A Small Parameter Method for Few-Body Problems

0.2 MB·English

by  V. D. Efros

A SMALL PARAMETER METHOD FOR FEW–BODY
PROBLEMS∗
V.D. Efros†
RRC ”Kurchatov Institute”
Abstract
A procedure to solve few–body problems which is based on an expansion over a small parameter
is developed. The parameter is the ratio of potential energy to kinetic energy in the subspace
of states having not small hyperspherical quantum numbers, K > K . Dynamic equations are
0
8
0 reduced perturbatively to those in the finite subspace with K ≤ K . The contribution from the
0
0
2
subspace with K > K is taken into account in a closed form, i.e. without an expansion over basis
0
n
a functions.
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∗ An extended version of the talk [1] at ”Nucleus–2007”,24–29 June 2007, Voronezh, Russia
† E-mail [email protected]
1
I. INTRODUCTION
Below an approach to solving few–body problems which is based on an expansion over
a small parameter is developed. The parameter is the ratio of potential energy to kinetic
energy for the states with hyperspherical numbers K exceeding some limiting value K .
0
Roughly speaking, the parameter is K−2. The method is a development of that of Ref. [2].
0
An expansion over the parameter K−2 has been given there for solving large systems of
0
linear equations that arise in bound–state problems in the framework of the hyperspherical–
hyperradial expansion.[14] The method [2] is efficient for this purpose [4, 5]. However,
for A>3 it is the calculating of matrix elements entering those systems of equations that
requires a massive computational effort. The difficulty stems from a swift rise of a number
of hyperspherical states with the same K as K increases, or a number of particles increases.
Selection of hyperspherical states to reduce the effort, see [2, 6, 7], is efficient for A=3 and
4 bound–state problems only. Such a selection is not justified in reaction calculations, in
particular. The problem isremoved inthe methodbelow since no expansion over basisstates
is employed here for K > K .
0
Recently a considerable progress in methods for solving few–body problems has been
achieved. However, those developments have limitations, and the latter are removed in
the present method. In particular, the well–known Green Function Monte Carlo method
to be mentioned in this connection is the method to calculate a bound state of a system,
and it is not suit to calculate reactions. (Although the simplest scattering problems may
be considered in it frames.) Besides, this method is not convenient in the respect that it
provides separate observables, such as an energy or a size, as a result of a calculation but it
does not provide the wave function of a bound state that could be employed in subsequent
calculations. Unlike this method, the method below is suitable for calculating reactions of
a general type. And when in its frames one needs to use a bound state wave function one
need not recalculate it completely each time.
Recently a way was found to extend the Faddeev–Yakubovsky A=4 calculations over
the energy range above the four–body breakup threshold [8]. However, Yakubovsky type
calculations require too much numerical effort even in the A=4 case. Amount of calculations
is considerably less in the scheme below.
At solving few–body problems with expansion methods convergence of expansions for
2
calculated quantities was accelerated with the help of the effective interaction approaches.
Such approaches were developed in the framework of the oscillator expansion [9] and the
hyperspherical expansion [10]. In their framework a true Hamiltonian is replaced with some
effective Hamiltonianacting ina subspace ofonlylowexcitations. When, formally, thelatter
subspace is enlarged up to coincidence with the total space an effective Hamiltonian turns to
a true one. An effective Hamiltonian is constructed from a requirement that its ingredients,
as defined in a subspace of low excitations, reproduce some properties of the corresponding
ingredients of a true Hamiltonian in the total space. It has been shown [9, 10] that this,
indeed, leads to an improvement of convergence of observables considered.
Higher excitations are disregarded in such type calculations. It is clear, however, that
correlation effects related to higher excitations cannot be reproduced by any state vector
lying in an allowed subspace of only low excitations. For example, let us consider the mean
value, hΨ |H|Ψ i, of such an ”observable” as a true Hamiltonian. It follows from the varia-
0 0
tional principle that an approximate state Ψ supplied with such a method provides poorer
0
approximation to the true hΨ |H|Ψ i value than Ψ obtained by the simple diagonalization
0 0 0
of a Hamiltonian in the same subspace of low excitations. And even the value of hΨ |H|Ψ i
0 0
obtained with the latter Ψ is a very poor approximation for realistic Hamiltonians. On the
0
contrary, the method given below provides an approximate state vector that is apparently
close to a true state vector both as to its low excitation component and its high excitation
component.
And speaking of reaction calculations in the framework of Eq. (40) below, (H−σ)Ψ˜ = q,
one should in addition take into account that a rateof convergence is determined not only by
properties oftheHamiltonianH but also bythose ofthesource–term q. But theseproperties
are apparently ignored at constructing effective Hamiltonians. Therefore one cannot expect
fast convergence in all the cases, especially for source–terms q corresponding to strong–
interaction induced reactions. On the contrary, the method described below provides state
vectors genuinely close to the true ones both for bound state problems and any reaction
problems.
In the next section the bound state case is considered. In Sec. 3 modifications to treat
reactions are listed and a numerical estimate of the rate of convergence of the method is
done. Some comments on computational aspects contain in Sec. 4.
3
II. BOUND STATES
We consider the eigenvalue problem
(H −E )Ψ = 0, (1)
λ λ
where H = T + V is an A–body Hamiltonian. We split the whole space of states into
the subspaces with K ≤ K and K > K and we denote Ψl and Ψh the components of
0 0 λ λ
the solution Ψ that lie, respectively, in these subspaces. At a proper choice of K kinetic
λ 0
energy T of a state belonging to the second of the subspaces is much larger than its potential
energy. Indeed,
h¯2 Kˆ2
T = T + ,
ρ
2m ρ2
where Kˆ2 is the hyperangular momentum operator acting on a hypersphere, ρ is the hyper-
radius, and T is the hyperradial energy operator. The eigenvalues of the Kˆ2 operator are
ρ
K(K +n−2) where n = 3A–3 is the dimension of a problem. Thus hTi is large for states
having large K and not too large space extension. We choose K in a way that for K > K
0 0
one has, in a rough sense,
h¯2 K2
[Ψ ] ≫ [(V +T −E )Ψ ] . (2)
2m ρ2 λ K ρ λ λ K
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Here [...] denotes a component of a state with a given K. Eq. (2) is to be fulfilled for all
K (cid:12) (cid:12)
configurations that contribute significantly to a solution. The corresponding ρ values range
within the configuration space extension of a solution. These ρ values are such that ρ2 is
less than, or about, Ahr2i, where r is the single–particle size of a system.
At these conditions one may express the component Ψh in terms of Ψl perturbatively and
λ λ
obtain equations for the latter component alone. Let us define projectors onto the K ≤ K
0
and K > K subspaces as P and Q , respectively. Let us express formally Ψh in terms
0 K0 K0 λ
of Ψl:
λ
Ψh = −Γ (E )VΨl, (3)
λ K0 λ λ
where
Γ (E) = [Q (H −E)Q ]−1 (4)
K0 K0 K0
is the Green function defined in the second subspace. It is taken into account in (3) that
kinetic energy is diagonal with respect to K. It is convenient to define Γ as acting in the
K0
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whole space and to rewrite it in the form
Q [Q (H −E)Q ]−1Q . (5)
K0 K0 K0 K0
Substituting Eq. (3) into the relationship
P (H −E )Ψl +VΨh = 0 (6)
K0 λ λ λ
(cid:2) (cid:3)
one gets the equation for Ψl alone,
λ
P (H −E )Ψl = P VΓ (E )VΨl. (7)
K0 λ λ K0 K0 λ λ
The quantity P VΓ (E )VP represents the exact effective interaction arising due to
K0 K0 λ K0
coupling of the complementary K > K subspace to the K ≤ K subspace.
0 0
We shall solve Eq. (7) perturbatively. We write in (5) H −E = L +U and we use an
expansion
Γ (E) = G −G U(E)G +G U(E)G U(E)G −..., (8)
K0 K0 K0 K0 K0 K0 K0
where G = Q (Q LQ )−1Q . With the choices of L below it has no non–zero matrix
K0 K0 K0 K0 K0
elements between the subspaces with K ≤ K and K > K , i.e. [Q ,L] = 0. Then
0 0 K0
G = Q L−1 = L−1Q .
K0 K0 K0
For performing calculations in the coordinate representation we choose G as follows,
K0
−1
h¯2 Kˆ2
G = +W(ρ) Q . (9)
K0 2m ρ2 K0
" #
It is convenient to represent (9) as a sum of contributions from various K values,
G = g . (10)
K0 K
KX>K0
Then
h¯2 K(K +n−2) −1 δ(ρ−ρ′)
hξ|g |ξ′i = +W(ρ) Y∗ (ξˆ)Y (ξˆ′). (11)
K 2m ρ2 ρn−1 Kν Kν
(cid:20) (cid:21) ν
X
Here ξ and ξ′ are n–dimensional space vectors, W(ρ) is a subsidiary interaction, ξˆ= ξ/ρ,
ξˆ′ = ξ′/ρ, and Y form a complete set of orthonormalized hyperspherical harmonics having
Kν
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the same K. The hyperangular factor entering here may be represented with the simple
expression (e.g. [11])
K +γ
Y∗ (ξˆ)Y (ξˆ′) = Γ(γ)Cγ(ξˆ·ξˆ′), (12)
Kν Kν 2·πn/2 K
ν
X
where Cγ(x) is the Gegenbauer polynomial.
K
The choice (11) of g is done to facilitate Monte–Carlo calculations of matrix elements.
K
At this choice one has in (8)
hξ|U(E)|ξ′i = hξ|V|ξ′i+δ(ξ −ξ′)[T −E −W(ρ)], (13)
ρ
h¯2 d2 n−1 d
T = − + . (14)
ρ 2m dρ2 ρ dρ
(cid:18) (cid:19)
To perform calculations in the momentum representation we suggest the expansion (8)
with a modified G , G = [Π2/(2m)−E +W(Π)]−1Q ,
K0 K0 0 K0
δ(π¯ −π¯′)−Π−(n−1)δ(Π−Π′) Y∗ (π¯)Y (π¯′)
hπ¯|G |π¯′i = K≤K0;ν Kν Kν
K0 Π2/(2m)−E +W(Π)
0P
δ(π¯ −π¯′)−Π−(n−1)δ(Π−Π′)(2·πn/2)−1 (K +γ)Cγ(πˆ ·πˆ′)
≡ K≤K0 K
Π2/(2m)−E +W(Π)
0 P
Π−(n−1)δ(Π−Π′)(2·πn/2)−1 (K +γ)Cγ(πˆ ·πˆ′)
= K>K0 K , (15)
Π2/(2m)−E +W(Π)
P0
hπ¯|U(E)|π¯′i = hπ¯|V|π¯′i−δ(π¯ −π¯′)[E −E +W(Π)]. (16)
0
Here π¯ and π¯′ are n–dimensional momentum vectors, Π = |π¯|, πˆ = π¯/Π, πˆ′ = π¯′/Π, and
W(Π) is a subsidiary interaction. The quantity E is a fixed energy chosen to be close to
0
E sought for.
λ
Roughly speaking, the expansion goes over K−2. As K increases relative contributions
0 0
to a solution from subsequent terms in the expansion (8) decrease. Taking K sufficiently
0
large we retain only the lower terms in the expansion.
The subsidiary interaction W(ρ) ≃ V¯(ρ) or W(Π) ≃ V¯(Π) is intended to accelerate
convergence of observables of interest when K increases. A better choice of subsidiary
0
interactions would be such that they include spin–isospin operators. Let us suppose that
calculations are performed in the coordinate representation. For a conventional NN inter-
action that includes static local central and tensor components V plus components that
loc
depend on angular and linear momentum a possible good choice is the following. Let us
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consider the K = 0 component in the expansion of V over hyperspherical harmonics. This
loc
component is the result of averaging V over a hypersphere. It has the structure F(ρ)Oˆ
loc
where Oˆ = Oˆ(ij) is an operator that depends on spin–isospin variables. The operator Oˆ
is symmetriPc with respect to particle permutations. Therefore it may be represented as
Oˆ = O θ[f]ihθ[f] ,
f µ µ
f µ
X X(cid:12) (cid:12)
(cid:12) (cid:12)
where f labels irreducible representations of the permutation group of A particles, µ labels
[f]
basis vectors belonging to a representation [f], {θ } is the corresponding orthonormalized
µ
set of basis functions, and O is defined as follows,
f
hθµ[f]|Oˆ|θµ[f′]′i = δff′δµµ′Of. (17)
We then choose W as
W = F(ρ) O θ[f]ihθ[f] .
f µ µ
f µ
X X(cid:12) (cid:12)
(cid:12) (cid:12)
At this choice, V and W cancel each other to a large degree in the difference V −W entering
U. This allows employing a smaller K value. The Green function G = g becomes
0 K0 K K
δ(ρ−ρ′) K +γ P
hξ|g |ξ′i = Γ(γ)Cγ(ξˆ·ξˆ′)
K ρn−1 2·πn/2 K
h¯2 K(K +n−2) −1
× +F(ρ)O θ[f]ihθ[f] . (18)
2m ρ2 f µ µ
f (cid:20) (cid:21) µ
X X(cid:12) (cid:12)
(cid:12) (cid:12)
The quantities O may also be varied around their values from (17). To simplify the pre-
f
sentation we did not include a spin–isospin dependence in the formulas above.
We set in (7)
Ψl = Ψl(n), E = E(n), (19)
λ λ λ λ
n n
X X
l(n) (n)
where Ψ and E correspond to the n–th order in the expansion over G U in (8). We
λ λ K0
then get from (7), (8)
(0) l(0)
P (H −E )Ψ = 0, (20)
K0 λ λ
(0) l(1) (1) l(0) l(0)
P (H −E )Ψ = E Ψ +P VG VΨ , (21)
K0 λ λ λ λ K0 K0 λ
(0) l(2) (1) l(1) (2) l(0) (0) l(0)
P (H −E )Ψ = E Ψ +E Ψ −P VG U(E )G VΨ
K0 λ λ λ λ λ λ K0 K0 λ K0 λ
l(1)
+P VG VΨ . (22)
K0 K0 λ
7
l(1) l(1) l(0)
If Ψ is a solution to Eq. (21) then Ψ +cΨ with an arbitrary c is also a solution.
λ λ λ
l(2)
The same holds true as to Ψ in (22). To get a unique solution it is sufficient to impose
λ
the normalization condition
hΨl|Ψli = hΨl(0)|Ψl(0)i. (23)
λ λ λ λ
This gives in the first and second order, respectively,
l(1) l(0) l(0) l(1)
hΨ |Ψ i+hΨ |Ψ i = 0, (24)
λ λ λ λ
l(2) l(0) l(0) l(2) l(1) l(1)
hΨ |Ψ i+hΨ |Ψ i+hΨ |Ψ i = 0. (25)
λ λ λ λ λ λ
Taking into account time reversal invariance of the operators entering (21), (22) it is seen
that the matrix elements in (24) and (25) are real. Therefore (24) and (25) turn to
l(0) l(1)
hΨ |Ψ i = 0, (26)
λ λ
l(0) l(2) l(1) l(1)
hΨ |Ψ i+(1/2)hΨ |Ψ i = 0. (27)
λ λ λ λ
l(0)
Taking scalar products of Eq. (21) and Eq. (22) with Ψ and making use of Eq. (20)
λ
we obtain, respectively,
l(0) l(0)
hΨ |VG V|Ψ i
E(1) = − λ K0 λ , (28)
λ l(0) l(0)
hΨ |Ψ i
λ λ
l(0) (0) l(0) l(1) l(1)
hΨ |VG U(E )G VΨ i−hΨ |VG V|Ψ i
E(2) = λ K0 λ K0 λ λ K0 λ . (29)
λ l(0) l(0)
hΨ |Ψ i
λ λ
To get (29) Eq. (26) was employed.
We seek for the component Ψl as an expansion over the hyperspherical basis. In the
λ
coordinate representation,
Ψl(ρ,ξˆ,σ ,τ ) = χ (ρ)F (ξˆ,σ ,τ ). (30)
λ zi zi Kν Kν zi zi
KX≤K0;ν
Here σ and τ are particle spin–isospin variables, F are basis functions that we consider
zi zi Kν
to be orthonormalized. They are combinations of basis hyperspherical harmonics and basis
spin–isospin functions. It is implied here and below that all the summations over K include
l(n)
only K values of a given parity. Let us write down similar expansions for Ψ ,
λ
l(n) ˆ (n) ˆ
Ψ (ρ,ξ,σ ,τ ) = χ (ρ)F (ξ,σ ,τ ),
λ zi zi Kν Kν zi zi
KX≤K0;ν
8
(n)
so that χ = χ . Eqs. (20), (21) turn into equations for the expansion coefficients
Kν n Kν
(n)
χ : P
Kν
(T −E(0))χ(0) + (Kν|V|K′ν′)χ(0) = 0. (31)
K λ Kν K′ν′
K′X≤K0;ν′
(T −E(0))χ(1) + (Kν|V|K′ν′)χ(1) = (Kν|VG V|Ψl(0))+E(1)χ(0). (32)
K λ Kν K′ν′ K0 λ λ Kν
K′X≤K0;ν′
(T −E(0))χ(2) + (Kν|V|K′ν′)χ(2)
K λ Kν K′ν′
K′X≤K0;ν′
l(1) (0) l(0) (1) (1) (2) (0)
= (Kν|VG V|Ψ )−(Kν|VG U(E )G V|Ψ )+E χ +E χ . (33)
K0 λ K0 λ K0 λ λ Kν λ Kν
Here T denotes the hyperradial operator of kinetic energy,
K
h¯2 K(K +n−2)
T = T + . (34)
K ρ 2m ρ2
In the notation above (Kν|...) ≡ (FKν|...) and (Kν|V|K′ν′) ≡ (FKν|V|FK′ν′). These
quantities are defined in a obvious way. We recall that the equations written down include
K values only within a finite range, K ≤ K . The zero order equations (31) are the
0
standard ones that arise when coupling to states with K > K is disregarded. The higher
0
order equations just take this coupling into account.
Eq. (24 reads as
ρn−1dρ χ(0)(ρ)χ(1)(ρ) = 0. (35)
Kν Kν
Z K′X≤K0;ν′
The condition (35) is to be added to Eqs. (32). Let us suppose that Eqs. (31) and (32)
(0) (1)
are solved via an expansion of χ and χ over the same hyperradial basis with the same
Kν Kν
number of basis functions retained. The linear equations arising in this case from Eqs. (32)
are linearly dependent. In general, one should remove one of these equations and replace it
with the linear equation to which Eq. (35) turns. Eq. (25) becomes
2
ρn−1dρ χ(0)(ρ)χ(2)(ρ)+(1/2) χ(1)(ρ) = 0. (36)
Kν Kν Kν
Z K′X≤K0;ν′ h i
This should be used similar to Eq. (35). If instead of (30) a hyperspherical expansion is
employed within a momentum representation calculation similar equations may be written
down proceeding from (20)–(22).
The complementary K > K component Ψh of a state sought for may be written as
0 λ
Ψh = Ψh(n) (37)
λ λ
n
X
9
h(n) h(0)
where Ψ signifies a contribution having the n–th order in G U, and Ψ = 0. Then
λ K0 λ
one has
h(1) l(0)
Ψ = −G VΨ , (38)
λ K0 λ
h(2) l(1) (0) l(0)
Ψ = −G VΨ +G U(E )G VΨ . (39)
λ K0 λ K0 λ K0 λ
The component Ψl has been obtained above in the form of a hyperspherical expansion.
λ
Therefore one may store it and use in various applications. The complementary component
Ψh then may be reconstructed as a simple quadrature (38), (39).
λ
If, for example, Ψ is calculated up to the n > 1 corrections, i.e. Ψappr = Ψl(0) +Ψl(1) +
λ λ λ λ
Ψh(1), then the average energy E¯ = hΨappr|H|Ψappri/hΨappr|Ψappri differs from the exact E
λ λ λ λ λ λ λ
value in terms only of the third order and higher in the expansion over G . In particular,
K0
the second order energy (29) is correctly reproduced with E¯ . Indeed, according to the
λ
variational principle the difference between E¯ and the exact E value includes the term
λ λ
hδΨ |H|δΨ i and powers of the term hδΨ |δΨ i. Here δΨ = Ψexact − Ψappr. We have
λ λ λ λ λ λ λ
δΨ ∼ (G )2 while presence of H in the above matrix element changes the net power in
λ K0
G from (G )4 to (G )3.
K0 K0 K0
Basing on Table 4 in Ref. [4] one infers the following. When only the above considered
n = 1 correction is retained the choice K = 14 ensures the correct binding energy at the
0
accuracy level better than 0.1 MeV in the A=4 bound state problem with a realistic NN
interaction that includes a strong core. The net number of HH with K ≤ 14 entering the
problem does not exceeds several hundreds which is acceptable.[15]
III. REACTIONS
1. We consider a dynamic equation of the form
(H −σ)Ψ˜ = q. (40)
Here σ is a subsidiary complex energy, and q is a given state. Reaction amplitudes may
be obtained from Ψ˜(σ) in a simple way, see e.g. the review [12]. The approach extensively
applied to perturbation induced reactions and proved to be very efficient. Any strong–
interaction induced reactions can also be treated with this approach.
The solution Ψ˜ is localized. Therefore the procedure quite similar to that described above
is applicable also here. One represents Ψ˜ as Ψ˜l+Ψ˜h and obtains these components as sums
10