i/2,
mi=ma==m-=h-i.
We know [4] that, under these conditions, problem (1.1’) (1.2), (1.7) has 14 radial solutions, 13, and they are known in analytical form.
1x=0,...,
The SPSOL program was used in the conditions z,,= 10e6; h, -0.00065,h,-0.00043 (h, is the basic step, h2 is the step used when obtaining the “lower bound of energy”); Cl = 0.005. Notice that, with the given hl and Cl, the quantities XK and N are computed by the program in such a way that step hl remains constant while the inequality
is satisfied. In Table 1 we quote the following quantities: the exact values of the energy ?? (see [4,5]); the approximate values r(l), found by the method described in [4] with h = 0.004,XK = 24.2; the approximate energies ?‘c2), found by the finite-difference method [5] with h = 0.03; the approximate energies ?c3), found by the method of rod splines with hl = 0.00065, and also the values of XK and N used by the SPSOL program. We also give in Table 1 the lower bounds (AZ(~))low, evaluated from the expression (Az(~)),~~ = IP) (h,) -z”(3) ( h2) ( /2.
(24
Obviously,
The coefficients al, @1,71, by means of which, using (1.4) (1.8), (1.1 l), (1.15), (1.17), and (1 .19) of [ 11, the approximate wave functions gt3) (r, z”(‘)), were evaluated, are given in Table 2.
L Aleksandrov,M. Drenska and D. Karadzhov
134
TABLE 1
‘, 0 1 2 3 4
I = I ;
; (2)
(1)
z
-49.457788728 -48.148430420 -46.290753954 -43.968318431 -41.232607772
N
i(3)
-49.451788728 -48 *148430419 -46.290753955
-49.461 -48.158 -46.310
-
I_::*;;: -38:11a -34.743
-38.122785096 -34.672 13205 -34.67%13206 -26 .a73448915 -30.912 447488 -30.912247489 -26.873448917 9 -22.588602251 10 --18.094688282 -13.436869039 -8.67608167Od :; -13.436869040 -8.6760816704 13 -3.90823248OE -3.908232479! i i
d’))
(hl)
-49.457697822 Et: -48.148354976 11086 -46.290696520
11086 -43.968281276 -41.232591580 ::i:: -38.122789293 12268 -34.672335569 12268 -30.912284284 12268 -26 .a73495043 13110 -22.588651438 13110 -18.094133390 13110 -13.436902606 14512 -8.676096775 14512 -3.908246487
:;:*g -22 : 708 -18.231 -13.590 -8.884 -4.086
HIIm
-
-
0.0000252 0.0000209 0.0ooo159
0.0ooo128 0.0000137 EEE 0: 0000042 O.OOo(~O51
ProblemB. To find the coupled states in the case of the Coulomb potential u(x) =-UQ/X,
u,=o.7,
(2.3)
with Z-0, iZ=‘f2, mi=m2=m=fi=1. TABLE 2
Ii.88265151 19 e85388229 25.63097061 29.61591470 31 .a9855914 32.49507102 31.40564487 28.63661721 24.21327797 18.19132820 10.67199900 1 .a2793751 -8.04156793 -13.63321714
3.381397382 5.779754796 7.719521811 9.336398640 10.66788469 11.72131879 12.48958877 12.95569235 13.09322624 12.86409991 12.21244540 11.05016072 9.2 1678804 7.04452679
0.4808161463 0.8329491694 1.134601602 1.408022327 1.661337814 1.898384405 2.121078143 2.330210855 2.525717674 2.706665062 2.870958012 3.014527920 3.129086833 3.290166413
TABLE 3
4-t
J 2 3
-0.1225(O)
--0.030625(O)
-0.0136(i) -0.00765625(O)
25.0 56.5 92.5 181.3
1000 2244 3672 7226
-0.122500000 -0.030625000 -0.013611111 -0.007656249929
8 2.4%-l”
1.5*10-‘8 9.3.10-s 9.7.10-Q 3.4.10-15
The potential (2.3) is unbounded at the left-hand end of the semi-axis X0. Solution of SchrGdinger’s equation with a potential of this type is in principle impossible by the perturbation
method of [4] ; the same is true for any potential which increase at infinity.
with unbounded
variation, including potentials
Solving the radial Schriidinger equation for coupled states
135
In the case of potential (2.3), the SPSOL program was used with the conditions r,=10-6; h,=0.025, h,=0.017; Cl=O.OOl. L=O, 1, 2, 3, the values?‘l,(3)
In Table 3 we give the exact energies i,, approximate
energies, the values
XK
and N, the lower bounds of the energies
found for the (AZ(~) (h,) ) low,
zalurted
from (2.2) and the upper bounds (AZ(~) (h,) ) Up, obtained from inequality (1.8) with z - z = 0 and in which, when evaluating constants MI and M3, we used, instead of the exact value of function (p (x), the value of the computed parabolic spline s2 (x). In this case, M2 = 1. In Table 4 we give the values of the coefficients
approximate
wave functions
,pl,
CYI
~1, by means of which the
gC3j(5, Zf3’)
are computed; the upper bounds (Ay(3&, , evaluated from the first term on the right-hand side of inequality (1.9), are also given. On comparing,
for examples A and B, our method of rod splines with the methods used in
[4] and [5], the following features are revealed. 1. In computing
time, our method is comparable
with those of [4,5].
2. It is more stable than the finite difference method
[5]. This is shown by the fact that we
can operate with a smaller step h, and in the final analysis, by the greater accuracy of the approximate
solutions obtained
this is the use of rod function
(both methods have a rate of convergence
0 (K’)). The reason for
exp [-k (z)xq] , which carries “extra information”
about the
solution. TABLE 4
I
‘z
al
0
:
-0.5 *IO-14 -0.9.10-‘5 0.000949808
n.4.10-15 -0.024423641 -0.025660990
0.414208026 0.104672749 0.146634231
1.8.1~‘6 3.8.10-j 1.8.10-j
3
0.000792086
-0.013588558
0.051765347
7.4.10-0
Bl
Unlike the methods described in [4,5],
Yl
(Ahepx
our method is furnished with effective upper bounds
(Az),,, and (Ar)un (see Tables 3 and 4). 3. When the number I, increases, the errors
1Ezt3)
1 of the eigenvalues do not increase
(see Table l), whereas the errors
(z-i”’ 1 , ~-Z’2’ I do increase (for the former, see Table V of [4], and for the latter, see our Table 1). In the case of the method proposed in [4] , this drawback is linked with the fact that the approximate function g(‘) (z, z(“) is not matched at the mesh nodes; this leads to deterioration
of the approximate
solution of the global Cauchy problem
[4] when Ix increases. 4. Unlike the method proposed in [4] , our method is applicable, singular at zero, and for potentials which increase as x + Q). 5. Our method is easily extended
to interpolation
both for potentials
which are
splines of higher degree (cubic, 4th degree,
etc.). The method of rod splines, using splines of the 4th degree (instead of parabolic [l I), has order of convergence 0 (hd), which is the same as the order of convergence of the method described in
[41.
L. Aleksandrov, M. Drenska and D. Karadzhov
136
6. In the case of increasing potentials (1.4b), bounds (AzXP and (Ay)uP do the assumption of Theorem 2 of [ 11, that
not
hold, since
r,
J
p,'(r,Z)p(x,Z)
[s~+‘qJ(z)]*dx+O,
511
is violated in this case. 7. Difficulties can arise when evaluating the integrals appearing in M3, e.g. in the case of too large values of the coefficients q, pi, yi of the parabolic spline for certain i= {I, 2, . . . , N). For instance, when solving problem A we have a~v”lO~~, Pablo’” i.N-lO”, which is the result of rapid damping of the rod exp [-k (z)x4] close to XK. To avoid this, we have to change the scale of the variable x. 8. The lower bound (Az)l,, (see (2.2) and Tables 1, 3) is in principle also applicable to finite-difference methods, It is of practical importance, since, when using fairly small hl and h2 (or in the case of secondary realization of the method with new, smaller hl and h2), a true picture is obtained of the present error ~z_~(3) 1 (see Tables 1 and 3). However, as we pointed out in property 2, the use of a fine mesh step is better adapted to the method of rod splines, than to finite-difference methods. Notice finally that a comparison of the method of rod splines with other methods, notably the new promising method proposed in [6], is desirable, The authors sincerely thank A. F. Nikiforov for valuable discussions. Translated by
D. E. Brown
REFERENCES 1. ALEKSANDROV, L., and KARADZHOV, D., A method for the approximate solution of eigenvalue problems for high-order linear differential equations, Zh. vychisl.Mar. mat Fiz., 20, No. 4, 923-938, 1980. 2. TIKHONOV, A. N., and SAMARSKII, A. A., The Sturm-Liouville Fiz., 1,No.5,784-805, 1961.
difference problem, Zh. vychisl.Mar. mat.
3. UVAROV, V. B., and ALDONYASOV, V. I., A phase method of finding the eigenvalues of Schrodinger’s equation, Zh. vFchis1.Mat. mat. Fiz., 7, No. 2,436-440, 1967. 4. ADAM, CH., IXARU, L. GR., and CORCIOVEI, A., A first-order perturbative numerical method for the solution of the radial Schrijdinger equation, J. Compur. Phys., 22, l-33, 1976. 5. AKISHIN, A. T., and PUZYNIN, I. V., Realization of Newton’s method in the Sturm-LiouviIle problem, Soobshch. OIYaI, S-10992, Dubna, 1977.
difference
6. NIKIFOROV, A. F., UVAROV, V. B., and NOVIKOV, V. G., Solution of Schrodinger’s equation for coupled states by method of backward iterations, Preprint IPM Akad. Nauk SSSR, No. 42, Moscow,1980. 7. ALEKSANDROV, L., et al, The SPSOL program for solving radial Schrodinger equations using the method of rod splines, Soobshch. OIYaI, Pll-80752, Dubna, 1980.
1982.
0041-5553/82/020130-07$07.50/O 01983. Pergamon Press Ltd.
APPLICATION OF ROD SPLINES FOR SOLVING THE RADIAL SCHRiiDINGER EQUATION FOR COUPLED STATES* L. ALEKSANDROV, M. DRENSKA and D. KARADZHOV Sofia, Bulgaria (Received 17 April 1980; revised 3 June 1981)
THE METHOD of rod splines is compared with other methods for solving the radial SchrGdinger equation,
and special features of the method are pointed out.
Introduction We described in [I ] a method of rod splines for solving eigenvalue problems for linear differential equations of high order, and we applied the method to radial Kadyshevskii equations (equations of infinite order). Here we shall apply the method to the radial Schr6dinger equation for coupled states a low-order equation); this equation can be solved by a variety of methods, see [2-61, etc. We shall compare our method with the finite difference method [5] and the method described in [4], taking the example of the Saxon-Woods potential (the optical potential) and the Coulomb potential.
1, Solution of Schriidinger’s equation for coupled states In the method of rod splines, the radial Schrijdinger
equation
(1.1) =x
s
[y(x,z)]*ds=l
(1.2)
XH
X= [z., s,] cXo= (0, m), where XH is taken fairly close to zero, while XK is fairly large, and in an interval of discrete energies Z = [ZH, ZK] , the choice of which depends on the behaviour of the potential v (x) as x + 00.
is considered in a finite interval
In Eq. (1.1) we use the following notation: 1 is the orbital momentum quantum number (1=0,1,. . . .), and fi is the relative mass, which in the system of units with fl = 1 has the form
R=m,m2/(m,+m,).
*Zh. vjkhisl. Mat. mat. Fiz., 22, 2, 375-381,
1982.
130
Solving the radial Schrbdinger equation for coupled states
131
To use the method of rod splines, we have to choose a rod function [l] ; to this end, we shall consider potentials v (x) E C (X0) with the following behaviour at the ends of the interval X0: as x+0,
u(x)+
=
UOO const
r<2,
(1.3) and asx+=, u(x)-+0
(1.4a)
or u(x) “VOX’,
uo, t=const>O.
(1.4b)
This class of functions u (x) defines the class of radial functions characterized by the asymptotic behaviour y
(x, 2) “0,x’+‘,
Y (x92) 42 exp [-k
Oj=const,
(z)4,
1/x+00,
&=const,
where, in case (I .4a), we have q=l, k(z) =(-~Ez)‘~, z&c have 4= (t+2) /2, k (z) = (2iiivo) ‘“Iq, ZEZC [0, =J).
(1.5)
x*00,
(-a~, 01,
(1.6)
and in case (1.4b) we
Relations (1.5), (1.6) show that, as the rod of the solution we can use the function p* (x, 2) =xl+l exp [-k(z)s’].
Using the identity a2 Z-,$+’
L(z+l) x2 -5+2
+ 2% [z-v (3) ]
>
(a++‘& (x, 2) )
1+x d --~+2iiiIz--u(r)l
We will solve the equivalent Schrijdinger equation x=X,
$+2
Z&,
(1.1’)
with the simpler rod
p2(x, z)=exp
[-k(z)sg].
(1.7)
L. Aleksandrov, M. Drensku and D. Karadzhov
132
In the case of problem (1 .l’), (1.2), (1.7), the main quantities [l] have the values ,jrr
Wzp:-Pp,,+D,(“),
P’&
of the method of rod splines
2,3,
d,,=qk (2) xy-’{gk(2)2,*-[2(1+1)+q--ll}exp[-k(z)qPl, dsj=O,
dz~=-2~k(z)xjp-‘exp[-Ic(z)qp], P*j’P2 (q,
where
(zj},
2),
2, . . . , N,
j=l,
1+1
Uj=2E[C_V(Zj)],
dij=2’-,
a:,=11 xj
nodes of the parabolic spline [ I] ,
is the mesh of interpolation
Let the pairs (I?, s(x, Z)), (z”, y”(5, 2) ) denote respectively the exact and approximate solutions (found by the method of rod splines) of problem (1 .l’), (1.2) and let the pair == considered on the semi-axis X0 (y, 2, Z”&), be the exact solution of problem (2, I/(x, 5, (1 .l’), (1.2). Also, let the conditions of Theorem 2 of [I] be satisfied. Then, as when solving equations of infinite order (see [ 1, Section 2]), we have
(1.8)
GM, (Jf,+M,M,) hZ+O (h’) + max I [q kc> -? (z> 1 sLp l;(5) {expL-k(Z)sql (OF,1
xe~p{-[k(T)-k(Z)]sq}l+
-
exp[-k(
Y)rq]}
(1.9)
I+
SUP
{exp[-k(Z)xQl-
I;(Z)
exp[-k(~)zqlI
1,
P$,-1
where h =
(XK -
is the mesh step of the parabolic spline,
XH)/N
Mi=2.5
max I q” (5) I,
Mp -
max Ip2(z, E) I, I
X
Ms= xx
XX
(1.10)
MI = marl sz(x) pzz’(X,z> 1, z
If lim max I q (5) -7 (5) I 4 XH4 x x,:-c= the second term on the right-hand side of (1.8), and the last three terms on the right-hand side of (1.9), can be made as small as desired by taking sufficiently small values of XH and h, and sufficiently large XK .
Solving the radial Schrbkiingerequation for coupled states
133
2. Numerical comparison of the method of rod splines with other methods
The method of rod splines was realized by means of the SPSOL subroutine in the type of construction of the basis spline described in [7]. The IBM 370/145 computer was used in the double accuracy mode.
The following two problems were solved. Roblem A. To find the coupled states in the case of the optical potential [4]
v(5)=where T-exp
[ (x-v~)/v~],
Vi
i+T
1
T
3
--;;(1ST) +
(2.1)
v,=-50, ~~-7,vs=0.6.
We wish to find the coupled states corresponding to I= 0 with Fii=i/2,
mi=ma==m-=h-i.
We know [4] that, under these conditions, problem (1.1’) (1.2), (1.7) has 14 radial solutions, 13, and they are known in analytical form.
1x=0,...,
The SPSOL program was used in the conditions z,,= 10e6; h, -0.00065,h,-0.00043 (h, is the basic step, h2 is the step used when obtaining the “lower bound of energy”); Cl = 0.005. Notice that, with the given hl and Cl, the quantities XK and N are computed by the program in such a way that step hl remains constant while the inequality
is satisfied. In Table 1 we quote the following quantities: the exact values of the energy ?? (see [4,5]); the approximate values r(l), found by the method described in [4] with h = 0.004,XK = 24.2; the approximate energies ?‘c2), found by the finite-difference method [5] with h = 0.03; the approximate energies ?c3), found by the method of rod splines with hl = 0.00065, and also the values of XK and N used by the SPSOL program. We also give in Table 1 the lower bounds (AZ(~))low, evaluated from the expression (Az(~)),~~ = IP) (h,) -z”(3) ( h2) ( /2.
(24
Obviously,
The coefficients al, @1,71, by means of which, using (1.4) (1.8), (1.1 l), (1.15), (1.17), and (1 .19) of [ 11, the approximate wave functions gt3) (r, z”(‘)), were evaluated, are given in Table 2.
L Aleksandrov,M. Drenska and D. Karadzhov
134
TABLE 1
‘, 0 1 2 3 4
I = I ;
; (2)
(1)
z
-49.457788728 -48.148430420 -46.290753954 -43.968318431 -41.232607772
N
i(3)
-49.451788728 -48 *148430419 -46.290753955
-49.461 -48.158 -46.310
-
I_::*;;: -38:11a -34.743
-38.122785096 -34.672 13205 -34.67%13206 -26 .a73448915 -30.912 447488 -30.912247489 -26.873448917 9 -22.588602251 10 --18.094688282 -13.436869039 -8.67608167Od :; -13.436869040 -8.6760816704 13 -3.90823248OE -3.908232479! i i
d’))
(hl)
-49.457697822 Et: -48.148354976 11086 -46.290696520
11086 -43.968281276 -41.232591580 ::i:: -38.122789293 12268 -34.672335569 12268 -30.912284284 12268 -26 .a73495043 13110 -22.588651438 13110 -18.094133390 13110 -13.436902606 14512 -8.676096775 14512 -3.908246487
:;:*g -22 : 708 -18.231 -13.590 -8.884 -4.086
HIIm
-
-
0.0000252 0.0000209 0.0ooo159
0.0ooo128 0.0000137 EEE 0: 0000042 O.OOo(~O51
ProblemB. To find the coupled states in the case of the Coulomb potential u(x) =-UQ/X,
u,=o.7,
(2.3)
with Z-0, iZ=‘f2, mi=m2=m=fi=1. TABLE 2
Ii.88265151 19 e85388229 25.63097061 29.61591470 31 .a9855914 32.49507102 31.40564487 28.63661721 24.21327797 18.19132820 10.67199900 1 .a2793751 -8.04156793 -13.63321714
3.381397382 5.779754796 7.719521811 9.336398640 10.66788469 11.72131879 12.48958877 12.95569235 13.09322624 12.86409991 12.21244540 11.05016072 9.2 1678804 7.04452679
0.4808161463 0.8329491694 1.134601602 1.408022327 1.661337814 1.898384405 2.121078143 2.330210855 2.525717674 2.706665062 2.870958012 3.014527920 3.129086833 3.290166413
TABLE 3
4-t
J 2 3
-0.1225(O)
--0.030625(O)
-0.0136(i) -0.00765625(O)
25.0 56.5 92.5 181.3
1000 2244 3672 7226
-0.122500000 -0.030625000 -0.013611111 -0.007656249929
8 2.4%-l”
1.5*10-‘8 9.3.10-s 9.7.10-Q 3.4.10-15
The potential (2.3) is unbounded at the left-hand end of the semi-axis X0. Solution of SchrGdinger’s equation with a potential of this type is in principle impossible by the perturbation
method of [4] ; the same is true for any potential which increase at infinity.
with unbounded
variation, including potentials
Solving the radial Schriidinger equation for coupled states
135
In the case of potential (2.3), the SPSOL program was used with the conditions r,=10-6; h,=0.025, h,=0.017; Cl=O.OOl. L=O, 1, 2, 3, the values?‘l,(3)
In Table 3 we give the exact energies i,, approximate
energies, the values
XK
and N, the lower bounds of the energies
found for the (AZ(~) (h,) ) low,
zalurted
from (2.2) and the upper bounds (AZ(~) (h,) ) Up, obtained from inequality (1.8) with z - z = 0 and in which, when evaluating constants MI and M3, we used, instead of the exact value of function (p (x), the value of the computed parabolic spline s2 (x). In this case, M2 = 1. In Table 4 we give the values of the coefficients
approximate
wave functions
,pl,
CYI
~1, by means of which the
gC3j(5, Zf3’)
are computed; the upper bounds (Ay(3&, , evaluated from the first term on the right-hand side of inequality (1.9), are also given. On comparing,
for examples A and B, our method of rod splines with the methods used in
[4] and [5], the following features are revealed. 1. In computing
time, our method is comparable
with those of [4,5].
2. It is more stable than the finite difference method
[5]. This is shown by the fact that we
can operate with a smaller step h, and in the final analysis, by the greater accuracy of the approximate
solutions obtained
this is the use of rod function
(both methods have a rate of convergence
0 (K’)). The reason for
exp [-k (z)xq] , which carries “extra information”
about the
solution. TABLE 4
I
‘z
al
0
:
-0.5 *IO-14 -0.9.10-‘5 0.000949808
n.4.10-15 -0.024423641 -0.025660990
0.414208026 0.104672749 0.146634231
1.8.1~‘6 3.8.10-j 1.8.10-j
3
0.000792086
-0.013588558
0.051765347
7.4.10-0
Bl
Unlike the methods described in [4,5],
Yl
(Ahepx
our method is furnished with effective upper bounds
(Az),,, and (Ar)un (see Tables 3 and 4). 3. When the number I, increases, the errors
1Ezt3)
1 of the eigenvalues do not increase
(see Table l), whereas the errors
(z-i”’ 1 , ~-Z’2’ I do increase (for the former, see Table V of [4], and for the latter, see our Table 1). In the case of the method proposed in [4] , this drawback is linked with the fact that the approximate function g(‘) (z, z(“) is not matched at the mesh nodes; this leads to deterioration
of the approximate
solution of the global Cauchy problem
[4] when Ix increases. 4. Unlike the method proposed in [4] , our method is applicable, singular at zero, and for potentials which increase as x + Q). 5. Our method is easily extended
to interpolation
both for potentials
which are
splines of higher degree (cubic, 4th degree,
etc.). The method of rod splines, using splines of the 4th degree (instead of parabolic [l I), has order of convergence 0 (hd), which is the same as the order of convergence of the method described in
[41.
L. Aleksandrov, M. Drenska and D. Karadzhov
136
6. In the case of increasing potentials (1.4b), bounds (AzXP and (Ay)uP do the assumption of Theorem 2 of [ 11, that
not
hold, since
r,
J
p,'(r,Z)p(x,Z)
[s~+‘qJ(z)]*dx+O,
511
is violated in this case. 7. Difficulties can arise when evaluating the integrals appearing in M3, e.g. in the case of too large values of the coefficients q, pi, yi of the parabolic spline for certain i= {I, 2, . . . , N). For instance, when solving problem A we have a~v”lO~~, Pablo’” i.N-lO”, which is the result of rapid damping of the rod exp [-k (z)x4] close to XK. To avoid this, we have to change the scale of the variable x. 8. The lower bound (Az)l,, (see (2.2) and Tables 1, 3) is in principle also applicable to finite-difference methods, It is of practical importance, since, when using fairly small hl and h2 (or in the case of secondary realization of the method with new, smaller hl and h2), a true picture is obtained of the present error ~z_~(3) 1 (see Tables 1 and 3). However, as we pointed out in property 2, the use of a fine mesh step is better adapted to the method of rod splines, than to finite-difference methods. Notice finally that a comparison of the method of rod splines with other methods, notably the new promising method proposed in [6], is desirable, The authors sincerely thank A. F. Nikiforov for valuable discussions. Translated by
D. E. Brown
REFERENCES 1. ALEKSANDROV, L., and KARADZHOV, D., A method for the approximate solution of eigenvalue problems for high-order linear differential equations, Zh. vychisl.Mar. mat Fiz., 20, No. 4, 923-938, 1980. 2. TIKHONOV, A. N., and SAMARSKII, A. A., The Sturm-Liouville Fiz., 1,No.5,784-805, 1961.
difference problem, Zh. vychisl.Mar. mat.
3. UVAROV, V. B., and ALDONYASOV, V. I., A phase method of finding the eigenvalues of Schrodinger’s equation, Zh. vFchis1.Mat. mat. Fiz., 7, No. 2,436-440, 1967. 4. ADAM, CH., IXARU, L. GR., and CORCIOVEI, A., A first-order perturbative numerical method for the solution of the radial Schrijdinger equation, J. Compur. Phys., 22, l-33, 1976. 5. AKISHIN, A. T., and PUZYNIN, I. V., Realization of Newton’s method in the Sturm-LiouviIle problem, Soobshch. OIYaI, S-10992, Dubna, 1977.
difference
6. NIKIFOROV, A. F., UVAROV, V. B., and NOVIKOV, V. G., Solution of Schrodinger’s equation for coupled states by method of backward iterations, Preprint IPM Akad. Nauk SSSR, No. 42, Moscow,1980. 7. ALEKSANDROV, L., et al, The SPSOL program for solving radial Schrodinger equations using the method of rod splines, Soobshch. OIYaI, Pll-80752, Dubna, 1980.