The Hall--Post inequalities provide lower bounds on $N$-body energies in
terms of $N'$-body energies with $N'<N$. They are rewritten and generalized to
be tested with exactly-solvable models of Calogero-Sutherland type in one and
higher dimensions. The bound for $N$ spinless fermions in one dimension is
better saturated at large coupling than for noninteracting fermions in an
oscillatorComment: 7 pages, Latex2e, 2 .eps figure

Auberson G

Avinash Khare

Calogero F

Dyson F J

Fisher M E

Forrester P J

Gonzalez-Garcia A

Hall R L

Jean-Marc Richard

Khare A

Krivnov V Ya

Lévy-Leblond J-M

Olshanetsky M

English

arXiv

The Hall--Post inequalities provide lower bounds on $N$-body energies in terms of $N'$-body energies with $N' N$. They are rewritten and generalized to be tested with exactly-solvable models of Calogero-Sutherland type in one and higher dimensions. The bound for $N$ spinless fermions in one dimension is better saturated at large coupling than for noninteracting fermions in an oscillato

Khare, A.

Richard, J.-M.

Hal - Université Grenoble Alpes

Testing hall-post inequalities with exactly solvable N-body problems

The Hall-Post inequalities provide lower bounds on N-body energies in terms of N '- body energies with N ' &lt; N. They are rewritten and generalized to be tested with exactly solvable models of Calogero-Sutherland type in one and higher dimensions. The bound for N spinless fermions in one dimension is better saturated at large coupling than for noninteracting fermions in an oscillator potential

Khare, Avinash

Richard, Jean-Marc

Publications of the IAS Fellows

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quant-ph/0105130
February 1, 2008
Testing Hall-Post Inequalities
With Exactly Solvable N-Body Problems
Avinash KHARE(1) and Jean-Marc RICHARD(2)
(1) Institute of Physics,
Sachivalaya Marg, Bhubaneswar 751005,India
(2)Institut des Sciences Nucle´aires, Universite´ Joseph Fourier–CNRS-IN2P3
53, avenue des Martyrs,F 38026 Grenoble, France.
Abstract
The Hall–Post inequalities provide lower bounds on N -body energies in terms of N ′-body
energies with N ′ < N . They are rewritten and generalized to be tested with exactly-solvable
models of Calogero-Sutherland type in one and higher dimensions. The bound for N spinless
fermions in one dimension is better saturated at large coupling than for noninteracting fermions
in an oscillator potential.
1
It is important to obtain good upper and lower bounds on the binding energy ofN -particle systems.
Since upper bounds are provided by variational estimates, it is natural that a major emphasis in recent
years has been on obtaining good lower bounds.
The Hall–Post inequalities consist of lower bounds to N -body energies in terms of N ′-body ener-
gies with N ′ < N and modified constituent masses or coupling constants. [1, 2, 3]. Applications of
these inequalities have been proposed for studying the thermodynamic limit of large systems and the
stability of matter [3], or the relation of baryons to mesons in hadron spectroscopy [2, 4].
So far, the Hall-Post inequalities have been tested in great detail for few-body systems, for which
accurate numerical calculations can be performed, or in the large N limit, within some approximation.
It seems appealing to test the inequalities for arbitrary N with energies which are calculated exactly.
The purpose of this note is to adapt and apply the Hall–Post inequalities to several exactly-solvable
N -body models.
Inspired by the seminal work of Calogero and Sutherland, in recent years, ground state energy and
at least a part of the excitation spectrum (if not the full spectrum) has been obtained for several new
N -body problems. Some of these Hamiltonians have already found applications in a variety of areas.
We broadly consider four types of N -body problems.
1. N -body problems in one dimension with two-body (and also may be one-body) interaction in
the presence of either two-body or one-body oscillator potential. Some examples of this class
are AN−1 [5], BCN , DN [6] models. The case of D dimensions is considered in [7].
2. N -body problems in one dimension but with periodic boundary conditions and with two-body
(and also possible one-body interaction). Again, some examples of this class are the AN−1 [8]
and BCN , DN [6] models.
3. N -body problems in one dimension with two-body inverse square interaction in the presence
of a hyper-Coulomb potential [9].
4. N -body problems of Calogero-type in two and higher dimensions with two and three-body
interactions in the presence of one or two-body oscillator potential [10, 11].
Our conclusions can be summarized as follows. While comparing the first and second type of
models with the Hall–Post inequalities, we find that contrary to our initial belief, the bound for
spinless fermions in one dimension is better saturated in the large coupling limit rather than for
N -noninteracting fermions in an oscillator potential. For the hyper-Coulomb case, using convex-
ity argument, we derive a slightly better lower bound than was known before. Finally, we show that
Hall–Post bounds also work reasonably well in models with both two and three-body interactions.
In this paper we shall consider models with N identical particles of mass m (which we shall put
equal to 1 without any loss of generality) and whose interaction does not depend on their spins. This
corresponds to the Hamiltonian
HN(m, g1, g2, g3) =
N∑
i=1
p
2
i
2m
+ g1V (ri) + g2
∑
i<j
V (rij) + g3
∑
i<j<k
V (rijk) . (1)
For this case, it has been shown that the corresponding N -particle bound-state energy EN satisfies the
bound [3, 4]
EN (m, g1, g2, g3) ≥
N
N − 1EN−1
(
m, g1,
N − 1
N − 2g2,
N − 2
N − 3g3
)
. (2)
2
In case the Hamiltonian (1) is translationally invariant, then this bound can be improved yielding the
new inequality [1, 2]
EN (m, g2, g3) ≥
N − 1
N − 2EN−1
(
m,
N
N − 1g2,
N(N − 2)
(N − 1)(N − 3)g3
)
. (3)
As a representative of the first type of models, we consider the original Calogero problem [5] for
which the N -body Hamiltonian is given by (h¯ = m = 1)
H = −1
2
N∑
i=1
d2
dx2i
+
N∑
i<j=1
[
ω2
4
(xi − xj)2 +
g
(xi − xj)2
]
. (4)
As shown by Calogero, the ground-state energy for this N -particle system is given by
EN =
√
N
8
[
N2 − 1 + (β − 1)N(N − 1)
]
ω , (5)
where
g = β(β − 1), i.e., β = 1
2
±
√
1 + 4g
2
, (6)
while the corresponding ground-state eigenfunction is given by
ψ =
N∏
i=1
(xi − xj)β × exp
[
− ω√
2N
N∑
i<j=1
(xi − xj)2
]
. (7)
Note that we have to choose the positive square root in eq. (6) so that g = 0 corresponds to β = 1 and
hence to fermions.
For translationally invariant Hamiltonians with two-body interaction, as the one given in eq. (4),
the bound (3) takes the form
EN (m,ω, g) ≥
N − 1
N − 2EN−1

m,ω
√
N
N − 1 ,
N
N − 1g

 . (8)
Using the bound-state energy expression (5) as derived for the Hamiltonian (4), it is easily seen
that
EN−1

m,ω
√
N
N − 1 ,
N
N − 1g

 =
√
N
8
[
N(N − 2) + (β′ − 1)(N − 1)(N − 2)
]
ω , (9)
where
β′ =
1
2
+
√
N(2β − 1)2 − 1
2
√
N − 1 . (10)
Using eqs. (5), (8) and (9) we then get
RN ≡
EN (m,ω, g)
N−1
N−2EN−1(m,ω
√
N
N−1 ,
N
N−1g)
≥ N + 1 + (β − 1)N
N + (β′ − 1)(N − 1) . (11)
In the non-interacting fermion limit (i.e., g = 0 or β = β′ = 1), the right-hand side is N+1N while
in the strong-coupling limit (i.e., g → ∞) the r.h.s. is
√
N√
N−1 . On the other hand, as N → ∞ in
3
2 4 6 8 g
1.12
1.16
1.20
R
Figure 1: Hall–Post ratio RN , as defined in eq. (11) for the Calogero model (4), in the case of N = 5
particles, as a function of the coupling constant g. The dotted line is the g →∞ limit.
one dimension, then the ratio goes to 1, for all values of g. Thus we find that contrary to our naive
expectation, the bound is better satisfied by strongly interacting rather than non-interacting fermions
in an oscillator potential.
Fig. 1 displays the ratio RN as a function of the coupling constant g, in case N = 5. One starts at
g = 0 from the value R5 = 6/5 corresponding to a pure oscillator. Then the ratio evolves regularly
towards its large g limit
√
5/4.
We have also examined how Hall-Post inequalities behave in similar models like BCN , DN , and
even periodic models like those of Sutherland. In all these cases we found a behaviour similar to that
given in Fig. 1. We might add here that many of these models are not translationally invariant and
instead of the lower bound (8), we have to use the simpler bound (2).
As a second example, we now consider a model of the third type [9], i.e.,
H = −1
2
N∑
i=1
d2
dx2i
+
N∑
i<j=1
g
(xi − xj)2
− α
2√∑N
i<j=1(xi − xj)2
, (12)
which contains a hypercentral Coulomb interaction. The N -particle ground-state binding energy has
been shown to be
EN = −
α2
N [N − 2 +N(N − 1)β]2 , (13)
with β and g being again related by eq. (6).
We now show that using a convexity argument, one can adapt the bound (3) to this case. In
particular, we use the fact that if a function f is a convex function, then
f
(
αa+ βb+ · · ·
α+ β + · · ·
)
≥ α
α+ β + · · ·f(a) +
β
α+ β + · · ·f(b) + · · · , (14)
for positive weight factors α, β, . . . . On using f(x) = −1/√x, it is easily shown that in this case the
Hall-Post inequality takes the form
EN (m, g, α) ≥
N − 1
N − 2EN−1
(
m,
N
N − 1g,
(N − 1)√(N − 2)
N3/2
α
)
. (15)
4
5 10 15 20 g
0.6
0.7
0.8
R
Figure 2: Ratio RN , as given in eq. (13), for the model of eq. (12), in the case of N = 5 particles, as
a function of the coupling constant g. The dotted line is the g →∞ limit.
How good is this inequality? We can test that by using the energy eigenvalue (13) and computing
EN−1 for the appropriate couplings. On noting the fact that the binding energy is negative, the Hall-
Post inequality (15) in our case takes the form
RN ≡
EN (m, g, α)
N−1
N−2EN−1(m,
N
N−1g,
(N−1)
√
(N−2)
N3/2
α)
≤ (N)
2[N − 3 + (N − 1)(N − 2)β′]2
(N − 1)2[N − 2 +N(N − 1)β]2 , (16)
where
β′ =
1
2
+
√
(N − 1)(2β − 1)2 − 1
2
√
N − 2 . (17)
We find that but for N → ∞, g arbitrary, this bound is not as good as (11) in the Calogero case. In
Fig. 2, we have plotted the ratio RN as a function of g in case N = 5 which gives an indication of
how good this inequality is for finite N .
As the third and the last example, we consider a model of fourth type [10, 11] which is a Calogero-
type model but in D-dimensions given by
H =
1
2
N∑
i=1
p
2
i +
ω2
4
N∑
i<j=1
r
2
ij + g
N∑
i<j=1
1
r
2
ij
+G
∑
i<j,k
k 6=i,j
rki.rkj
r
2
kir
2
kj
, (18)
where one also requires a three-body potential in addition to the two-body potential. The N -boson
ground-state energy in
EN (m, g,G(g)) =
√
N
8
[D(N − 1) +N(N − 1)β]ω , (19)
provided the two-body coupling g and the three-body coupling G are related by
g = G+ (D − 2)
√
G , G = β2 , (20)
this defining G(g). Note that unlike in one-dimension case, here β = 0 corresponds to g = G = 0
and hence to bosons in a pure oscillator potential.
5
Let us now examine how the Hall-Post inequality (3) fares in this case. It is not possible to test
this inequality directly since g and G have a specific relation G(g) between them which in general
will not be satisfied if g and G are changed to NN−1g and
N(N−2)
(N−1)(N−3)G, respectively. However, we
make use of the fact that both the two and three body terms in eq. (18) are positive in any dimension
D(≥ 2) and hence we can write an appropriate inequality. For example we obviously have
EN (m,ω, g,G(g)) ≥
N − 1
N − 2EN−1

m,ω
√
N
N − 1 ,
N
N − 1g,
N(N − 2)
(N − 1)(N − 3)G(g)


≥ N − 1
N − 2EN−1

m,ω
√
N
N − 1 ,
N
N − 1g,G(
N
N − 1g)

 , (21)
since
N(N − 2)
(N − 1)(N − 3)G(g) ≥ G(
N
N − 1g). (22)
How good is this inequality? Using the exact N -particle binding energy (19) it is easily seen that
EN (m,ω, g,G(g))
N−1
N−2EN−1(m,ω
√
N
N−1 ,
N
N−1g,G(
N
N−1g))
≥ D +Nβ
D + (N − 1)β′ , (23)
where
β′ = −D − 2
2
+
√
N(2β +D − 2)2 − (D − 2)2
2
√
N − 1 . (24)
It is interesting that this bound is saturated for noninteracting bosons (β = 0 i.e., g = G = 0) in
an oscillator potential as well as for large number of particles, no matter what the coupling is. For
large coupling but finite N , the ratio is
√
N/(N − 1). The bound is also saturated at large D, as seen
explicitly in eq. (23). This property of the large D limit was noted by Gonzalez-Garcia [12].
Final remarks are in order:
1. It would be interesting to extend the investigation to the case of unequal masses. Some of
the exactly-solvable models considered here can be generalized as to accommodate different
constituent masses [13]. The bound for unequal masses was discussed in the case of N = 3 or
N = 4 particles interacting through simple pairwise potentials [14].
2. Exactly solvable models are also available for particles with residual interaction with nearest
and next-to-nearest neighbour only [15]. It would be interesting to extend the Hall-Post inequal-
ities to this situation.
3. One of the striking feature of some of the models considered here in one or two dimensions
is that the statistics evolves continuously as one changes the coupling constant. Still, if one
looks at the EN to EN−1 ratio measuring how far the inequalities are from saturation, it evolves
with a rather smooth and monotonic behaviour. Fermions can approach saturation provided the
coupling and the number of particles are large enough.
We hope to address some of the open issues in the near future.
Acknowledgements
AK would like to thank the members of the Institut des Sciences Nucle´aires, Grenoble for warm
hospitality during his trip there as a part of the Indo-French Collaboration Project CEFIPRA 1501-
1502, supported by the Indo-French Centre for Promotion of Advanced Research.
6
References
[1] R. L. Hall and H. R. Post, Proc. Phys. Soc. 90 (1967) 381.
[2] J.-L. Basdevant, A. Martin and J.-M. Richard, Nucl. Phys. B 343 (1990) 60, 69.
[3] M.E. Fisher and D. Ruelle, J. Math. Phys. 7 (1966) 260; F.J. Dyson and A. Lenard, J. Math.
Phys. 8 (1967) 423; J.-M. Le´vy-Leblond, J. Math. Phys. 10 (1969) 806.
[4] J.-P. Ader, J.-M. Richard and P. Taxil, Phys. Rev. D 25 (1982) 2370; S. Nussinov, Phys. Rev.
Lett. 51 (1983) 2081; J.-M. Richard, Phys. Lett. B 139 (1984) 408.
[5] F. Calogero, J. Math. Phys. 10 (1969) 2191, 2197; ibid. 12 (1971) 419.
[6] M. Olshanetsky and M. Perelomov, Phys. Rep. 71 (1981) 313, ibid. 94 (1983) 6.
[7] P.K. Ghosh, Phys. Lett. A 229 (1997) 203.
[8] B. Sutherland, Phys. Rev. A 4 (1971) 2019.
[9] A. Khare, J. of Phys. A 29 (1996) L45.
[10] F. Calogero and C. Marchioro, J. Math. Phys. 14 (1973) 182.
[11] A. Khare and K. Ray, Phys. Lett. A 230 (1997) 139; A. Khare, Phys. Lett. A 245 (1998) 14.
[12] A. Gonzalez-Garcia, Few-Body Systems 8 (1990) 73; 10 (1991) 73.
[13] V.Ya. Krivnov and A.A. Ovchinnikov, Theor. Math. Phys. 50 (1982) 100; P.J. Forrester, J. Phys.
A 25 (1992) L607, 5547; D. Sen, Nucl. Phys. B 479 (1997) 554.
[14] J. Basdevant, A. Martin, J.-M. Richard and T. T. Wu, Nucl. Phys. B 393 (1993) 111;
A. Benslama, A. Metatla, A. Bachkhaznadji, S.R. Zouzou, A. Krikeb, J.-L. Basdevant and
T.T. Wu, Few-Body Systems 24 (1998) 39.
[15] S.R. Jain, and A. Khare, Phys. Lett. A 262 (1999) 35; G. Auberson, S.R. Jain and A. Khare,
J. Phys. A 34 (2001) 695.
7


Testing Hall-Post inequalities with exactly solvable N-body problems

HAL: Hyper Article en Ligne

HAL-IN2P3

Crossref

Testing Hall-Post inequalities with exactly solvable
                    N

Indian Academy of Sciences

ar
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:q
ua
nt
-p
h/
01
05
13
0v
1 
 2
5 
M
ay
 2
00
1
IP/BBSR/01-12
ISN-01-48
quant-ph/0105130
February 1, 2008
Testing Hall-Post Inequalities
With Exactly Solvable N-Body Problems
Avinash KHARE(1) and Jean-Marc RICHARD(2)
(1) Institute of Physics,
Sachivalaya Marg, Bhubaneswar 751005,India
(2)Institut des Sciences Nucléaires, Université Joseph Fourier–CNRS-IN2P3
53, avenue des Martyrs,F 38026 Grenoble, France.
Abstract
The Hall–Post inequalities provide lower bounds onN -body energies in terms ofN ′-body
energies withN ′ < N . They are rewritten and generalized to be tested with exactly-solvable
models of Calogero-Sutherland type in one and higher dimensions. The bound forN spinless
fermions in one dimension is better saturated at large coupling than for noninteracting fermions
in an oscillator potential.
1
It is important to obtain good upper and lower bounds on the binding energy ofN -particle systems.
Since upper bounds are provided by variational estimates, it is natural that a major emphasis in recent
years has been on obtaining good lower bounds.
The Hall–Post inequalities consist of lower bounds toN -body energies in terms ofN ′-body ener-
gies withN ′ < N and modified constituent masses or coupling constants. [1, 2, 3]. Applications of
these inequalities have been proposed for studying the thermodynamic limit of large systems and the
stability of matter [3], or the relation of baryons to mesonsin hadron spectroscopy [2, 4].
So far, the Hall-Post inequalities have been tested in greatd t il for few-body systems, for which
accurate numerical calculations can be performed, or in thelarg N limit, within some approximation.
It seems appealing to test the inequalities for arbitraryN with energies which are calculated exactly.
The purpose of this note is to adapt and apply the Hall–Post inequalities to several exactly-solvable
N -body models.
Inspired by the seminal work of Calogero and Sutherland, in recent years, ground state energy and
at least a part of the excitation spectrum (if not the full spectrum) has been obtained for several new
N -body problems. Some of these Hamiltonians have already found applications in a variety of areas.
We broadly consider four types ofN -body problems.
1. N -body problems in one dimension with two-body (and also may be one-body) interaction in
the presence of either two-body or one-body oscillator potential. Some examples of this class
areAN−1 [5], BCN ,DN [6] models. The case ofD dimensions is considered in [7].
2. N -body problems in one dimension but with periodic boundary conditions and with two-body
(and also possible one-body interaction). Again, some examples of this class are theAN−1 [8]
andBCN ,DN [6] models.
3. N -body problems in one dimension with two-body inverse square interaction in the presence
of a hyper-Coulomb potential [9].
4. N -body problems of Calogero-type in two and higher dimensions with two and three-body
interactions in the presence of one or two-body oscillator potential [10, 11].
Our conclusions can be summarized as follows. While comparing the first and second type of
models with the Hall–Post inequalities, we find that contrary to our initial belief, the bound for
spinless fermions in one dimension is better saturated in the large coupling limit rather than for
N -noninteracting fermions in an oscillator potential. For the hyper-Coulomb case, using convex-
ity argument, we derive a slightly better lower bound than was known before. Finally, we show that
Hall–Post bounds also work reasonably well in models with bot two and three-body interactions.
In this paper we shall consider models withN identical particles of massm (which we shall put
equal to 1 without any loss of generality) and whose interaction does not depend on their spins. This
corresponds to the Hamiltonian
HN(m, g1, g2, g3) =
N
∑
i=1
p
2
i
2m
+ g1V (ri) + g2
∑
i<j
V (rij) + g3
∑
i<j<k
V (rijk) . (1)
For this case, it has been shown that the correspondingN -particle bound-state energyEN satisfies the
bound [3, 4]
EN (m, g1, g2, g3) ≥
N
N − 1
EN−1
(
m, g1,
N − 1
N − 2
g2,
N − 2
N − 3
g3
)
. (2)
2
In case the Hamiltonian (1) is translationally invariant, then this bound can be improved yielding the
new inequality [1, 2]
EN (m, g2, g3) ≥
N − 1
N − 2
EN−1
(
m,
N
N − 1
g2,
N(N − 2)
(N − 1)(N − 3)
g3
)
. (3)
As a representative of the first type of models, we consider the original Calogero problem [5] for
which theN -body Hamiltonian is given by (̄h = m = 1)
H = −
1
2
N
∑
i=1
d2
dx2i
+
N
∑
i<j=1
[
ω2
4
(xi − xj)2 +
g
(xi − xj)2
]
. (4)
As shown by Calogero, the ground-state energy for thisN -particle system is given by
EN =
√
N
8
[
N2 − 1 + (β − 1)N(N − 1)
]
ω , (5)
where
g = β(β − 1), i.e., β =
1
2
±
√
1 + 4g
2
, (6)
while the corresponding ground-state eigenfunction is given by
ψ =
N
∏
i=1
(xi − xj)β × exp
[
−
ω√
2N
N
∑
i<j=1
(xi − xj)2
]
. (7)
Note that we have to choose the positive square root in eq. (6)so thatg = 0 corresponds toβ = 1 and
hence to fermions.
For translationally invariant Hamiltonians with two-bodyinteraction, as the one given in eq. (4),
the bound (3) takes the form
EN (m,ω, g) ≥
N − 1
N − 2
EN−1

m,ω
√
N
N − 1
,
N
N − 1
g

 . (8)
Using the bound-state energy expression (5) as derived for the Hamiltonian (4), it is easily seen
that
EN−1

m,ω
√
N
N − 1
,
N
N − 1
g

 =
√
N
8
[
N(N − 2) + (β′ − 1)(N − 1)(N − 2)
]
ω , (9)
where
β′ =
1
2
+
√
N(2β − 1)2 − 1
2
√
N − 1
. (10)
Using eqs. (5), (8) and (9) we then get
RN ≡
EN (m,ω, g)
N−1
N−2EN−1(m,ω
√
N
N−1 ,
N
N−1g)
≥
N + 1 + (β − 1)N
N + (β′ − 1)(N − 1)
. (11)
In the non-interacting fermion limit (i.e.,g = 0 or β = β′ = 1), the right-hand side isN+1N while
in the strong-coupling limit (i.e.,g → ∞) the r.h.s. is
√
N√
N−1 . On the other hand, asN → ∞ in
3
2 4 6 8
g
1.12
1.16
1.20
R
Figure 1: Hall–Post ratioRN , as defined in eq. (11) for the Calogero model (4), in the case of N = 5
particles, as a function of the coupling constantg. The dotted line is theg → ∞ limit.
one dimension, then the ratio goes to 1, for all values ofg. Thus we find that contrary to our naive
expectation, the bound is better satisfied by strongly interacting rather than non-interacting fermions
in an oscillator potential.
Fig. 1 displays the ratioRN as a function of the coupling constantg, in caseN = 5. One starts at
g = 0 from the valueR5 = 6/5 corresponding to a pure oscillator. Then the ratio evolves regularly
towards its largeg limit
√
5/4.
We have also examined how Hall-Post inequalities behave in similar models likeBCN , DN , and
even periodic models like those of Sutherland. In all these cases we found a behaviour similar to that
given in Fig. 1. We might add here that many of these models arenot translationally invariant and
instead of the lower bound (8), we have to use the simpler bound (2).
As a second example, we now consider a model of the third type [9], i.e.,
H = −
1
2
N
∑
i=1
d2
dx2i
+
N
∑
i<j=1
g
(xi − xj)2
−
α2
√
∑N
i<j=1(xi − xj)2
, (12)
which contains a hypercentral Coulomb interaction. TheN -particle ground-state binding energy has
been shown to be
EN = −
α2
N [N − 2 +N(N − 1)β]2
, (13)
with β andg being again related by eq. (6).
We now show that using a convexity argument, one can adapt thebound (3) to this case. In
particular, we use the fact that if a functionf is a convex function, then
f
(
αa+ βb+ · · ·
α+ β + · · ·
)
≥
α
α+ β + · · ·
f(a) +
β
α+ β + · · ·
f(b) + · · · , (14)
for positive weight factorsα, β, . . . . On usingf(x) = −1/
√
x, it is easily shown that in this case the
Hall-Post inequality takes the form
EN (m, g, α) ≥
N − 1
N − 2
EN−1
(
m,
N
N − 1
g,
(N − 1)
√
(N − 2)
N3/2
α
)
. (15)
4
5 10 15 20
g
0.6
0.7
0.8
R
Figure 2: RatioRN , as given in eq. (13), for the model of eq. (12), in the case ofN = 5 particles, as
a function of the coupling constantg. The dotted line is theg → ∞ limit.
How good is this inequality? We can test that by using the energy igenvalue (13) and computing
EN−1 for the appropriate couplings. On noting the fact that the binding energy is negative, the Hall-
Post inequality (15) in our case takes the form
RN ≡
EN (m, g, α)
N−1
N−2EN−1(m,
N
N−1g,
(N−1)
√
(N−2)
N3/2
α)
≤
(N)2[N − 3 + (N − 1)(N − 2)β′]2
(N − 1)2[N − 2 +N(N − 1)β]2
, (16)
where
β′ =
1
2
+
√
(N − 1)(2β − 1)2 − 1
2
√
N − 2
. (17)
We find that but forN → ∞, g arbitrary, this bound is not as good as (11) in the Calogero case. In
Fig. 2, we have plotted the ratioRN as a function ofg in caseN = 5 which gives an indication of
how good this inequality is for finiteN .
As the third and the last example, we consider a model of fourth type [10, 11] which is a Calogero-
type model but inD-dimensions given by
H =
1
2
N
∑
i=1
p
2
i +
ω2
4
N
∑
i<j=1
r
2
ij + g
N
∑
i<j=1
1
r
2
ij
+G
∑
i<j,k
k 6=i,j
rki.rkj
r
2
kir
2
kj
, (18)
where one also requires a three-body potential in addition to the two-body potential. TheN -boson
ground-state energy in
EN (m, g,G(g)) =
√
N
8
[D(N − 1) +N(N − 1)β]ω , (19)
provided the two-body coupling and the three-body couplingG are related by
g = G+ (D − 2)
√
G , G = β2 , (20)
this definingG(g). Note that unlike in one-dimension case, hereβ = 0 corresponds tog = G = 0
and hence to bosons in a pure oscillator potential.
5
Let us now examine how the Hall-Post inequality (3) fares in th s case. It is not possible to test
this inequality directly sinceg andG have a specific relationG(g) between them which in general
will not be satisfied ifg andG are changed to NN−1g and
N(N−2)
(N−1)(N−3)G, respectively. However, we
make use of the fact that both the two and three body terms in eq. (18) are positive in any dimension
D(≥ 2) and hence we can write an appropriate inequality. For example we obviously have
EN (m,ω, g,G(g)) ≥
N − 1
N − 2
EN−1

m,ω
√
N
N − 1
,
N
N − 1
g,
N(N − 2)
(N − 1)(N − 3)
G(g)


≥
N − 1
N − 2
EN−1

m,ω
√
N
N − 1
,
N
N − 1
g,G(
N
N − 1
g)

 , (21)
since
N(N − 2)
(N − 1)(N − 3)
G(g) ≥ G(
N
N − 1
g). (22)
How good is this inequality? Using the exactN -particle binding energy (19) it is easily seen that
EN (m,ω, g,G(g))
N−1
N−2EN−1(m,ω
√
N
N−1 ,
N
N−1g,G(
N
N−1g))
≥
D +Nβ
D + (N − 1)β′
, (23)
where
β′ = −
D − 2
2
+
√
N(2β +D − 2)2 − (D − 2)2
2
√
N − 1
. (24)
It is interesting that this bound is saturated for noninteracting bosons (β = 0 i.e., g = G = 0) in
an oscillator potential as well as for large number of particles, no matter what the coupling is. For
large coupling but finiteN , the ratio is
√
N/(N − 1). The bound is also saturated at largeD, as seen
explicitly in eq. (23). This property of the largeD limit was noted by Gonzalez-Garcia [12].
Final remarks are in order:
1. It would be interesting to extend the investigation to thecase of unequal masses. Some of
the exactly-solvable models considered here can be generalized as to accommodate different
constituent masses [13]. The bound for unequal masses was discussed in the case ofN = 3 or
N = 4 particles interacting through simple pairwise potentials[14].
2. Exactly solvable models are also available for particleswith residual interaction with nearest
and next-to-nearest neighbour only [15]. It would be interesting to extend the Hall-Post inequal-
ities to this situation.
3. One of the striking feature of some of the models considered here in one or two dimensions
is that the statistics evolves continuously as one changes the coupling constant. Still, if one
looks at theEN toEN−1 ratio measuring how far the inequalities are from saturation, it evolves
with a rather smooth and monotonic behaviour. Fermions can appro ch saturation provided the
coupling and the number of particles are large enough.
We hope to address some of the open issues in the near future.
Acknowledgements
AK would like to thank the members of the Institut des Sciences Nucléaires, Grenoble for warm
hospitality during his trip there as a part of the Indo-French Collaboration Project CEFIPRA 1501-
1502, supported by the Indo-French Centre for Promotion of Advanced Research.
6
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7


Testing Hall-Post Inequalities With Exactly Solvable N-Body Problems

Testing Hall-Post Inequalities With Exactly Solvable N-Body Problems

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