Extension of the Factorization MethodMayer Humi Citation: J. Math. Phys. 9, 1258 (1968); doi: 10.1063/1.1664706 View online: http://dx.doi.org/10.1063/1.1664706 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v9/i8 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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1258 PIERRE C. SABA TIER

which yields

Q(z) = 2cx(1 + 2cxz2)-l,

U(z) = 4cx(1 - cxz2)(1 + 2cxz2r 2,

X-:;:(z) = (1 + 2cxz2r!s;.(z),

x1(z) = [(1 + 2cxz2)! _A_ A+l

+ (1 + 2cxz2)-! _l_-Js.b). A+l

(3.99)

It is pleasant to see in this example most of the singularities studied above. It is straightforward and tedious to verify the SchrOdinger equation (3.91).

Correspondence with the Figure

The correspondence of the zero-base machinery with Fig. 1 is as follows. The input functions are J±(z, z'). The properties (1) are either (3.26) and (3.75) or (3.27). The output generator is K±(z, z'). The

JOURNAL OF MATHEMATICAL PHYSICS

properties (2) are (3.89) and (3.78). Relations A are (3.5), (3.7), and (3.8). Relation B is (3.91). C is the set (3.88), (3.89). D is (3.3).

The correspondence of the general base machinery with the figure is as follows. The input functions are g±(z, z'). The property (1) is (3.47). The output generator is K±(z, z'). The properties (2) are (3.16), (3.17), and (3.18). Relations A are (3.19), (3.20), (3.23), and (3.8), with reference to (3.17) and (3.18). Relation B is (3.91). C follows from (2.35), (2.57), and (2.58). D is (3.15).

ACKNOWLEDGMENTS

It is a pleasure for the author to thank Professor R. G. Newton for fruitful discussions. He would also like to thank Professor E. J. Konopinski and Professor R. G. Newton for many stylistic improvements in the manuscript, and he wishes to express his appreciation to the theoretical physics group of Indiana University for their warm hospitality.

VOLUME 9, NUMBER 8 AUGUST 1968

Extension of the Factorization Method

MAYER HUM! Department of Applied Mathematics, The Weizmann Institute of Science, Rehovot, Israel

(Received 13 September 1967)

We show that it is possible to extend the formalism ofthe factorization method for any displacement in the spectrum space of any second-order differential equation. Following this, we show that we can extend, at least formally, the formalism for some nth-order ordinary differential equations.

I. INTRODUCTION

It is well known that both electromagnetic and quantum theory lead to equations of the type

d2

~ + rex, m)y + AY = 0 (1) dx

The well-known factorization methodl sets

d R = k(x, m + 1) - -,

1.- = k(x, m) + ~ , dx

dx

(3)

(or can be transformed to this form). An outstanding problem related to the solution of these equations is the problem of raising and lowering operators; i.e., assuming that we deal with the discrete spectrum of (1) so that we can label the solutions by yeA, m), then we want to find first-order differential operators which connect the solution yeA, m) with yeA, m + 1):

and discusses under what conditions the following equations hold:

[k(X, m + 1) - :x}(A, m)

= [A - L(m + 1)]!Y(A, m + 1),

[ k(x, m) + :x}(A, m)

RY(A, m) = CX(A, m + 1) yeA, m + 1), = [A - L(m)]!Y(A, m - 1). (4)

LY(A, m) = CX(A, m) yeA, m - 1). (2) 1 L. Infeld and T. E. Hull, Rev. Mod. Phys. 23, 21 (1950).

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EXTENSION OF THE FACTORIZATION METHOD 1259

The explicit form for the functionsk(x, m) andL(m) is, of course, related to the particular equation (1) with which we are dealing.

Thus the factorization method equips us with a procedure for performing integer displacements on the m-spectrum line (assuming m real). This is sufficient as long as the interest is focused on the discrete spectrum of the above equations. However, in recent years use has been made of solutions of Eq. (1) in which m and A. assume continuous (real or complex) values. This is the case, e.g., in Regge-pole theory where one uses Legendre polynomials with contin­uous indices. In what follows we shall therefore discuss a generalization of the factorization method for arbitrary displacement Am. One may look on this generalization as an analytic continuation of the operators Rand L in (3) above.

II. METHOD

In order to gain generality we assume that we deal with the complex differential equation

d2y(z) -;w:- + r(z, m)y(z) + A.y(z) = O. (5)

As a first step of our extension, we find operators which displace y(A., m) to y(A., m + Am), where Am is assumed to 'be real, while m may be complex. We denote the operators that displace m from y(A., m) to y(A., m + Am) by Hm.t1m 2 (Am is a positive or negative real number, jAmj > 0) and try to find a solution for Hm.t1m of the form3

Hm,tJ.m = [k(Z, m + t + ~m) - sgn(Am) :z} (6)

so that

[k(Z' m + t + A;) - sgn(Am) :z}(A., m)

= [A. - L(m + t + A;)]\(A., m + Am). (7)

One notes immediately that when Am = ± 1, the operators (7) coincide in their form with those of (4).

We want to fix k(z, m) so that

[HmHm,-tJ.mHm,tJ.m]y(A., m)

= [A. - L( m + t + A;)}(A., m) (8)

2 We note that we may restrict ourselves to I Llml < I, since for Llm > 1 we may perform the displacement in steps.

3 The form of the operators Hm.tJ.m and L is not arbitrary. In fact, if we try to write

d Hm.tJ.m = k[z, m + f(Llm)] - g(Llm) dz' L(m, Llm) = L[m + h(Llm)],

then it is easy to show that the only form that might work is the form chosen by us.

coincides with the original equation (5) for y(A., m),4 i.e.,

[k(Z, m + t + A;) - sgn(-Am) :J x [k(z,m+t+A;) -Sgn(Am):z}(A.,m)

(9)

Expanding, we get

[k(Z, m + t + A;r - sgn (-Am)

:x ~ k(Z m + 1. + Am) - ~]Y·(A. m) dz ' 2 2 dz 2 '

= [). - L( m + t + A;)]y(A., m). (10)

Equating (10) with (5), we get

k(Z' m + t + A2mr - sgn (-Am)

X :z k(z, m + t + A2m) + L(m + t + A2m)

= -r(z, m). (11)

If Am> 0, we then get

k( z, m + t + A2mr + :z k (z, m + t + A2m)

+ L( m + t + A2m) = -r(z, m); (12)

while if Am < 0, let us substitute Am' = -Am and drop the prime after the substitution (so that in both cases now Am = jAm!):

k (z m + 12 - Am) - ~ k (z m + 1. _ Am) , 2 dz ' 2 2

+ L( m + t - A;) = -r(z, m). (13)

Subtracting (13) from (12), we get

k(Z' m + t + A;r - k(Z' m + t - A;r + ~ k(Z' m + t + Am) + ~ k(Z' m + t _ Am)

dz 2 dz 2

=L(m+t-A;) -L(m+t+A;). (14)

• This requirement is needed since one might look on Eq. (7) as a definition of y(A, m + Llm). Equation (9) assures us that y(A, m + Llm) satisfies Eq. (5) for (A, m + Llm).

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1260 MAYER HUMI

Let us assume now that5

k(z, m, Llm)

= ko(z, Llm) + (m + t + Ll2m) kb, Llm). (15)

[This form of k(z, m, Llm) is compatible with the above explicit form of k(z, m + t + Llm/2) since Llm is always positive in our notation.] Substituting (15) into (14), we get

[(m + t + Ll;)kl + koT

- [ko + (m + t - ~m) kl] 2 + k~

+ (m + t + ~;) k{ + k~ + (m + t - Ll;) k{

=L(m+t-~m) -L(m+t+Ll2m). (16)

Note now that the coefficient of k~ is

( m + t + Ll;) + (m + t _ Ll;)

G( Llm)2 ( Llm)2] 1 = ~ in + t + 2 - m + t - 2 Llm'

while the coefficient of k~ is

Thus we may rewrite (16) as

{(m + t + Ll;)Tk~ + Ll~ k~J

+ 2(m + i + Ll;)[kokl + LlL k~]}

- {(m + i - Ll2m)Tk~ + Ll~ k{]

+ 2(m + i - Ll;) [kokl + Ll~ k~J}

=L(m+t-Ll;) -L(m+i+Ll2m). (17)

Replacing z by x = zLlm and denoting the derivative of k with respect to x by k, we get for the general

• Here, and in other places, we shall not give the full formal justification of our steps. This has been done already in Ref. I and might be generalized easily in our case.

solution of Eq. (17)

L(m+i+~m)

= -{(m + t + Ll;r[K~ + kd

+ 2 (m +t + Ll;) [kok1 + ko]}. (18)

This must hold for all values of m and therefore

k~ + kl = a, (19)

a =F 0,

a = 0, (20)

where a, b, and c are independent of z, m, and Llm. The form of Eqs. (19) and (20) is identical with that of Eqs. (3.15) in Ref. 1.

Let us now turn to the case of imaginary displace­ment of the spectrum iLlm (where Llm is real).

We denote once again the operators that displace fromY(A, m) to yeA, m + iLlm) by Hm.i~m(ILlml > 0), and try to find a solution of the form

./\ [( i iLlm) d ] Hm,. m = k z, m + 2 + 2 - sgn (Llm) dz '

so that

[ ( i iLlm) d] k z, n:t + 2 + 2 - sgn (Llm) dz yeA, m)

[ ( i iLlm)]t . = A - L m + 2 + 2 yeA, m + ILlm).

By similar reasoning, we are led to the equation

k z m+-+- -k z m+----( i iLlm)2 ( i iLlm)2

, 2 2 '2 2

+-k z m+-+-d ( i iLlm) dz ' 2 2

+-k z m+----d ( i iLlm) dz ' 2 2

Let us assume now that (Llm > 0)

k(z, m, iLlm)

(21)

(22)

= ko(z, iLlm) + (m + ~ + i~m) k1(z, iLlm). (24)

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EXTENSION OF THE FACTORIZATION METHOD 1261

This leads to the equation

{(m + £ + iLlm)2[k2 + _1 k'J 2 2 1 iLlm 1

( i iLlm) [ l'J} + 2 m + '2 + -2- kokl + iLlm ko

_ {(m + £ - iLlm)2[k2 + _1 k'J 2 2 1 iLlm 1

+ 2(m + £ - iLlm)[k k + _1 k'J} 2 2 0 1 iLlm 0

= L(m + ~ - iLl2m) - L(m + ~ + iLl2m). (25)

Substituting x = izLlm and denoting the derivative of k with respect to x by k, we get for the general solution of (25)

L(m + ~ + i~m)

= -{(m + ~ + ib2m)2[k~ + kd

+ 2 (m + ~ + i~m) [kok1 + ko]}, (26)

which leads to Egs. (19) and (20), and hence to the same solutions.

The same method described above is applicable to perform displacements on the A. plane. This can be done by solving the second-class problem (in the nomenclature of Ref. 1) of Eq. (5). We denote the operators that displace from (A., m) to (A., m') or (A.', m) by Hu.m).U.m') and H(A.m).p:.m), respectively.

Until now we have dealt with operators in which the displacement Llm was different from zero. When Llm equals zero, it is natural to define sgn(O) = 0 and k(z, m, 0) = 1 so that

HU,m),(A,m) = 1. (27)

The above definitions are natural since zero is the mean of the jump in the values of the sign function at this point. On the other hand, since k(z, m, Llm) depends on z through the form zLlm, it follows that k is independent of z when Llm = O. Therefore, if we deal with normalized operators, we must fix k(z, m, 0) = 1.

Once we have found operators which displace the real and imaginary axis of the m plane, we may use these operators in order to perform displacements in an arbitrary direction. This can be done by multiplying two operators of the above kinds. Thus if we want to perform a displacement Llm such that

Llm = Re Llm + i 1m Llm, (28)

then the desired operator, which will be denoted by D(A.m),(A.m+llm), is

D().,m),().,m+dm)

= H(A,m),(A,m + Re dm) H().,m),(A,m+i 1m dm). (29)

However, once we find these displacement operators, we can perform other operations in the spectrum space. We illustrate the method for rotation operators in the real A.m plane (other operators can be easily inferred).

To find these operators we use the polar-coordinate system (re) in the spectrum space:

A. = r cos e, m = r sin e. (30)

A rotation (r, e) ~ Cr, e + Lle) means that we are looking for operators that displace from CA., m) to (A.', m'), where

A' = r cos (e + Lle), m' =J sin (e + Lle). (31)

Therefore the desired operator, which is denoted by R(r.O).(r.6+M), is

R(r,O),(r,O+M) = D[r cos8,rsin (8+d8)].[r COs (O+d8),rsin (8-!-dO)]

X D(rcosO,rsinO),[rcos8,rsin(6+M)]. (32)

We notice, however, that a rotation by Lle may be accomplished in two ways: either by Lle or Lle - 21T; the two operators do not coincide. This is due to the discontinuity in the passage from positive-displace­ment operators to negative ones. This discontinuity can be understood if we view our procedure as a method for calculating the square root of a differential operator, which leads, as usual, to a two-sheeted solution of the problem. In the following we may choose to work in either sheet as convenient.

In the above discussion we confine ourselves to ordinary second-order differential equations. Never­theless, the whole discussion is valid for partial­differential equations which are separable. This is the case for the n-dimensional Laplacian. We observe, however, that in this case only two of the eigenvalues can be treated exactly as above. In order to displace other eigenvalues we must use combined first- and second-class operators. This follows from the fact that these eigenvalues appear in two ordinary equa­tions which result from the separation of the partial­differential equation (see Sec. IV).

m. GLOBAL TRANSFORMATIONS IN THE SPECTRUM SPACE

Until now we have been dealing with operators in the z space which transform a specified function y(A., m) to another one y(A.', m') [thus inducing a transformation in the spectrum space from (A., m) to (A.', m')]. In this sense the operators found until now

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1262 MAYER HUMI

are point operators since they usually depend on the starting spectrum point (A, m). We are looking now for operators which are global. This means that these operators will act on any function yeA, m) or a com­bination of these functions to produce a desired change in the dependence of these functions on the spectrum points. To this end we must define at first the space of functions upon which our global operators will operate.

At first let us observe that the integral

I y(z, A, m)*y(z, A', m') dz (33)

for arbitrary (A, m) might be divergent. Therefore we must use the technique of regularization6 in order that (33) will converge and be zero for (A'm') :;t. (A, m). It was shown in Ref. 6 that this is possible if the functions yeA, m) are not too steep. Since our interest will be focused on spherical harmonics, this will be always possible. A detailed study of the regularization technique for the spherical harmonics is given in Refs. 7 and 8. So let us define

[yeA, m), yeA', m')]

= reg I y(A, m)*Y(A', m')p(z) dz; (34)

where p(z) is a weighting function such that

[yeA, m), Y(A'm'») = ex(A, m)I5(A, X)b(m, m'), (35)

where ex is not necessarily positive. However, we may assume that the y's are normalized so that ex(A, m) = ± 1. Denote those functions yeA, m) for which the bilinear form (35) defined above is positive [negative) by yeA, m, + ) [y(A, m, - )]. In the following we deal with {yeA, m, + )}, but {yeA, m, -)} can be treated exactly in the same way.

Let us now define the space of functions

S+ = {f;f = ~l aiY(A;, m;, +), n finite}. (36)

S+ is a normal space in which one may define a scalar product in a natural way. According to a weH­known theorem in the theory of Hilbert spaces, it is possible to embed such a space into a Hilbert space Je+ so that S+ is dense everywhere in Je+. Therefore, for any I E Je+, it is possible to write

f = L L a(A, m)Y(A, m, +). (37) ). m

In a similar way, we can construct Je-.

6 I. M. Gel'fand and G. E. Shiloy, Generalized Functions (Aca­demic Press Inc., New York, 1964), Vol. I.

7 J. Fischer and R. Raczka, International Centre for Theoretical Physics (Trieste) preprint IC/66/101.

8 S. Sannikoy, Yad. Fiz. 2, 570 (1965) [SOY. J. Nuc. Phys. 2, 407 (1966)].

In the space Je+ EB Je-, one may write, for any function,

fez) = IIdA dm a(A, m) y(z, A, m), (38)

where the above equality is strong (in the norm) and

ex(A, m) = reg If*(Z)Y(Z, A, m)p(z) dz. (39)

Having defined the space of functions and its topology, we will now write the global-transformation opera­tors.

A. Translation Operators

Translations in the m plane:

I DAm = II dA dmD()',m),<A.,mHm) A~,

where

A~f = a(A, m)Y(A, m);

in the A plane we can write similarly

IDA), = II dA dmD().,m),().H)',m) A~,

(40)

(41)

(42)

and in a similar way we can write translation opera­tors in any direction by combining A- and m-transla­tion operators.

B. Rotation Operators

We write down, as an example, the openitors that rotate the real (A, m) plane:

IRAiJ = II dr dOR(r,6)(r,6H6)A;, (43)

where

A~f = a(r, O)y(r, 0). (44)

In a similar way we may write other rotation opera­tors.

The operators discussed above operate on the space Je+ EB Je-; however, each of these operators induces an appropriate transformation in the spectrum space. This is a result of the one-to-one correspondence between the points of the spectrum space and the functions {y(z, A, m)}. Thus, if we denote by I the map

I: (A, m) ---+ y(z, A, m), (45)

then the appropriate operator that displaces each point (A, m) to (A, m + ~m) is l-lID!!."'f In the following, we shall not write I explicitly and view the operators ID, IR, etc., as operators which operate on the spectrum space directly. In this way we are able to operate on the spectrum space by using operators which depend on the z-space coordinates. By means of

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EXTENSION OF THE FACTORIZATION METHOD 1263

the above procedures we can generate any regular transformation on the spectrum space. Thus we can express any transformation that belongs to GL(2, c) on the spectrum space by means of z-space operators. It is easily seen that we can generalize our results to the n-dimensional case for which we get the group GL(n, c).

IV. EXAMPLES

In the following, we give some examples in order to illustrate the method discussed above. We shall also deal with some algebraic aspects of our operators.

A. Associated Spherical Harmonics

Class I Problem

The differential equation is

[ _1_ ~(sin e~) -~ + ).Jp = O. (46) sin e de de sin2 e

We bring this equation to the standard form by means of the substitution

Y = sin! ep, (47)

and the differential equation which results is

d2

y _ (m2

- t) y + (). + t)y = o. de2 sin2 e

(48)

The solution we are looking for is given by

k(e, m, ~m) = (m + ~2m) cot (e~m),

( ~m) ( ~m)2 L m+t+T = m+T' (49)

Class II Problem

We introduce /(l + 1) for)' and replace -m2 by).. Equation (48) then becomes

[ _.1_ ~ (sin e.!!:...) + 1(1 + 1) + +Jp = O. (50) sm e de de sm e

By changing variables z = log tan (eI2), we obtain for pee)

d2P + 1(1 + 1) P + ).P = O.

dz2 cosh2 Z

(51)

The solution is then

k(z, I, ~l) = (I + t + ~l) tanh (z~l),

( ~I) ( ~1)2 L 1+t+2 =- l+t+ 2 · (52)

One can easily see now that for any ~m > 0 the

operators

and

(53)

with (54)

form an algebra which is isomorphic to 0(3). This can be easily seen by using the fact that

(55)

so that by (8) we get

L = [H+ H-] = _1_Jdm d)' o , (~m)2

x [Hm-Am,+AmHm.-Am _ Hm+Am.-AmHm,Am]A~

=(~~)2J[L(m+t+~;) -L(m+t-~2m)J x A~ dm d)' = 2 JmA~ d)' dm (56)

~m and

[H+ L] =Jdm d)' Hm,Am AlI 2m' A l ', dm' d)" , 0 ~m m ~m m

- J dm' d)" ~: A!;, J dm d)',

--A = dmd)' - - __ Al Hm

.Am

l J [2m 2(m + ~m)JHm'Am ~m m ~m ~m ~m m

= -2H+, (57) while

[H-, Lo] = 2H-. (58)

It is to be remarked that we could deal with the alge­bra {H~m}Am=_oo . This algebra is an infinite-param­eter Lie algebra. It plays no role in the following sections, although it may have some physical implica­tions which will be dealt with elsewhere.

B. 0(3, 1) Spherical Harmonics

We deal with the following differential equation:

{ _ 1 ~ (COSh2 ()l~) + 1 _1_ cosh2

()l ael a()l cosh2 el sin e2

X ~(sin () ~) - _1_ ~}p =),P (59) O()2 2 O()2 sin2 ()2 o()~ .

In order to separate the equation, let us substitute

(60)

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1264 MAYER HUMI

The differential equations which result are

-- - cosh2 ()1 - + 12(12 + 1) tp {I O( 0) 1 }

cosh2 ()1 O()1 O()1 cosh2

()1

= 11(11 + 2)tp, (61)

(62)

(63)

We see that Eqs. (62) and (63) are the same as for the associated spherical harmonics. Nevertheless, it is to be noticed that in order to perform a displacement in 12 we must combine first-class operators [for (61)] with second-class operators [for (62)]. Class I operators for (61) are

k«()I' 11' ~12) = (12 + t + ~;2) tanh «()1~()' (64)

from which

L(12 + ! + ~;2) = - (12 + t + ~;2r (65)

Class II operators are

k«()I' 11' ~Il) = (11 + ! + ~;1) tan «()1~11)' (66)

from which

L(11 + ! + ~;1) = (II + ! + ~;lr (67)

It has been shown by Raczka et af.9 that the solu­tions of (59) for fixed A provide a set of basis functions for the most degenerate representations of Ot3, 1). Thus we see that one can build in the spectrum space of 0(3, I) the groups GL(3, C) by space-time oper­ators. We shall return to this point later.

v. INFINITESIMAL OPERATORS

Let us remark at first that a suitable set of matrices which build the algebra GL(n, C) is given by

Enl = (E;)kl = (j:(jk, k, I = 1 , ... , n, (68)

and iEkl' which we denote by E~. The commutation relation between these operators is then

[Eij , Ek1 ] = - (jkjEi! + (jilEki'

[Eij, E:a = 0, (69)

[E~, E:1] = -( -(jkjEi! + (jilEki)·

It is to be noted that the n2 operators {Eil } generate the algebra U(n).

In order to build the desired operators, let us see the geometrical meaning of the operators Ek1 • When we

8 R. Raczka, N. Limic, and J. Niederle, J. Math. Phys. 7, 1861 (1966).

apply Ekl to a vector x = (Xl' ... , xn), this operator destroys all components of x except the kth one, which is transformed into the Ith component. The operators that we write down in order to build GL(n, C) will have essentially the same meaning. In the following, we write down the infinitesimal operators for the algebras GL(2, C) and GL(3, C); the general case can be inferred easily from these two cases.

1. GL(2, C). Let us define

E12 = dm (A?n - (j~(j;;'), (70) I H(O,O)(m,Ol H(O,m)(O,Ol

()( P

E2l = dl (AJ - (jJ(j;;'), I H(O,O)(O,!l HU,O)(O,Ol

y (j (71)

HI = I dl(A~ - (j~(j;;'), (72)

H2 = I dm(A?n - (jJ(j;;'), (73)

where ()(, p, y, (j are normalization constants. It is easy to verify that these operators satisfy the desired commutation relations.

2. GL(3, C). In the same way as above we are able to define operators that satisfy the commutation relation of GL(3, C). Thus E13 is given by

VI. METHOD OF EXTENSION FOR nth­ORDER EQUATIONS

The factorization method for second-order differ­ential equations looks for first-order differential­operator solutions. However, it is clear that there exist other trivial solutions. These are second- and zero-order differential operators. Thus, we have for second-order differential equations three independent solutions. In the general nth-order eigenvalue problem,

dn

- y + rex, m)y + AY = o. (74) dxn

We expect therefore to find n + 1 independent solu­tions to our problem in the nth order, of which n - 1 will be nontrivial.

The form which we impose on our operators is

Rm = K(x, m + 1) - D,

Lm = K(x,m) + D, (75) where

i = 1··· [~} (76)

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EXTENSION OF THE FACTORIZATION METHOD 1265

so that

RmY(A, m) = [A - L(m + 1)]tY(A, m + 1),

LmY(A, m) = [A - L(m)]tY(A, m - 1). (77)

We say that Eq. (74) is factorized by Rm , Lm , if it can be replaced by each of the following two equations:

Lm+1RmY(A, m) = [A - L(m + 1)]Y(A, m), (78)

Rm_1LmY(A, m) = [A ...:.. L(m)]Y(A, m). (79)

These equations lead to

{ dn 1 d2i

k2(x m + 1) + Dk(x m + 1) - - - --, , , dx" 2 dx2•

1 d2(n-i)} - 2: dx2(,.-i) yeA, m)

= [A - L(m + 1)]Y(A, m), (80)

{ dn 1 d2i

k2(x m) - Dk(x m) - - - --. , 'dxn 2 dx2'

1 d2(n-i)} - 2: dx2(n-;) yeA, m)

= [A - L(m)]Y(A, m). (81)

Using Eq. (74), we are then led to

2 1 d2i 1 d2( n-i)

k (x, m + 1) + Dk(x, m + 1) - --2' - --2(-') 2 dx' 2 dx n-.

+ L(m + 1) = -rex, m), (82)

2 1 d2i 1 d2( n-;)

k (X, m) - Dk(x, m) - --2' - - 2( I) + L(m) 2 dx ' 2 dx 11-

= -rex, m). (83)

Subtracting (83) from (82), we then get

k2(X, m + 1) - k2(X, m) + Dk(x, m - 1)

+ Dk(x, m) = -L(m + 1) + L(m). (84)

Let us now substitute

and denote

We then have

[em + 1)2(k~ + k{) + 2(m + 1)(kok1 + km

- [m2(k~ + kD + 2m(kokl + k~)]

(85)

= L(m) - L(m + 1). (86)

The solution to this equation is

L(m) = -{m2(k~ + kD + 2m(kokl + k~)}, (87)

since L(m) is a function ofm alone. We must then have

1 [ di

dn

-i

] 2 2 2 dxi + dxn-i kl + kl = a , (88)

![.!f.- dn

-i Jk + k k = {-ca i.f a =;t. 0, (89)

2 dXi + dxn- i 0 0 1 b If a = O.

We shall deal with the solution of Eqs. (88) and (89) in the next section.

Until now we found only [n/2] solutions to our factorization method; however, it is easy to infer the other solutions. These have the same form as in Eqs. (76), but in this case

i [d i d1l

-

i J D = .J2 dXi - dx1l- i •

The same procedure as in the preceding case follows. In the above discussion we dealt with integer dis­placements of the eigenfunctions of Eq. (74); how­ever, it has been shown in Sec. I that the factorization method can be generalized to noninteger displacement (for second-order ordinary differential equations). The same methods apply in this more general case. We shall not dwell on the details.

vn. ON THE SOLUTION OF EQ. (88)

Equations (88) and (89) are the basic equations which we must solve in order that the procedure developed above will have a practical use.to However, this is not an easy task. The main difficulty lies in the fact that Eq. (88) is not linear. Thus it is possible to solve this equation in special cases only.

As an illustration, we deal with a fourth-order equation and take i = n, i.e., i = 2, so that

d2

D = - (90) dx2

(dropping unimportant numerical factors). Equation (88) then turns out to be

d2

k k 2 2 -2 1 + 1 = -a, dx

which lead to the solution

x = C2 ± f (2c1 - a2k1 - l-k3r i dk.

(91)

This integral can be solved explicitly only for special values of a. For higher-order differential operators we face, of course, greater difficulties.

ACKNOWLEDGMENT

The author wishes to express his most sincere thanks to Professor B. Kaufman and Professor Y. Ne'eman for their interest, advice, and encouragement.

10 It is to be noticed that Eqs. (82) and (83) are operator equations. Therefore, even if we can solve Eq, (88), we can find the desired raising and lowering operators only if we impose on the solutions of (74) th.e subsidiary conditions

( cf2i cf2(n-i») dX2i + d 2(n-i) y = O.

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