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Spectral singularities of the quartic anharmonic oscillator
Volume 117, number 4
PHYSICS LETTERS A
11 August 1986
SPECTRAL SINGULARITIES OF THE QUARTIC ANHARMONIC OSCILLATOR Paul E. SHANLEY Department of Physics, Unioersity of Notre Dame, Notre Dame, IN 46556, USA Received 10 March 1986; revised manuscript received 2 June 1986; accepted for publication 10 June 1986
The singularities of the eigenvalues of the quartic anharmonic oscillator hamiltonian are studied numerically as a function of the quartic coupling parameter. Their location is found to be consistent with previous WKB calculations but the identification of which levels are singular is found to differ.
The analytic properties of the eigenvalues of quantum hamiltonians as a function of a coupling parameter have been studied in a limited number of cases such as Mathieu's and related equations [1], a linear potential problem [2], a field theory model [3], and band problems involving solids [4]. The most intensively studied problem of this sort is the familiar quartically perturbed simple harmonic oscillator for which many results are available in the papers o f Bender and Wu [5,6] and Simon [7]. Using WKB methods, Bender and Wu (BW) located some of the singularities of the eigenvalues of the anharmonic oscillator and further argued that they are of square-root type and of infinite number. Our purpose in this paper is to re-investigate the location and properties of these singularities using no approximations other than aumerical ones. The starting point is the hamiltonian H =p2 + xx2 + flx 4, p = - i d / d x , with nth eigenvalue E,(a, fl). A scaling of the variables [7] leads to :he relation
L(1, t ) = ~ 1 / 3 E n ( ~ - 2 / 3 , 1).
O)
thus the original spectrum is related to one in,olving the .quadratic perturbation of a quartic ~scillator. This reversal of the roles of the perurbation has been used by previous authors [8] as L basis for variational and perturbative calculaions. Letting "t = fl-2/3, we study H ' =p2 + ./x 2
+ x 4 with eigenvalue E,(y, 1) - E,(y). The eigenvalue problem reads
E., x)= E.(v),.(v, E., x), E., ±oo)=0.
(2)
The scaled problem has the advantages that no rotation of the x-contour [5] is necessary, H ' has only a discrete spectrum [7], and that E.(~,) is locally analytic at , / = 0 [7]. It is the nature of such problems that E. (,/) is singular when two or more energy levels cross or become degenerate at some complex y. The numerical problem is to devise a scheme that locates these points of degeneracy and also identifies which levels are degenerate. A simple procedure involving a finite basis is described in ref. [2]. As applied to our problem, the method amounts to diagonalizing the perturbation ,/x 2 in a finite basis ( - 2 0 ) of eigenstates of p 2 + X 4. The condition that an eigenvalue exist and that it be doubly degenerate is that the usual secular determinant vanish together with its first partial derivative with respect to E. The procedure is initialized by choosing "t and E as random complex numbers and then by searching iteratively for a solution. We do not display the results o f this study here [9] but only mention that the crossing points appear in a regular pattern in the "t-plane and that a more refined method is 161
V o l u m e 117, n u m b e r 4
PHYSICS LETTERS A
I
needed to determine their position with more precision. We have developed such a method [9] based on the numerical solution of the SchrSdinger equation. To extract both 7 and E for a crossing point requires an additional constraint. A necessary and sufficient condition [5,7] for crossing is that
a(.t, e) = f =dx
e, x)]2= 0
•
x
x x x
I
I
= x
15.0 Trn 'r
•
'
•
x" x P '
x
x
x
x
x
x
x• • x•
x , x• x•
o
x
x
10.0
x •
x
•
x
x x•
I
,
-20.0
x
•
•
x
(3)
We use an iterative procedure [9] t o solve the differential equation (2) by a tenth-order A d a m s - M o u l t o n method with the solution subject to (3) as an added condition. Since the energy is a singular function of .t at the crossing point, the situation is rather delicate. We use the results of the finite basis method described above to initialize the procedure. Given a starting estimate .t~ and E a for the coupling and energy at the crossing point, we consider E~ fixed and iteratively improve .tl by numerically solving the SchrSdinger equation out from the origin and in from large x using asymptotic estimates for the wave function and matching at an intermediate point in the usual way. This yields an improved Yz as well as A(y2, E l ) by evaluation of (3). The rate of change of a(.t, E ) with respect to E is next studied by evaluating it at four points in the complex neighborhood of El, with the Sehr&tinger equation solved at each point for the corresponding "t, ~, and A. This allows a move to a new E 2 with a lower I A I and the SchrSdinger equation gives the appropriate .t. This multiple iteration is repeated until I zal is sufficiently small ( = 10-~5). In fig. 1 we show the resulting distribution of crossing points obtained for I-tl < 20 in both even and odd parity. A point on the plot indicates that a pair of levels is degenerate at that coupling. The relation E * ( . t ) = E,(.t*) implies that mirror
11 A u g u s t 1986
-15.0
x x•
Xo
x
x
~ ,
I
/
-I0.0 Re
-5.0
0.0
7"
Fig. 1. Positions in 7 for b r a n c h p o i n t s with [ 7 1 ,< 20. Even p a r i t y are circles. O d d p a r i t y are crosses.
singularities appear for Im y < 0 and in what follows we ignore the odd parity case. Loeffel and Martin [10] have shown that E,(.t) is analytic in l arg 71 < 2,~/3 and the singularities of fig. 1 are consistent with their result. To proceed further we must determine which levels are singular at the points of fig. 1 and adopt some convention for forming a Riemann surface for .t. To address the latter problem we imagine that for Im .t = 0, we solve (2) for the real values of E , ( 7 ) for all Re "t and all n. We then make n copies of the "t-plane and define the Riemann surface so that the real, non-singular E , ( 7 ) lie over the real axis on the n th sheet. This convention rules out branch points on the real axis as well as branch cuts crossing it. To decide which levels are singular we map out E . ( 7 ) by starting on the real axis on each sheet, where n is known by node counting, and then by proceeding to Im .t > 0 by numerical solution of the SchriSdinger
Table 1 Positions a n d energies of the first three b r a n c h points. n, n + 2
Re 7
Im7
Re E ,
ImE,
0, 2 2, 4 2, 4
-4.193684 - 6.843163 - 5.711325
2.169740 1.894506 5.570935
0.404164 0.292425 4.205075
2.935855 3.585266 13.420065
162
5.0
Volume 117, number 4
PHYSICS LETTERS A
11 August 1986
15.0
'"'~
// --x._
//: ~ 2 s "
'
,,
".<
.," ImT
/
I
/
7" ".~
~
X--*° \
o.\"-,
-15.0
I0.0
/~
/
t
i
,/',
~
/
z/o
~
-I0.0
\,
/
;.'o
-5.0
,.~
5.0
0.0
ReY Fig. 2. Contours of constant Re E 0 (solid lines) and constant Im E 0 (broken lines) for n = 0 sheet of 7-
equation to yield E.()'). Based on m a n y calculations of this sort we show results for E0(Y) and E2()' ) in figs. 2 and 3 by plotting contours of constant Re E. and I m E. over two sheets of the underlying Riemann surface of )'. All branch cuts have been chosen to lie on radial lines to infinity. Numerical solution of (2) on small circular loops around each branch point shows that E . ( ) ' ) has a period of 40, indicating square-root singularities. The positions and energy for the n = 0 and n = 2 cases are given in table 1. We note that the singularity closest to )' = 0 is common to E0()' ) and E2()' ). E2()') has an additional pair that is comm o n to E4()' ) (not shown). This pattern persists on the higher sheets. In the even parity case and counting the mirror singularities, we find that sheet n contains n + 2 branch points that are making their first appearance, as well as n branch points that also appeared on sheet n - 2. These sequences of branch points fall approximately on circular arcs in the ),-plane whose radii increase with increasing n. We see in fig. 1 that the singularities become closer as f)'l increases and we thus expect that, as n --, 0o, an infinite number of
branch points will accumulate at infinity on the asymptotic sheet. This distribution of singularities implies that the radius of convergence of the Rayleigh-Schr~dinger series for .the energy about y = 0 for a quadratic perturbation of the quartic oscillator becomes infinite as n --, oo. We wish to compare our results with those of BW both in the position of the branch points in the coupling variable as well as in the identification of which levels are crossing at a given branch point. They have located thirty singularities [5] in what is essentially the fl variable, with arg fl near 3 0 / 2 and I fll small. This transforms to arg )' near o with Re 7 << 0. We take the position of the branch point they label as N = 78 that is nearest to 3 0 / 2 and transform to our variable, obtaining Yaw = -81.371 + i0.730. We see from fig. 1 that our calculation must be considerably extended to the left to reach this singularity. By such an extension we find that the 78th branch point on the curve with smallest I m 7 gives 7 = - 8 1 . 3 6 8 + i0.813, with E , ( ) ' ) = 0.041 + i10.986 at the singularity. We have checked the other BW positions and also find good agreement. Where we diverge 163
Volume 117, number 4
PHYSICS LETTERS A
11 August 1986 15.0
\
\\
~
/
/
~
/
,2
Imp' I0.0
j/
/'~
..-
/ / ~ 0 ,,
/
ii
6
/
//
I
/
5.0
-, o -~ - - -Y~ -
Y
~
~I==-==?-T
-15.0
:,o :
-
:
_
-~o i_ _ .~ _ _ ~_,
7" ~\~
....
-I0.0
~_ - 4 -
o,
---/~
- - I - - - ~- - ,
~
-5.0
I
0.0
ReT'
Fig. 3. Contours of constant Re E 2 (solid lines) and constant lm
f r o m BW is in the identification of which levels are crossing. We find that contiguous levels cross on = circular arcs in 7 whereas BW find the crossing o n -- radial lines in ft. T h e y identify the N = 78 b r a n c h point discussed above as c o m m o n to n = 0 and n = 2 while we find that it involves a highly excited state. F o r Re 7 = - 8 1 . 3 6 8 , the potential is that of a very deep double well with an easily estimated g r o u n d state energy of Re E 0 -- - 1 6 6 0 . O u r result of Re E . = 0.041 cannot be near the g r o u n d state. By singularity counting out from V = 0, the 78th b r a n c h point in our scheme is associated with n = 154 and 156 and is thus highly excited. O u r procedure of choosing a R i e m a n n surface and of assigning b r a n c h points to various levels leads to a significant difference from BW in that we find no explicit accumulation of singularities of E.(1') at 7 = oo on any finitely n u m b e r e d sheet, but only on the asymptotic one. T h e same would be true for E.(1, fl) at fl = 0. T o study the point at infinity of 1' on each sheet, we re-write (1) in terms of y to obtain E,(1', 1 ) = 1'1/2E,(1, 1'--3/2). N o w letting 1' ~ oo, the h a r m o n i c oscillator result 164
E2
(broken lines) for n = 2 sheet of 7.
En(T, 1)---~y1/2(2n
+ 1) is obtained, giving a square-root singularity on each sheet. These singular points cannot be considered ordinary squareroots Since Simon [7] has shown that 7 = oo is not an isolated singularity of E , ( 7 ) . If one starts on a sheet with finite n and attempts to circle the point at infinity, the presence of the b r a n c h cuts takes one to all other sheets. We therefore argue that the accumulation on the asymptotic sheet m a y be considered "close" to the point of infinity on all sheets. Helpful c o m m e n t s have been kindly provided b y C.M. Bender, B. Simon, A.J. Sommese, and C.M. Stanton. The computing center at the University of N o t r e D a m e is also thanked for making their facilities available.
References [1] C. Hunter and B. Guerrieri, Stud. Appl. Math. 64 (1981) 113; 66 (1982) 217; G. Blanch and D.S. Cloture, Math. Comp. 23 (1969) 97.
Volume 117, number 4
PHYSICS LETTERS A
[2] C.M. Bender, J.J. Happ and B. Svetitsky, Phys. Rev. D 9 (1974) 2324. [3] T.W. Ruijgrok, Nucl.Phys. B 39 (1972) 616. [4] J. Avron and B. Simon, Ann. Phys. (NY) 110 (1978) 85; M. Ferrari, V. Grecchi and F. Zirorti, J. Phys. C 18 (1985) 5825. [5] C.M. Bender and T.T. Wu, Phys. Rev. 184 (1969) 1231. [6] C.M. Bender and T.T. Wu. Phys. Rev. D 7 (1973) 1620. [7] B. Simon, Ann. Phys. (NY) 58 (1970) 76.
11 August i986
[8] S.I. Chan, D. Stelman and L.E. Thompson, J. Chem. Phys. 41 (1964) 2828; F.T. Hioe and E.W. MontrolL J. Math. Phys. 16 (1975) 1945; J. Killingbeck, J. Phys. A 13 (1980) 49. [9] P.E. Shanley, to be published. [10] J.J. Loeffel and A. Martin, CERN Report No. Th-l167 (1970), unpublished
165
PHYSICS LETTERS A
11 August 1986
SPECTRAL SINGULARITIES OF THE QUARTIC ANHARMONIC OSCILLATOR Paul E. SHANLEY Department of Physics, Unioersity of Notre Dame, Notre Dame, IN 46556, USA Received 10 March 1986; revised manuscript received 2 June 1986; accepted for publication 10 June 1986
The singularities of the eigenvalues of the quartic anharmonic oscillator hamiltonian are studied numerically as a function of the quartic coupling parameter. Their location is found to be consistent with previous WKB calculations but the identification of which levels are singular is found to differ.
The analytic properties of the eigenvalues of quantum hamiltonians as a function of a coupling parameter have been studied in a limited number of cases such as Mathieu's and related equations [1], a linear potential problem [2], a field theory model [3], and band problems involving solids [4]. The most intensively studied problem of this sort is the familiar quartically perturbed simple harmonic oscillator for which many results are available in the papers o f Bender and Wu [5,6] and Simon [7]. Using WKB methods, Bender and Wu (BW) located some of the singularities of the eigenvalues of the anharmonic oscillator and further argued that they are of square-root type and of infinite number. Our purpose in this paper is to re-investigate the location and properties of these singularities using no approximations other than aumerical ones. The starting point is the hamiltonian H =p2 + xx2 + flx 4, p = - i d / d x , with nth eigenvalue E,(a, fl). A scaling of the variables [7] leads to :he relation
L(1, t ) = ~ 1 / 3 E n ( ~ - 2 / 3 , 1).
O)
thus the original spectrum is related to one in,olving the .quadratic perturbation of a quartic ~scillator. This reversal of the roles of the perurbation has been used by previous authors [8] as L basis for variational and perturbative calculaions. Letting "t = fl-2/3, we study H ' =p2 + ./x 2
+ x 4 with eigenvalue E,(y, 1) - E,(y). The eigenvalue problem reads
E., x)= E.(v),.(v, E., x), E., ±oo)=0.
(2)
The scaled problem has the advantages that no rotation of the x-contour [5] is necessary, H ' has only a discrete spectrum [7], and that E.(~,) is locally analytic at , / = 0 [7]. It is the nature of such problems that E. (,/) is singular when two or more energy levels cross or become degenerate at some complex y. The numerical problem is to devise a scheme that locates these points of degeneracy and also identifies which levels are degenerate. A simple procedure involving a finite basis is described in ref. [2]. As applied to our problem, the method amounts to diagonalizing the perturbation ,/x 2 in a finite basis ( - 2 0 ) of eigenstates of p 2 + X 4. The condition that an eigenvalue exist and that it be doubly degenerate is that the usual secular determinant vanish together with its first partial derivative with respect to E. The procedure is initialized by choosing "t and E as random complex numbers and then by searching iteratively for a solution. We do not display the results o f this study here [9] but only mention that the crossing points appear in a regular pattern in the "t-plane and that a more refined method is 161
V o l u m e 117, n u m b e r 4
PHYSICS LETTERS A
I
needed to determine their position with more precision. We have developed such a method [9] based on the numerical solution of the SchrSdinger equation. To extract both 7 and E for a crossing point requires an additional constraint. A necessary and sufficient condition [5,7] for crossing is that
a(.t, e) = f =dx
e, x)]2= 0
•
x
x x x
I
I
= x
15.0 Trn 'r
•
'
•
x" x P '
x
x
x
x
x
x
x• • x•
x , x• x•
o
x
x
10.0
x •
x
•
x
x x•
I
,
-20.0
x
•
•
x
(3)
We use an iterative procedure [9] t o solve the differential equation (2) by a tenth-order A d a m s - M o u l t o n method with the solution subject to (3) as an added condition. Since the energy is a singular function of .t at the crossing point, the situation is rather delicate. We use the results of the finite basis method described above to initialize the procedure. Given a starting estimate .t~ and E a for the coupling and energy at the crossing point, we consider E~ fixed and iteratively improve .tl by numerically solving the SchrSdinger equation out from the origin and in from large x using asymptotic estimates for the wave function and matching at an intermediate point in the usual way. This yields an improved Yz as well as A(y2, E l ) by evaluation of (3). The rate of change of a(.t, E ) with respect to E is next studied by evaluating it at four points in the complex neighborhood of El, with the Sehr&tinger equation solved at each point for the corresponding "t, ~, and A. This allows a move to a new E 2 with a lower I A I and the SchrSdinger equation gives the appropriate .t. This multiple iteration is repeated until I zal is sufficiently small ( = 10-~5). In fig. 1 we show the resulting distribution of crossing points obtained for I-tl < 20 in both even and odd parity. A point on the plot indicates that a pair of levels is degenerate at that coupling. The relation E * ( . t ) = E,(.t*) implies that mirror
11 A u g u s t 1986
-15.0
x x•
Xo
x
x
~ ,
I
/
-I0.0 Re
-5.0
0.0
7"
Fig. 1. Positions in 7 for b r a n c h p o i n t s with [ 7 1 ,< 20. Even p a r i t y are circles. O d d p a r i t y are crosses.
singularities appear for Im y < 0 and in what follows we ignore the odd parity case. Loeffel and Martin [10] have shown that E,(.t) is analytic in l arg 71 < 2,~/3 and the singularities of fig. 1 are consistent with their result. To proceed further we must determine which levels are singular at the points of fig. 1 and adopt some convention for forming a Riemann surface for .t. To address the latter problem we imagine that for Im .t = 0, we solve (2) for the real values of E , ( 7 ) for all Re "t and all n. We then make n copies of the "t-plane and define the Riemann surface so that the real, non-singular E , ( 7 ) lie over the real axis on the n th sheet. This convention rules out branch points on the real axis as well as branch cuts crossing it. To decide which levels are singular we map out E . ( 7 ) by starting on the real axis on each sheet, where n is known by node counting, and then by proceeding to Im .t > 0 by numerical solution of the SchriSdinger
Table 1 Positions a n d energies of the first three b r a n c h points. n, n + 2
Re 7
Im7
Re E ,
ImE,
0, 2 2, 4 2, 4
-4.193684 - 6.843163 - 5.711325
2.169740 1.894506 5.570935
0.404164 0.292425 4.205075
2.935855 3.585266 13.420065
162
5.0
Volume 117, number 4
PHYSICS LETTERS A
11 August 1986
15.0
'"'~
// --x._
//: ~ 2 s "
'
,,
".<
.," ImT
/
I
/
7" ".~
~
X--*° \
o.\"-,
-15.0
I0.0
/~
/
t
i
,/',
~
/
z/o
~
-I0.0
\,
/
;.'o
-5.0
,.~
5.0
0.0
ReY Fig. 2. Contours of constant Re E 0 (solid lines) and constant Im E 0 (broken lines) for n = 0 sheet of 7-
equation to yield E.()'). Based on m a n y calculations of this sort we show results for E0(Y) and E2()' ) in figs. 2 and 3 by plotting contours of constant Re E. and I m E. over two sheets of the underlying Riemann surface of )'. All branch cuts have been chosen to lie on radial lines to infinity. Numerical solution of (2) on small circular loops around each branch point shows that E . ( ) ' ) has a period of 40, indicating square-root singularities. The positions and energy for the n = 0 and n = 2 cases are given in table 1. We note that the singularity closest to )' = 0 is common to E0()' ) and E2()' ). E2()') has an additional pair that is comm o n to E4()' ) (not shown). This pattern persists on the higher sheets. In the even parity case and counting the mirror singularities, we find that sheet n contains n + 2 branch points that are making their first appearance, as well as n branch points that also appeared on sheet n - 2. These sequences of branch points fall approximately on circular arcs in the ),-plane whose radii increase with increasing n. We see in fig. 1 that the singularities become closer as f)'l increases and we thus expect that, as n --, 0o, an infinite number of
branch points will accumulate at infinity on the asymptotic sheet. This distribution of singularities implies that the radius of convergence of the Rayleigh-Schr~dinger series for .the energy about y = 0 for a quadratic perturbation of the quartic oscillator becomes infinite as n --, oo. We wish to compare our results with those of BW both in the position of the branch points in the coupling variable as well as in the identification of which levels are crossing at a given branch point. They have located thirty singularities [5] in what is essentially the fl variable, with arg fl near 3 0 / 2 and I fll small. This transforms to arg )' near o with Re 7 << 0. We take the position of the branch point they label as N = 78 that is nearest to 3 0 / 2 and transform to our variable, obtaining Yaw = -81.371 + i0.730. We see from fig. 1 that our calculation must be considerably extended to the left to reach this singularity. By such an extension we find that the 78th branch point on the curve with smallest I m 7 gives 7 = - 8 1 . 3 6 8 + i0.813, with E , ( ) ' ) = 0.041 + i10.986 at the singularity. We have checked the other BW positions and also find good agreement. Where we diverge 163
Volume 117, number 4
PHYSICS LETTERS A
11 August 1986 15.0
\
\\
~
/
/
~
/
,2
Imp' I0.0
j/
/'~
..-
/ / ~ 0 ,,
/
ii
6
/
//
I
/
5.0
-, o -~ - - -Y~ -
Y
~
~I==-==?-T
-15.0
:,o :
-
:
_
-~o i_ _ .~ _ _ ~_,
7" ~\~
....
-I0.0
~_ - 4 -
o,
---/~
- - I - - - ~- - ,
~
-5.0
I
0.0
ReT'
Fig. 3. Contours of constant Re E 2 (solid lines) and constant lm
f r o m BW is in the identification of which levels are crossing. We find that contiguous levels cross on = circular arcs in 7 whereas BW find the crossing o n -- radial lines in ft. T h e y identify the N = 78 b r a n c h point discussed above as c o m m o n to n = 0 and n = 2 while we find that it involves a highly excited state. F o r Re 7 = - 8 1 . 3 6 8 , the potential is that of a very deep double well with an easily estimated g r o u n d state energy of Re E 0 -- - 1 6 6 0 . O u r result of Re E . = 0.041 cannot be near the g r o u n d state. By singularity counting out from V = 0, the 78th b r a n c h point in our scheme is associated with n = 154 and 156 and is thus highly excited. O u r procedure of choosing a R i e m a n n surface and of assigning b r a n c h points to various levels leads to a significant difference from BW in that we find no explicit accumulation of singularities of E.(1') at 7 = oo on any finitely n u m b e r e d sheet, but only on the asymptotic one. T h e same would be true for E.(1, fl) at fl = 0. T o study the point at infinity of 1' on each sheet, we re-write (1) in terms of y to obtain E,(1', 1 ) = 1'1/2E,(1, 1'--3/2). N o w letting 1' ~ oo, the h a r m o n i c oscillator result 164
E2
(broken lines) for n = 2 sheet of 7.
En(T, 1)---~y1/2(2n
+ 1) is obtained, giving a square-root singularity on each sheet. These singular points cannot be considered ordinary squareroots Since Simon [7] has shown that 7 = oo is not an isolated singularity of E , ( 7 ) . If one starts on a sheet with finite n and attempts to circle the point at infinity, the presence of the b r a n c h cuts takes one to all other sheets. We therefore argue that the accumulation on the asymptotic sheet m a y be considered "close" to the point of infinity on all sheets. Helpful c o m m e n t s have been kindly provided b y C.M. Bender, B. Simon, A.J. Sommese, and C.M. Stanton. The computing center at the University of N o t r e D a m e is also thanked for making their facilities available.
References [1] C. Hunter and B. Guerrieri, Stud. Appl. Math. 64 (1981) 113; 66 (1982) 217; G. Blanch and D.S. Cloture, Math. Comp. 23 (1969) 97.
Volume 117, number 4
PHYSICS LETTERS A
[2] C.M. Bender, J.J. Happ and B. Svetitsky, Phys. Rev. D 9 (1974) 2324. [3] T.W. Ruijgrok, Nucl.Phys. B 39 (1972) 616. [4] J. Avron and B. Simon, Ann. Phys. (NY) 110 (1978) 85; M. Ferrari, V. Grecchi and F. Zirorti, J. Phys. C 18 (1985) 5825. [5] C.M. Bender and T.T. Wu, Phys. Rev. 184 (1969) 1231. [6] C.M. Bender and T.T. Wu. Phys. Rev. D 7 (1973) 1620. [7] B. Simon, Ann. Phys. (NY) 58 (1970) 76.
11 August i986
[8] S.I. Chan, D. Stelman and L.E. Thompson, J. Chem. Phys. 41 (1964) 2828; F.T. Hioe and E.W. MontrolL J. Math. Phys. 16 (1975) 1945; J. Killingbeck, J. Phys. A 13 (1980) 49. [9] P.E. Shanley, to be published. [10] J.J. Loeffel and A. Martin, CERN Report No. Th-l167 (1970), unpublished
165