- Email: [email protected]
Applications of matrix Padé approximants in potential theory
ANNALS
OF PHYSICS
Applications
114, 290-295 (1978)
of Matrix
Pad4 Approximants
in Potential
Theory
P. R. GRAVES-MORRIS Mathematical
Institute, University of Kent, Canterbury, Kent, England Received July 7, 1977
The method of variational matrix Pade approximants is seen to unify the Green’s function method and the variational principle for calculating scattering amplitudes. The surprising result that exact answers can be obtained for square well potentials is thereby proved, and the remarkable accuracy of the matrix Pad6 method for more general potentials is explained.
1. INTRODUCTJON
Matrix Padt approximants were introduced by Bessis,Gammel, and co-workers [l, 21 in the context of the nucleon-nucleon interaction. The concept of the operator Pad& approximant in the context of Green’s functions has been enthusiastically investigated by Bessisand co-workers [3], and various matrix PadC approximants which represent this elusive concept have been useful. In particular, the method of matrix Padt approximants for the T-matrix (scattering matrix) has been notably successful, especially when combined with a variational principle [2,4]. Recently, it has been empirically discovered that exact results for scattering lengths and other T-matrix elements may be obtained by the method of variational matrix PadC approximants for the square well potential [5]. This result is amazing, because the Born approximation to the scattering length, an inherently inexact approximation, is an apparently essentialpart of the calculation, yet an exact result is finally yielded by the method. In Section 2 we prove the validity of the observation. Moreover, generalizations of such results by using piecewisesquare well potentials as approximations of realistic potentials might lead to convergence proofs of the variational matrix PadC method in potential theory. In turn, this would establish a securer basis for the application of such methods in hypothetical field theories, such as the u-model. The previous derivation [3] of matrix Pad& approximants begins with a variational principle, variation being with respect to matrices. It is unclear whether this is the method implied by Nuttall [6] or that implied by Benofy et al. [2]. In this paper, we follow Turchetti’s derivation [7], based on the original Schwinger principle. This enables us to prove the conjecture of Fratamico et al. [5] for square well potentials and to speculate about efficient methods for general potentials. It is clear that matrix Pade approximants will become an essentialfeature of the construction of the T-matrix from its perturbation series, computational difficulties having been resolved by Starkand’s subprogram [S]. 290 0003-4916/78/l 142-0290$05.00/O Copyright All rights
0 1978 by Academic Press, Inc. of reproduction in any form reserved.
291
PAD6 APPROXIMANTS
2. MATRIX
PADI? APPROXIMANTS
FROM SCHWINGER'S
PRINCIPLE
Using Turchetti’s method [7], we show that the N x N matrix Pad6 approximants of the T-matrix are generated by Schwinger’s variational principle starting with N independent basis functions in the Born approximation matrix. We consider wavefunctions (Hil’bert space elementsideally) g, , g, ,..., g, and conjugate wavefunctions (adjoint spaceelements)h, , h, ,..., h, . From these we construct the Born approximation matrix, with elementsgiven by
The T-matrix elementsare given formally by the expansion Tz T,, = 13 (h, / (VG)’ I’ 1gBB:= i A’“(/?, 1(UG)i CJ1go) = f hi+‘T,(;, i==o i-0 i=O
(2)
where h is the coupling strength, U(v) is a potential of unit strength on a convenient scale, and V(Y) = AU(r). It is usually the case that (2) is convergent for j h 1 < A0 for some positive A, and T,, is meromorphic in h for fixed h, and g, . The ordinary Pad6 method consists of taking the first 2M terms of the seriesfor the T,, matrix element given by (2) and forming the [M - l/M] Pad6 approximant for T&. This has been prowzd [9] to be the equivalent of using the Schwinger variational principle with trial functions which are linear combinations of
and II,*, GVh’h,*,(GV)” h,*,..., (GQ”-l
h:.
The matrix Padt method consistsof forming approximants to the matrix series T/X = f
Xi(h
1 (lJG)i
U 1 g>.
i=O
We will prove that Tasderived by this method is the same as that derived from the Schwinger principle for T,, using trial functions which are linear combinations of g,
, g,
>...,
gN,
GUg, . ..)
,...>
G&N,
(GU)“-l
(GW2
g,
g, ,..., (GU)“-l
>...,
(Gu)2g,,
gN
and
(31 h:, h,*,..., h,$, GUh,* ,..., GUh,$, (GU)’ h: ,..., (G(J)’ h;, ...( (GU)“-lh;,...,
(GU)+l
h;.
292
P.
R.
GRAVES-MORRIS
The method of proof is the same for arbitrary N as for N = 2; for ease of presentation of the matrices we take N = 2. Thus our trial wave function are M-l I #,>
=
M-l
1 i=O
d,!l’(GU)i
I gl>
+
1
d,‘“‘(GU>” I gz>
i=O
and M-l (t,b,'
1 =
M-i
1 i=O
We consider the Schwinger
dp'
j
bivariational
(UC)”
+
1 i=O
dj%~~
I
(UG)t
(4)
functional
(5) where 01,/? = 1, 2. The functional
is stationary
when
for i = 0, l,..., M - 1. These lead to the equations (h, i U(GW
I gl? - (#I I (U - AUGU)(GU)i
j g,) = 0,
(6)
I gz> - <#; I (U - AUGU)(GU)i
I g,) = 0,
(7)
(h, 1(UG)iUlg,) (h, / (UGYU
-
(h, / (UG)i(U
- XUGU)
1 &> = 0,
(8)
/ go) -
(h, I (UG)t(U
- AUGU)
I #,) = 0,
(9)
which hold for i = 0, I,..., A4 T,';'
_
M-l 1
&d{T,t;+i)
_
1. Equations (6) and (7) simplify to j+l+j+'))
j=O
_
M-l 1
#d{+f+j)
_
jjT,(l+j+'))
=
0,
j=O
which are 2M equations for dj’), dj”, j = 0, l,..., M - I. They can be written veniently in block matrix form as T,, - AT,‘:’ T,, - XT,‘;’
T,, - h Tz, - AT.22
where the elements of the M x M block matrices are (T,Jij
= T$+j)
and
(Ta(i))ii = T,(p+‘),
i, j = O,..., M -
1.
con-
PADi:
293
APPROXIMANTS
Hence T,, - AT,‘:’ (d(l) ii’“‘) = (T,, T,,) ( T,, - XT;‘:’ Similarly
T,, - AT,;’ Tf:,, - XT:;’
(10)
we may derive from (8) and (9) that
td(7i
AT,‘;’ T,, - AT,‘:’ T,, -
d(2) =
T,? - AT,‘,+’ -’ T,, T,, - AT,?’ ( )) (TQ,)’ _
(11)
Substituting (lo), (I I) in (4) (5) the stationary value of [T,,] is given by T,, - AT,‘;’ - AT,‘:’
T,, - XT,‘,+’ -’ T,, T,, - AT?, (+)I (T,,)’
[Tmalst= (Tel T,,) ( T,,
(12)
This is the Nuttall compact form of the 2 x 2 matrix Pad6 approximant. The N x N generalized form of this formula is entirely obvious, and the result for genera1N follows by this proof. We are now in a position to understand the significance of the matrix Pad& method and why it is proving so powerful. First, we can prove the empirical result of Fratamico et al. [5]. This is that the variational 2 x 2 matrix Pad6 method is exact for scattering off a square well potential. We use g, = h: a sin kr for the S-wave on-shell scattering wavefunction, and g, = A,* a sin gr for the S-wave off-shell wavefunction, depending on the variational parameter q. Let V(r) = M(b - r) be a square well potential of strength X and range b. With units in which fi = 2m = 1, the momentum q inside the well is given by q2 =
k2 -
A.
(13)
With this value of q, g,(q) is the exact wavefunction within the well. However, (5) is independent of the wavefunction outside the well, and consequently g, is effectively the exact wavefunction if q is given by (13). The Schwinger principle assertsthat {T,,] is the exact scattering amplitude and that 1I,!+> is the exact wavefunction either if the basisis complete or if the exact wavefunction is a linear combination of the basis wavefunctions. Therefore [Tao] has a turning point and takes the exact value of the scattering amplitude when q is given by (13). This proves the conjecture of Fratamico et al. Note that we do not assertthat (13) gives the only turning point of [TJ, and also note that the order of Pad6 approximation is immaterial: the lowest-order [O/l] approximant suffices for an exact result.
294
P. R. GRAVES-MORRIS
3. CONCLUSION In the past, binding energies were determined to good accuracy from T-matrix expansions using off-shell momenta as variational parameters in the Pad6 method [4]. The off-shell values of the momenta at the turning points were significantly different from the on-shell values of the momenta of the scattering process. This led to a difficulty of principle about how the Pad6 method should be used to calculate the on-shell T-matrix elements themselves. The moral of the previous section is that a 2 x 2 matrix Pad6 approximant, using the physical momentum in one channel and an expectation value of the momentum within the potential well in the other channel is probably as efficient as any other simple method for a single Yukawa-like potential [lo]. What is best for the realistic sign-changing nuclear potentials is not so clear, but we hope to have shed somelight on this problem. Finally, let us reconsider the conjecture of Bessiset al. [l 11,that a piecewisesquare well potential with p-wells is solved exactly by a variational [p - l/p] matrix Pad.6 approximant in a (p + I)-dimensional space, the p off-shell momenta being given by the classicalmomenta of the particle in each of the wells. The theory of Section 2 indicates that a complete basisrequires 2,~basis states (sin qlr; sin q2r, cos q2r; sin q3r, cos qg;... sin qpr, cos qpr; sin kr), where q1 is the momentum in the innermost well and k is the on-shell momentum. Further, we speculate that the [O/l] matrix Pad6 is sufficient for an exact result using variational (2~ x 2~) matrix Padt approximants for this problem, if an exact result is attainable using methods of this type.
ACKNOWLEDGMENT
1 am grateful to Dr. D. Bessis and Dr. P. Moussafor discussions and hospitalityat Saclay: I apologizefor my previousskepticismof some speculations in Ref. [12]. Note added in proof. Sincesubmission of this paper, the following relevant paper has appeared: L. P. BENOFY AND J. L. GAMMEL, “Proceedings of the Florida Conference on Rational Approximation” (E. B. Saff and R. S. Varga, Eds.), p. 339, Academic Press, New York, 1977.
REFERENCES I. D. BESSIS,G. TURCHETTI, AND W. R. WORTMAN, Phys. Lett. B 39 (1972), 601; R. H. BARLOW AND M. C. BERG~RE, Nuouo Cimento A 11 (1972), 557; J. FLEISCHER,J. L. GAMMEL, AND M. T. MENZEL, Phys. Rev. D 8 (1973), 1545; J. L. GAMMEL AND M. T. MENZEL, Phys. Rev. D 11 (1975), 963.
L. P. BENOFY, J. GAMMEL, AND P. MERY, Phys. Rev. D 13 (1976), 3111. 3. D. BESSIS,“Padt Approximants” (P. R. Graves-Morris, Ed.), p. 19, Institute of Physics, London, 1973; D. BESSIS,“Pad6 Approximants and their Applications” (P. R. Graves-Morris, Ed.), p. 275, Academic Press, London, 1973. 2.
PAD6
7. 8. 9. 10. 11. 12.
295
P. BUTERA, AND G. M. PROSPERI, Nuclear Phys. B 31 (1971), 141; 42 (1972), 493; 46 (1972), 593. G. FRATAMIC’O, F. ORTOLANI, AND G. TRUCHETTI, Lett. Nuovo Cimento 17 (1976), 582. J. NUTTALL, “The Pad& Approximant in Theoretical Physics” (G. A. Baker, Jr., and J. L. Gammel, Eds.), p. 224, Academic Press, New York, 1970. G. TURCHETTI, Lett. Nuovo Cimento 15 (1976), 129. Y. STARKANI), Comput. Phys. Comm. 11 (1976), 325. J. NUTTALL, Phys. Reu. 157 (1967), 1312. D. BESSISAND G. TURCHETTI, Variational matrix PadC approximations in potential scattering and low energy Lagrangian field theory, Saclay, preprint, Dec. 1976. D. BESSIS,P. MERY, AND G. TURCHETTI, Variational bounds from matrix Padt approximants in potential scattering, Saclay preprint, Nov. 1976. P. MERY, “Proceedings of the Florida Conference on Rational Approximation” (E. B. Saff and R. S. Varga, Eds.), Academic Press, New York, 1977.
4. C. ALABISO, 5. 6.
APPROXIMANTS
OF PHYSICS
Applications
114, 290-295 (1978)
of Matrix
Pad4 Approximants
in Potential
Theory
P. R. GRAVES-MORRIS Mathematical
Institute, University of Kent, Canterbury, Kent, England Received July 7, 1977
The method of variational matrix Pade approximants is seen to unify the Green’s function method and the variational principle for calculating scattering amplitudes. The surprising result that exact answers can be obtained for square well potentials is thereby proved, and the remarkable accuracy of the matrix Pad6 method for more general potentials is explained.
1. INTRODUCTJON
Matrix Padt approximants were introduced by Bessis,Gammel, and co-workers [l, 21 in the context of the nucleon-nucleon interaction. The concept of the operator Pad& approximant in the context of Green’s functions has been enthusiastically investigated by Bessisand co-workers [3], and various matrix PadC approximants which represent this elusive concept have been useful. In particular, the method of matrix Padt approximants for the T-matrix (scattering matrix) has been notably successful, especially when combined with a variational principle [2,4]. Recently, it has been empirically discovered that exact results for scattering lengths and other T-matrix elements may be obtained by the method of variational matrix PadC approximants for the square well potential [5]. This result is amazing, because the Born approximation to the scattering length, an inherently inexact approximation, is an apparently essentialpart of the calculation, yet an exact result is finally yielded by the method. In Section 2 we prove the validity of the observation. Moreover, generalizations of such results by using piecewisesquare well potentials as approximations of realistic potentials might lead to convergence proofs of the variational matrix PadC method in potential theory. In turn, this would establish a securer basis for the application of such methods in hypothetical field theories, such as the u-model. The previous derivation [3] of matrix Pad& approximants begins with a variational principle, variation being with respect to matrices. It is unclear whether this is the method implied by Nuttall [6] or that implied by Benofy et al. [2]. In this paper, we follow Turchetti’s derivation [7], based on the original Schwinger principle. This enables us to prove the conjecture of Fratamico et al. [5] for square well potentials and to speculate about efficient methods for general potentials. It is clear that matrix Pade approximants will become an essentialfeature of the construction of the T-matrix from its perturbation series, computational difficulties having been resolved by Starkand’s subprogram [S]. 290 0003-4916/78/l 142-0290$05.00/O Copyright All rights
0 1978 by Academic Press, Inc. of reproduction in any form reserved.
291
PAD6 APPROXIMANTS
2. MATRIX
PADI? APPROXIMANTS
FROM SCHWINGER'S
PRINCIPLE
Using Turchetti’s method [7], we show that the N x N matrix Pad6 approximants of the T-matrix are generated by Schwinger’s variational principle starting with N independent basis functions in the Born approximation matrix. We consider wavefunctions (Hil’bert space elementsideally) g, , g, ,..., g, and conjugate wavefunctions (adjoint spaceelements)h, , h, ,..., h, . From these we construct the Born approximation matrix, with elementsgiven by
The T-matrix elementsare given formally by the expansion Tz T,, = 13 (h, / (VG)’ I’ 1gBB:= i A’“(/?, 1(UG)i CJ1go) = f hi+‘T,(;, i==o i-0 i=O
(2)
where h is the coupling strength, U(v) is a potential of unit strength on a convenient scale, and V(Y) = AU(r). It is usually the case that (2) is convergent for j h 1 < A0 for some positive A, and T,, is meromorphic in h for fixed h, and g, . The ordinary Pad6 method consists of taking the first 2M terms of the seriesfor the T,, matrix element given by (2) and forming the [M - l/M] Pad6 approximant for T&. This has been prowzd [9] to be the equivalent of using the Schwinger variational principle with trial functions which are linear combinations of
and II,*, GVh’h,*,(GV)” h,*,..., (GQ”-l
h:.
The matrix Padt method consistsof forming approximants to the matrix series T/X = f
Xi(h
1 (lJG)i
U 1 g>.
i=O
We will prove that Tasderived by this method is the same as that derived from the Schwinger principle for T,, using trial functions which are linear combinations of g,
, g,
>...,
gN,
GUg, . ..)
,...>
G&N,
(GU)“-l
(GW2
g,
g, ,..., (GU)“-l
>...,
(Gu)2g,,
gN
and
(31 h:, h,*,..., h,$, GUh,* ,..., GUh,$, (GU)’ h: ,..., (G(J)’ h;, ...( (GU)“-lh;,...,
(GU)+l
h;.
292
P.
R.
GRAVES-MORRIS
The method of proof is the same for arbitrary N as for N = 2; for ease of presentation of the matrices we take N = 2. Thus our trial wave function are M-l I #,>
=
M-l
1 i=O
d,!l’(GU)i
I gl>
+
1
d,‘“‘(GU>” I gz>
i=O
and M-l (t,b,'
1 =
M-i
1 i=O
We consider the Schwinger
dp'
j
bivariational
(UC)”
+
1 i=O
dj%~~
I
(UG)t
(4)
functional
(5) where 01,/? = 1, 2. The functional
is stationary
when
for i = 0, l,..., M - 1. These lead to the equations (h, i U(GW
I gl? - (#I I (U - AUGU)(GU)i
j g,) = 0,
(6)
I gz> - <#; I (U - AUGU)(GU)i
I g,) = 0,
(7)
(h, 1(UG)iUlg,) (h, / (UGYU
-
(h, / (UG)i(U
- XUGU)
1 &> = 0,
(8)
/ go) -
(h, I (UG)t(U
- AUGU)
I #,) = 0,
(9)
which hold for i = 0, I,..., A4 T,';'
_
M-l 1
&d{T,t;+i)
_
1. Equations (6) and (7) simplify to j+l+j+'))
j=O
_
M-l 1
#d{+f+j)
_
jjT,(l+j+'))
=
0,
j=O
which are 2M equations for dj’), dj”, j = 0, l,..., M - I. They can be written veniently in block matrix form as T,, - AT,‘:’ T,, - XT,‘;’
T,, - h Tz, - AT.22
where the elements of the M x M block matrices are (T,Jij
= T$+j)
and
(Ta(i))ii = T,(p+‘),
i, j = O,..., M -
1.
con-
PADi:
293
APPROXIMANTS
Hence T,, - AT,‘:’ (d(l) ii’“‘) = (T,, T,,) ( T,, - XT;‘:’ Similarly
T,, - AT,;’ Tf:,, - XT:;’
(10)
we may derive from (8) and (9) that
td(7i
AT,‘;’ T,, - AT,‘:’ T,, -
d(2) =
T,? - AT,‘,+’ -’ T,, T,, - AT,?’ ( )) (TQ,)’ _
(11)
Substituting (lo), (I I) in (4) (5) the stationary value of [T,,] is given by T,, - AT,‘;’ - AT,‘:’
T,, - XT,‘,+’ -’ T,, T,, - AT?, (+)I (T,,)’
[Tmalst= (Tel T,,) ( T,,
(12)
This is the Nuttall compact form of the 2 x 2 matrix Pad6 approximant. The N x N generalized form of this formula is entirely obvious, and the result for genera1N follows by this proof. We are now in a position to understand the significance of the matrix Pad& method and why it is proving so powerful. First, we can prove the empirical result of Fratamico et al. [5]. This is that the variational 2 x 2 matrix Pad6 method is exact for scattering off a square well potential. We use g, = h: a sin kr for the S-wave on-shell scattering wavefunction, and g, = A,* a sin gr for the S-wave off-shell wavefunction, depending on the variational parameter q. Let V(r) = M(b - r) be a square well potential of strength X and range b. With units in which fi = 2m = 1, the momentum q inside the well is given by q2 =
k2 -
A.
(13)
With this value of q, g,(q) is the exact wavefunction within the well. However, (5) is independent of the wavefunction outside the well, and consequently g, is effectively the exact wavefunction if q is given by (13). The Schwinger principle assertsthat {T,,] is the exact scattering amplitude and that 1I,!+> is the exact wavefunction either if the basisis complete or if the exact wavefunction is a linear combination of the basis wavefunctions. Therefore [Tao] has a turning point and takes the exact value of the scattering amplitude when q is given by (13). This proves the conjecture of Fratamico et al. Note that we do not assertthat (13) gives the only turning point of [TJ, and also note that the order of Pad6 approximation is immaterial: the lowest-order [O/l] approximant suffices for an exact result.
294
P. R. GRAVES-MORRIS
3. CONCLUSION In the past, binding energies were determined to good accuracy from T-matrix expansions using off-shell momenta as variational parameters in the Pad6 method [4]. The off-shell values of the momenta at the turning points were significantly different from the on-shell values of the momenta of the scattering process. This led to a difficulty of principle about how the Pad6 method should be used to calculate the on-shell T-matrix elements themselves. The moral of the previous section is that a 2 x 2 matrix Pad6 approximant, using the physical momentum in one channel and an expectation value of the momentum within the potential well in the other channel is probably as efficient as any other simple method for a single Yukawa-like potential [lo]. What is best for the realistic sign-changing nuclear potentials is not so clear, but we hope to have shed somelight on this problem. Finally, let us reconsider the conjecture of Bessiset al. [l 11,that a piecewisesquare well potential with p-wells is solved exactly by a variational [p - l/p] matrix Pad.6 approximant in a (p + I)-dimensional space, the p off-shell momenta being given by the classicalmomenta of the particle in each of the wells. The theory of Section 2 indicates that a complete basisrequires 2,~basis states (sin qlr; sin q2r, cos q2r; sin q3r, cos qg;... sin qpr, cos qpr; sin kr), where q1 is the momentum in the innermost well and k is the on-shell momentum. Further, we speculate that the [O/l] matrix Pad6 is sufficient for an exact result using variational (2~ x 2~) matrix Padt approximants for this problem, if an exact result is attainable using methods of this type.
ACKNOWLEDGMENT
1 am grateful to Dr. D. Bessis and Dr. P. Moussafor discussions and hospitalityat Saclay: I apologizefor my previousskepticismof some speculations in Ref. [12]. Note added in proof. Sincesubmission of this paper, the following relevant paper has appeared: L. P. BENOFY AND J. L. GAMMEL, “Proceedings of the Florida Conference on Rational Approximation” (E. B. Saff and R. S. Varga, Eds.), p. 339, Academic Press, New York, 1977.
REFERENCES I. D. BESSIS,G. TURCHETTI, AND W. R. WORTMAN, Phys. Lett. B 39 (1972), 601; R. H. BARLOW AND M. C. BERG~RE, Nuouo Cimento A 11 (1972), 557; J. FLEISCHER,J. L. GAMMEL, AND M. T. MENZEL, Phys. Rev. D 8 (1973), 1545; J. L. GAMMEL AND M. T. MENZEL, Phys. Rev. D 11 (1975), 963.
L. P. BENOFY, J. GAMMEL, AND P. MERY, Phys. Rev. D 13 (1976), 3111. 3. D. BESSIS,“Padt Approximants” (P. R. Graves-Morris, Ed.), p. 19, Institute of Physics, London, 1973; D. BESSIS,“Pad6 Approximants and their Applications” (P. R. Graves-Morris, Ed.), p. 275, Academic Press, London, 1973. 2.
PAD6
7. 8. 9. 10. 11. 12.
295
P. BUTERA, AND G. M. PROSPERI, Nuclear Phys. B 31 (1971), 141; 42 (1972), 493; 46 (1972), 593. G. FRATAMIC’O, F. ORTOLANI, AND G. TRUCHETTI, Lett. Nuovo Cimento 17 (1976), 582. J. NUTTALL, “The Pad& Approximant in Theoretical Physics” (G. A. Baker, Jr., and J. L. Gammel, Eds.), p. 224, Academic Press, New York, 1970. G. TURCHETTI, Lett. Nuovo Cimento 15 (1976), 129. Y. STARKANI), Comput. Phys. Comm. 11 (1976), 325. J. NUTTALL, Phys. Reu. 157 (1967), 1312. D. BESSISAND G. TURCHETTI, Variational matrix PadC approximations in potential scattering and low energy Lagrangian field theory, Saclay, preprint, Dec. 1976. D. BESSIS,P. MERY, AND G. TURCHETTI, Variational bounds from matrix Padt approximants in potential scattering, Saclay preprint, Nov. 1976. P. MERY, “Proceedings of the Florida Conference on Rational Approximation” (E. B. Saff and R. S. Varga, Eds.), Academic Press, New York, 1977.
4. C. ALABISO, 5. 6.
APPROXIMANTS