expansion and perturbation methods for the gamow shell model. · expansion and perturbation methods...
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Expansion and perturbationmethods for the Gamow Shell Model.
Gaute Hagen
Centre of Mathematics for Applications,
University of Oslo, Norway
With Morten-Hjorth Jensen and Jan S. Vaagen
Expansion and perturbation methods for the Gamow Shell Model. – p.1/53
Contents
Motivation
Why momentum space ? Momentum spaceSchrödinger equation.
Contour deformation in momentum space
Single particle Berggren basis
The Gamow shell model
Construction of effective interaction by the Lee-Suzukisimilarity transformation.
The Möller-Plesset multireference perturbation method.
Three particle resonances, and convergence studies.
Outlook and conclusions.
Expansion and perturbation methods for the Gamow Shell Model. – p.2/53
Physical motivation.
In subatomic physics the expansion of many-bodywavefunctions on single particle bases has beencommon practice.
In the traditional shell model the harmonic oscillator hasserved as a paradigmatic tool for describing boundnuclei.
The recent exploration of nuclear driplines has pushedtraditional single particle methods to their limits ofapplicability.
In the description of weakly bound nuclei, coupling tounbound nuclear states such as resonances and virtualstates has to be accounted for.
This may be accomplished by analytic continuation ofthe Newton completeness relation to the second energysheet. Expansion and perturbation methods for the Gamow Shell Model. – p.3/53
Why momentum space ?
Realistic NN - interactions derived from field theorygiven explicity in momentum space
Boundary conditions automatically built into the integralequations
Gamow states are non-oscillating decreasing functionsin momentum
Numerical procedures often easy to implement andcheck
Convergence obtained by increasing number ofintegration points
Expansion and perturbation methods for the Gamow Shell Model. – p.4/53
Momentum space Schrödinger equation.
In natural units
� � � � �
, the two-body momentum space Schrödingerequation in a partial wave decomposition reads
� ����
� � � � � ��
��
��� � � � � ��� � � � � � � � � � � � ���
� � ��� � � �
is given by a double Fourier-Bessel transform of the potential in
�-space:
� � ��� � � � ��
�� � � �
��
� � � � � ��� � � � �� � � � � � � � � �� � � ��� (1)
Hermitian Hamiltonian real eigenvalues.
Complete set of states forming a Hilbert space (
! �
- space)
" �
# � � $% � # � ��
�& �
� � � � # � � � $% � � � # �
Expansion and perturbation methods for the Gamow Shell Model. – p.5/53
Momentum space Schrödinger equation.
Resonant and virtual states can never be obtained froma Hermitian Hamiltonian.
The Hermitian version of the Schrödinger equation isdefined on the physical energy sheet.
The momentum is a multivalued function of energy;
� � � � � ��� �
. Multivalued functions may berepresented by Riemann surface or branch cuts.
Riemann surface with two sheets:1. The physical energy sheet is a mapping of the upperhalf
�
-plane.2. The non-physical sheet is a mapping of the lower half�
-plane.
Expansion and perturbation methods for the Gamow Shell Model. – p.6/53
Analytic continuation.
To reach into the non-physical energy sheet whereanti-bound and resonant states are located, one has toanalytic continue the scattering equations through thecut along the real energy axis and into the lower halfcomplex energy plane.
Analytic continuation of the scattering equations into thenon-physical energy sheet may be done by the contourdeformation method.
Expansion and perturbation methods for the Gamow Shell Model. – p.7/53
Contour deformation method
Im k
Re k
C
C−
+
The most general Berggren completeness relation on an arbitraryinversion symmetric contour
� � � � � � &can be written as
" � � �
# � $ % � � # �� �
� � � � # � � � $% �� � � # �
Inversion symmetric contours gives
���
� � � � # � � � $% �� � � # �� �
� � � � # � � � $% �� � � #
Expansion and perturbation methods for the Gamow Shell Model. – p.8/53
Contour deformation method
From the general Berggren completeness relation one may deduce thecorresponding eigenvalue equation.
� ����
� � � � � �� � �
� � � � � � � � ��� � � � � � � � � � � � � � ���
General rule: Continuing an integral in the complex plane, the movingsingularities of the integrand must not intercept the integration contour.The only moving singularties are given by the interaction potential� � ��� � � �
. Choice of
� �
determined by analytic properties of
� � ��� � � �
Non-hermitian Hamiltonian complex eigenvalues
Bound states invariant under this transformation
Eigen states form a biorthogonal set.
Normalization given by the generalized �-product,
% � � # � � $ � �� � .
Expansion and perturbation methods for the Gamow Shell Model. – p.9/53
Contour deformation method
�������� �� �� ����
�������������
kmax
Im k
Re k
L1
L2
L3
BS
DS
BS
CS
VS
VS
The line segment
! is given by a rotation � � � � � �� � � �
and
! � is given
by a translation � � � � � � � ��� ��� � �. May give virtual states in the spectrum.
Expansion and perturbation methods for the Gamow Shell Model. – p.10/53
The Malfliet-Tjon interaction.
The Malfliet-Tjon interaction in a partial wave decomposition reads:
� � ��� � � � � ����
� � � � � � �� � � � ����
� � � � � � � � �(2)
Here
� � � �
is a Legendre function of the second kind. The arguments ofthe Legendre functions are � �
�� � � � � � � � � � ��
�� �� � � � �
. Analyticcontinue Schrödinger equation to the non-physical energy sheet. TheLegendre functions are singular for � � � �
, and this condition determinethe analyticity region of the interaction in the complex
�
-plane.
� � � � � � � � ���
� � � (3)
which gives:
� � � � � � � � � � � ��� � � � � � � ��
� � ��� � � � � �
Expansion and perturbation methods for the Gamow Shell Model. – p.11/53
Continuation in the 3’rd quadrant .
Continuation in the third quadrant :If the translation into the complex
�
-plane exceeds � � �
it is not possible toconstruct overlapping domains of analyticity which contain the distortedcontour.
−µ
µ
C
complex k−plane
R+T−µ/2
� ����
� � � � �� � �
� � � � � � � � ��� � � � � � � � � � � � � ���
Expansion and perturbation methods for the Gamow Shell Model. – p.12/53
Continuation in the 3’rd quadrant.
Continuation in the third quadrant :If the translation into the complex
�
-plane is less than � � �it is possible to
construct overlapping domains of analyticity which contain the distortedcontour.
C
complex k−plane
−µ
µ
R+T
−µ/2
� ����
� � � � �� � �
� � � � � � � � ��� � � � � � � � � � � � � ���
Expansion and perturbation methods for the Gamow Shell Model. – p.13/53
Virtual states in the M-T interaction.
A B C
� � Energy Energy Energy
-2.6047 -0.066674 -0.066653 -0.066653
-2.5 -0.310114 -0.310115 -0.310115
-2.3 -1.229845 -1.229845 -1.229845
-2.1 -2.679069 -2.678979 -2.678979
Calculation of the neutron-proton virtual state as function of increasingattractive strength � � for the �-wave Malfliet-Tjon interaction. Column Aused N1 = 30, N2 = 50 integration points, column B used N1 = 100, N2 =100 integration points and column C used N1 = 150, N2 = 250 integrationpoints.
Expansion and perturbation methods for the Gamow Shell Model. – p.14/53
Gamow shell model.
Construct antisymmetric many-particle basis based on the singleparticle Berggren basis.
The unperturbed many-particle energy is the tensorial product of allsingle particle energies. May be problematic to locate physicalresonances.
Choose a suitable contour which separates the many-particleresonances from the dense distribution of complex continuum states.
� � � � is a good choice.
� � � � allows for continuation in the third quadrant, i.e. anti-boundstates treated on equal footing with bound- and resonant states.
Expansion and perturbation methods for the Gamow Shell Model. – p.15/53
A model example:
�
He.
Unbound nucleus, with two known resonant states moving in ��� � �
and � � � orbitals.
The single particle motion may be described phenomenologicallywith the SBB - potential, consisting of Gaussian terms.
The SBB potential in momentum space takes the analytic form,
� � ��� � � � � � � ���� �
�� � � � � � ��
� � � � � �� � � � � � � �
� � ��� � � (4)
� � � � � � is a Bessel function of the first kind.
The SBB-potential diverges exponentially for
# �� � � � # # � � � � # . � ��
� � � �� �
.
A contour of type� �� � � allows for continuation in the third quadrant.
Expansion and perturbation methods for the Gamow Shell Model. – p.16/53
A model example:
�
He.
0 2 4 6 8 10−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
1
Re[E] MeV
Im[E
] M
eV
p1/2
resonance in 5He
−1 −0.5 0 0.5 1 1.5 2 2.5 3−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Re[E] MeV
Im[E
] M
eV
p3/2
resonance in 5 He
� � � � Re[E] Im[E] Re[E] Im[E]
10 10 0.752321 -0.329830 2.148476 -2.912522
12 12 0.752495 -0.327963 2.152992 -2.913609
20 20 0.752476 -0.328033 2.154139 -2.912148
30 30 0.752476 -0.328033 2.154147 -2.912162
40 40 0.752476 -0.328033 2.154147 -2.912162
Expansion and perturbation methods for the Gamow Shell Model. – p.17/53
Two particle resonances in
�
He.
Expand the two-particle shell model equation in the unperturbed singleparticle basis:
� � � � � � � � �� � � �� � � � �� � � �� � � ��(5)
� � �� � � �� �� � �
� ��
���
� � � ���
� � � ��(6)
Here
� ���
� � � ��
is an antisymmetric two-particle basis-state in j-j coupling.
� ���
� � � �� �
�� � � �� � � �
� � ���
� � � �� � � � � ��� � ��� &� � ���� � � �� �
(7)
The two-particle Berggren completentess reads
� �� � �
# � ���
� � � �� $ % � ���
� � � �� � #
(8)
Expansion and perturbation methods for the Gamow Shell Model. – p.18/53
2-n interaction
The residual n-n interaction,
� � , is for simplicity chosen to be ofGaussian type, and separable in � � � � .
� � � � � � � � � �� � �� � � � � � � �� � ��
� � � ��� � � �� �
We fit the interaction strength to reproduce the
�
binding energy( � � � ��
MeV ) in
�
He in both cases.
Observe that position of
� �
resonance energy is dependent on therange � of the Gaussian
The many-particle resonant spectrum is dependent on the radialshape of the interaction, which stresses the need for a realisticderived n-n interaciton.
Expansion and perturbation methods for the Gamow Shell Model. – p.19/53
in
�
He, � ��� s.p. motion.
� � ��
� ��� Re[E] Im[E]
12 12 300 -0.980067 -0.000759
20 20 820 -0.979508 0.000000
25 25 1275 -0.979509 0.000000
Convergence of
�
bound state energy in�
He as the number of integra-
tion points along the rotated
� � and the translated part
�� of the contour
increases.
� ��� gives the dimension of the two-particle anti symmetrized
basis.
Expansion and perturbation methods for the Gamow Shell Model. – p.20/53
resonance in
�
He, � ��� s.p. motion.
� � ��
� ��� Re[E] Im[E]
12 12 300 1.215956 -0.267521
20 20 820 1.216495 -0.267745
25 25 1275 1.216496 -0.267745
Convergence of
� �
resonant state energy in
�
He as the number of integra-
tion points along the rotated
� � and the translated part
�� of the contour
increases.
� ��� gives the dimension of the two-particle anti symmetrized
basis.
Expansion and perturbation methods for the Gamow Shell Model. – p.21/53
� energy spectrum in
�
He.
−2 0 2 4 6 8 10−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
1
Re[E] MeV
Im[E
] MeV
6He 0+2 resonance
6He 0+1 bound state
,0 2 4 6 8 10
−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
1
Re[E] MeV
Im[E
MeV
6He 2+ resonances
The choice of contour (
� � � � ) makes transparent all physical two-particle
states. One may also do a search for which state that has largest over-
lap with the unperturbed state where both particles move in single particle
resonant states.Expansion and perturbation methods for the Gamow Shell Model. – p.22/53
in
�
He, � ����
� ��� s.p. motion.
� � ��
� ��� Re[E] Im[E] Re[E] Im[E]
12 12 600 -0.980111 -0.000497 4.289194 -3.882119
20 20 1640 -0.979148 -0.000000 4.286186 -3.882878
25 25 2550 -0.979148 0.000000 4.286181 -3.882876
Convergence of
� bound and the
�� resonant state energy in
�
He as the
number of integration points along the rotated
� � and the translated part
�� of the contour increases.
� ��� gives the dimension of the two-particle
anti symmetrized basis.
Expansion and perturbation methods for the Gamow Shell Model. – p.23/53
��
� in
�
He, � ����
� ��� s.p. motion.
� � ��
� ��� Re[E] Im[E] Re[E] Im[E]
12 12 876 1.149842 -0.203052 2.372295 -2.122474
20 20 2420 1.150527 -0.203060 2.372818 -2.123253
25 25 3775 1.150527 -0.203060 2.372817 -2.123254
Convergence of the
� � and
� �� resonance energy in
�
He as the number
of integration points along the rotated
� � and the translated part
�� of
the contour increases.
� ��� gives the dimension of the two-particle anti
symmetrized basis.
Expansion and perturbation methods for the Gamow Shell Model. – p.24/53
strength distribution in
�
He.
Expansion coefficients of the
��� bound and resonant state in
� � �. Here�� � � and � � � single particle states are included.
# � �� � � $ # � � � � $Re[
� �
] Im[
� �
] Re[� �
] Im[
� �
]
RR 1.10488 -0.83161 0.22620 -0.16120
RC -0.06036 0.88137 -0.19842 0.22423
CC -0.09716 -0.04974 0.02486 -0.06305
# � �� � � $ # � � � � $
Re[
� �] Im[
� �
] Re[
� �
] Im[
� �
]
RR -0.01136 -0.08003 0.90189 0.33029
RC 0.04282 -0.03939 0.05966 -0.24478
CC 0.00617 0.00494 0.00082 0.02896Expansion and perturbation methods for the Gamow Shell Model. – p.25/53
strength distribution in
�
He.
Expansion coefficients of the
� ��� resonances in
� � �. Here �� � � and � � �
single particle states are included.
# � �� � � $ # �� � � � � � $Re[
� �
] Im[
� �
] Re[
� �] Im[
� �
]
RR 0.96962 0.05539 0.11394 -0.00494
RC -0.05247 -0.02727 -0.03250 -0.01265
CC -0.00089 -0.00282 0.00229 -0.00772
# � �� � � $ # �� � � � � � $
Re[
� �] Im[
� �
] Re[
� �
] Im[
� �
]
RR 0.08911 -0.03742 0.88847 -0.03742
RC -0.00175 -0.02524 0.01170 0.02338
CC 0.00115 -0.00220 0.01131 -0.00447Expansion and perturbation methods for the Gamow Shell Model. – p.26/53
Model example:
�
He ( � ) resonance.
Three-particle shell model equations in j-j coupling:
� � ��
�� �� �
�� � � �� � � �� � � � � � �� � � �� � � �� � �
(9)
The wave function is expanded in a three-particle antisymmetric Berggrenbasis � � � � � �� � � �
� � � ���� ����
�� � � ��
�� � � � �� � �
where the completeness reads:
� �� � � ���
# � ���
� � � � �� � � $% � � ���
� � � � �� � � #
and
� �� � � � �
� � ���
� � � �
Expansion and perturbation methods for the Gamow Shell Model. – p.27/53
Model example:
�
He ( � ) spectrum.
−1.5 −1 −0.5 0 0.5 1 1.5 2−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
Re[E] MeV
Im[E
] M
eV
0+ threshold
2+ threshold J = 3/2 resonance
,−1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
Re[E] MeV
Im[E
] M
eV
The
� � �� �
energy spectrum of
�
He obtained in a full diagonalization,
using only �� � � single particle motion. The shell-model dimension is in this
case
� � � � � �
.
� � �� �resonance energy:
� � � � � � � � � � � � ��
MeV.
Expansion and perturbation methods for the Gamow Shell Model. – p.28/53
Model example:
�
He ( � ) resonance.
# ��� � � $
Re[
� �
] Im[
� �
]
RRR 1.295549 -0.986836
RRC -0.184544 1.099729
RCC -0.115738 -0.110375
CCC 0.004733 -0.002518
Expansion coefficients of the
�� � &ground state in
� � �. Here only �� � �
single particle states are included.Full diagonalization:
� � � � � � � � � � � � ��
MeV.
Diag. within
#� � � $ � #� � � $ � #� � � $�� � � � � � � � � � � �� �
MeV.
Expansion and perturbation methods for the Gamow Shell Model. – p.29/53
Lee-Suzuki similarity transformation.
For � � �
one has to handle the extreme growth of the shell modeldimension.
In j-j coupling one has typically,
� � � � � � � � ��� � � � � � �� � � � � �
.
As continuum-resonance and continuum-continuum couplings playan important role, the configuration mixings in the full-space has tobe accounted for in some way.
Construction of a two-particle effective in a reduced space.
The Lee-suzuki similarity transformation for complex interactions.
The multireference perturbation method.
Expansion and perturbation methods for the Gamow Shell Model. – p.30/53
Lee-Suzuki similarity transformation.
Two-body Schröginer equation:
� � � � � � # $ � � # $We introduce projection operators
� � � � � � � ��� � � ��
� ���
# � � $% � � # � � ���
# � � $ % � � #
The two-particle states follows the Berggren metric,
% � � # � $ � % � � � # � $ � � ���
�
And the projection operators fulfill the relations,� � � �� � � � � � � � � �
� � � � � � � � � � � � � � � (10)
Expansion and perturbation methods for the Gamow Shell Model. – p.31/53
Lee-Suzuki similarity transformation.
Construct an effective two-body interaction within the model space,reproducing exactly the � � eigenvalues of the full Hamiltonian. This canbe accomplished by a similarity transformation,
�� � � &� � � �
� is defined by � � � � �
, and it follows that � � � � � � � � � �
and
� � � � � � � �. The two-body Schrödinger equation can then be writtenin a 2x2 block structure,
� �� � � �� �
� �� � � � �
� ��
� �
� ��
If
� � �
is to be the two-particle effective interaction the decoupling condi-
tion
� �� � � � � � � must be fulfilled.
Expansion and perturbation methods for the Gamow Shell Model. – p.32/53
Lee-Suzuki similarity transformation.
The decoupling condition:
� � � � � � � � � � � � � � � � � � � �
� acts as a transformation P-space to Q-space:
% � � # $ ���
% � � # � # � � $% � � # $ �
No unique solution to �. � may be obtained as long as the matrix
% � � # $
is invertible and non-singular. Based on this we choose those � � exactsolutions with the largest overlap with the two particle model space. Withthe solution � the non-symmetric effective interaction
�
is given by,
� � � � � � �� � � � � � � � � � � � � �� (11)
Expansion and perturbation methods for the Gamow Shell Model. – p.33/53
Lee-Suzuki similarity transformation.
It would be preferable to obtain a complex symmtric effective interaction,to take advantage of the anti-symmetrization of the two-particle basis.This may be accomplished by a complex orthogonal transformation:
��� � � � � & � � � � � � � � � �
here
�
is complex orthogonal, i.e.
� � � � � � � � � � �� � � �� � � � � �
� � � � � � � � � � � � � � &
Complex orthogonal transformations preserves the Berggren metric � � �
of any vector � in
�
. This gives the complex symmetrix interaction:
��� � � � � � � � � � � � � � � � � � � � � � � � � � � & � � � � �
Expansion and perturbation methods for the Gamow Shell Model. – p.34/53
Lee-Suzuki similarity transformation.
To obtain the complex symmetric effective interaction,
�� � � , one has to
find the matrix square root of
� � � � � � � � . In the case of�
being realand positive semi-definite,
� � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � �
� � � �� � � � � �� � � � � � � � � � �� � � � �
It can be shown that the matrix square root is related to the matrix signfunction (N.J.Higham). In the case of A being complex and having all
eigenvalues in the open right half complex plane, iterations based on thematrix sign are generally more stable:
� ��� �
��
�
��
�
� � �
� & � �
Expansion and perturbation methods for the Gamow Shell Model. – p.35/53
Lee-Suzuki similarity transformation.
One stable iteration scheme for the matrix sign was derived by Denmanand Beavers, as a special case of a method for solving the algebraicRiccati equation:
� � � �� � � � � (12)
��� � �
�� ��
� � � &�
�
(13)
�� � �
�� �
� � � &�
� � � � � � � �� � � � � (14)
provided
�
has no non-positive eigenvalues this iteration has quadratic rate
of convergence. Typically in our calcuations convergence is obtained after
2-3 iterates.
Expansion and perturbation methods for the Gamow Shell Model. – p.36/53
Model example:
�
He resonance.
0 500 1000 1500 2000−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
np
Re[
E]
MeV
nsp
= 11
,0 500 1000 1500 2000
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
np
Im[E
] M
eV
nsp
= 10
# �� � � � � �
�� �$ ��
� # �� � � � � ��� � � �
�� �$ �
�� # �� � � � �� �� � � �
�� �$
Convergence of the� � �� �
resonance in
�
He, using the Lee-Suzuki
similarity transformation (solid line), compared with the bare interaction
(dashed line).�
� � � � � � � �
Expansion and perturbation methods for the Gamow Shell Model. – p.37/53
Multireference perturbation method.
Extensively used in quantum chemistry (Ex.: C. Buth et.al (2003) andChen et al (2002)). Define a suitable n-particle reference (model) space
�
which describes most of the many-body correlations, and gives a weakcoupling between the reference and the complement space.
� � � � � �
� � � � � �
� �
� �
� �
Then we write:
� � � � � �
� � � � � � �
� � �
� � �
� � � �
� � � �� � � � � � � �
Here
� � �
is the diagonal part of
� � �
and
�� � �
the off-diagonal part.
Expansion and perturbation methods for the Gamow Shell Model. – p.38/53
Multireference perturbation method.
Diagonalize
� �
and obtain the zero-order correction to the exact energy
and wavefunction.
� � � � � � � � �
and
� � � � � � � � �
. We define
� � � � � � �
which gives the orthogonal transforamtion of the couplingblock. The energy corrections up to third order may then be shown to be:
� �� � � �� � � � � �� � � �
� �� �
�
��� �� �
� ���
�� �� � � ��
�� (15)
� � � �
�
��
� � � �
� ��
� � � ���
�
� ��
��
� �� � � ���
� � � �� � � ��
��
� � � � �
(16)
(17)
Expansion and perturbation methods for the Gamow Shell Model. – p.39/53
Multireference perturbation method.
3-particle model space used in the multireference perturbationcalculations:
�� � � � #� � � $ � #� � � $ � #� � � $ � � (18)
� �� � � � � �� � � � � � ��� � � � (19)
��� �� � � � � �� � �� � � ��� � �� (20)
0 5 10 15 20−20
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
Re[E] MeV
Im[M
eV]
P1
P2
P3
Expansion and perturbation methods for the Gamow Shell Model. – p.40/53
Multireference perturbation method.
0 500 1000 1500 2000−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
np
Re[
E]
MeV
,0 500 1000 1500 2000
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
np
Im[E
] M
eVConvergence of real (left) and imaginary (right) part of
�� � &
resonance in
�
He The dashed line uses the effective interaction generated by the Lee-
Suzuki similarity transformation method, and the solid line uses the mul-
tireference perturbation method (mrpm).
Expansion and perturbation methods for the Gamow Shell Model. – p.41/53
Effective interaction scheme for GSM.
Choose an optimal set of ��� � single particle states, definingmodelspaces:
� � � � �� � � � � �
Construct a two-particle effective interaction (Lee-Suzuki similaritytransform) within
� �� .
Within
�� � we define a new three-particle model (and complement)space:
� �� � � � #� � � $ � #� � � $ � #� � � $ � � � �� � � � # � � � $ �
Use multireference perturbation theory to obtain energy correctionsto a specific order.
Start from top with a larger set
� � until convergence is reached.
Expansion and perturbation methods for the Gamow Shell Model. – p.42/53
Lee-suzuki + multireference perturbation
0 100 200 300 400 500 600−0.25
−0.2
−0.15
−0.1
−0.05
0
np
Re[
E]
MeV
,0 100 200 300 400 500 600
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
np
Im[E
] M
eVConvergence of real and imaginary part of
�� � &
resonance in
�
He The solid
line uses the effective interaction generated by the Lee-Suzuki similarity
transformation method and subsequently the mcpm, and the dashed line
uses the multirefernce within the full space.
� � � � � �
.Expansion and perturbation methods for the Gamow Shell Model. – p.43/53
Conclusions.
The choice of contour considered, allows for a clear distinctionbetween physical resonant states and the remaningcomplex-continuum states, even for � �
.
The shell-model dimension increases dramatically when severalvalence particles and partial waves are included.
The clear distinction of unperturbed resonances from the rest, allowsfor a perturbation treatment of the unperturbed resonant state, whenconfiguration mixing is taken into account.
Have seen that MCPM combined with the similarity transformationmethod reduces the dimension of the full problem to about
� � � �
.
Expansion and perturbation methods for the Gamow Shell Model. – p.44/53
Ongoing work and perspectives.
Why include anti-bound states in many-particle problems? Treatingthe many-particle problem in some pertubative scheme, we need todefine a reference (model) space which describes most of themany-body correlations.
The scheme outlined here, allows for calculation of anti-bound states,and a pertubative treatment many-body states in which anti-boundstates are included (Ex.
Li ).
Self consistent Hartree-Fock single particle energies and realisticeffective interactions (the G-matrix), using the Berggren ensemblemay give an understanding of many-body resonances from amicroscopic viewpoint.
Solve the problem of vector-transformation brackets for complexmomenta.
Expansion and perturbation methods for the Gamow Shell Model. – p.45/53
CDBonn interaction.
The charge dependent Bonn potential (CD-Bonn):
A realistic nonlocal nucleon - nucleon interaction.
One boson-exchange interaction Yukawa - terms.
Given explicitly in momentum space CDM appropriate.
� � � � � �
�� � ��
� ��
��
��
�� �
� � � � � � � � � � � � � �
� ����
Both
� � � � � � and
� � � � � � �
� �� contain Yukawa terms�
� � � � � � � ��
The poles of the interaction are determined by the various meson masses
�and cut-off masses� �.
Expansion and perturbation methods for the Gamow Shell Model. – p.46/53
Virtual/bound states in CDBonn;
�
��
�
�
Calculation of
� � � � � bound and
� �virtual states using a rotated +translated contour.
N1 N2 Energy
20 50 -2.224573
50 100 -2.224575
50 150 -2.224575
CDM EFR
a:20,30 b:20,50 c:30,80
��� Energy Energy Energy Energy
-1 (pp) -0.11766 -0.11761 -0.11761 -0.11763
1 (nn) -0.10070 -0.10069 -0.10069 -0.10070
0 (np) -0.06632 -0.06632 -0.06632 -0.06632
Expansion and perturbation methods for the Gamow Shell Model. – p.47/53
Green’s function.
The resolvent (Green’s operator) �� � � � and the corresponding free
Green’s operator �� � � � � are defined:
�� � � � � �
�� � � �� (21)
�� � � � �
�� � � � (22)
They are related through the Dyson equation
�� � � � � �� � � � � � �� � � � � �
�� � � � (23)
The physical interpretation of the Green’s functions is that �� � �describes the
propagation of two noninteracting particles, and �� �
describes the propa-
gation of two interacting particles in free space.
Expansion and perturbation methods for the Gamow Shell Model. – p.48/53
Spectral represenation.
By expanding the unit operator on a complete set of physical eigenstatesof
�
, we can write the interacting Green’s operator in a spectraldecomposition:
�� � � � �
�# � $ % � #
� � � � � ��
� ��# � $% � #
� � �� � (24)
The Green’s function is analytic on the entire physical energy sheetexcept at the spectrum of the Hamiltonian, and a branch cut along thepositive energy axis:The discontinuity of the Green’s function across the cut is
�� � � � ��� � � �� � � � ��� � � � � �� # � $ % � #
(25)
Expansion and perturbation methods for the Gamow Shell Model. – p.49/53
Green’s function in Berggren basis.
Analytic continuation can be used to extend the valuesof an analytic function across a branch cut in thecomplex plane.
Expanding unity on a complete set of Berggren states,the Green’s function becomes analytic through the cut,and is extended into the non-physical energy sheet.
The branch cut along the real axis is then transformedonto the deformed contour
� �.
� � � ��� � ��
�� � � � �
� � � ��
�� ��� �
�� � � � � � �
� � ��� � � (26)
� � [a,b,c,d], � � denotes complex continuum states.
Expansion and perturbation methods for the Gamow Shell Model. – p.50/53
T-matrix in Berggren basis.
The
�
-matrix is defined in terms of the interacting Green’s function:
� � � � � � ��
� � � � �� (27)
Using the Berggren representation for �� �
, and decomposing into partialwaves, the
�
-matrix reads:
� � ��� � � � � � � � � ��� � � � � ��� � � � �
��� ��� � � � � � � � � ��� � ��
� � � � � �� � � � � � � � � � � �
In free space scattering, � is defined on the energy shell, � � � �
, here
��� � �
are real. By an appropriate choice of contour CDM gives an alternative
to the standard principal value prescription in solving numerically for the
�
-matrix.
Expansion and perturbation methods for the Gamow Shell Model. – p.51/53
T-matrix for M-T interaction.
� � ��� � � � � � � � � ��� � � � � ��� � � � �
��� ��� � � � � � � � � ��� � ��
� � � � � �� � � � � � � � � � � �
−µ
µ
C2 C1
Kmax Re k
Im k
For
��� � � �� �� the contour
� � will pass through the singularity of theinteraction
� � ��� � � �
.�� �� � � �� � � �
� ��� �
Expansion and perturbation methods for the Gamow Shell Model. – p.52/53