Con inuum disc e iza ion in a basis o ans o med ha monic-oscilla o s a es
F. Pe
´ ez-Be nal,1I. Ma el,1J. M. A ias,2and J. Go
´mez-Camacho2
1Depa amen o de Fı
´sica Aplicada e Ingenie ı
´a Ele
´c ica, Uni e sidad de Huel a, 21071 Huel a, Spain
2Depa amen o de Fı
´sica A o
´mica, Molecula y Nuclea , Facul ad de Fı
´sica, Uni e sidad de Se illa, Apa ado 1065,
41080 Se illa, Spain
共Recei ed 10 No embe 2000; published 18 Ap il 2001兲
The inclusion o he con inuum in he s udy o weakly bound sys ems is discussed. A ans o med ha monic-
oscilla o basis is in oduced o p o ide an app op ia e disc e e and ini e basis o ea ing he con inuum pa
o he spec um. As examples o applica ion o he me hod he one-dimensional Poeschl-Telle and Mo se
po en ials a e wo ked ou . The s eng h unc ions co esponding o di e en ope a o s ha couple he g ound
s a e o he con inuum a e in es iga ed. I is ound ha he ene gy momen s o hose dis ibu ions a e accu a ely
ep oduced wi h a small basis se .
DOI: 10.1103/PhysRe A.63.052111 PACS numbe 共s兲: 03.65.Ca, 21.60.⫺n, 31.15.⫺p, 34.10.⫹x
I. INTRODUCTION
A gene al ime-independen quan um-mechanical po en-
ial gi es ise o a Hamil onian wi h bo h bound and unbound
eigens a es. Usually, he Hamil onian o he sys em has a
ini e numbe o bound eigens a es while he unbound ones
o m a con inuum. The e o e, a calcula ion o he sys em
p ope ies in e ms o eigen unc ions o Hin ol es a sum-
ma ion o e he disc e e s a es as well as an in eg a ion o e
he con inuum ones. The las one is an in ol ed ask and
no mally he p ope ies o he bound sys em a e analyzed by
jus using he bound eigens a es, while he con inuum ones
a e o special ele ance o dispe sion p ocesses. Howe e ,
he s udy o he e ec o he con inuum pa o he spec um
o ea ing p ope ies o he bound sys em has a long adi-
ion in physics 共con e sely, he e ec o he bound s a es on
dispe sion p ocesses has been widely in es iga ed oo兲. Re-
cen examples can be ound in nuclea 关1–6兴, molecula
关7–9兴, and a omic physics 关10–13兴. In pa icula , in nuclea
physics he ad en o he adioac i e beam acili ies has p o-
ided a a ie y o new nuclea s uc u e p oblems 关14兴 ha
include halo nuclei and neu on and p o on ich nuclei close
o he d ip lines. All hese sys ems a e weakly bound and
hei p ope ea men equi es he inclusion in some way o
he con inuum pa o he spec um. This has been done in
se e al ways, each one ha ing i s own ad an ages and d aw-
backs. Among hem we ci e:
共i兲The R-ma ix me hod 关15兴in which he basic idea is o
sol e he many-body p oblem in a box and hen make he
ma ching wi h he adequa e bounda y condi ions.
共ii兲The use o a S u mian basis 关16–18兴, whe e one uses
bound s a es o scaled po en ials, which a e o hogonal when
weigh ed wi h he po en ials.
共iii兲The Siege pseudos a e o mula ion 关19兴, which p o-
ides a ini e basis ep esen a ion o he ou going wa e so-
lu ions o he adial Sch o
¨dinge equa ion o cu o po en-
ials.
共i 兲The use o Gamow s a es 关20兴, which a e nonno mal-
izable solu ions o he Sch o
¨dinge equa ion co esponding
o ou going bounda y condi ions cha ac e ized by complex
ene gies.
共 兲The me hod o con inuum disc e iza ion coupled chan-
nels 共CDCC兲关21兴in which he con inuum is disc e ized by
means o aking ixed in e als, o bins, o k- alues in he
con inuum s a es.
共 i兲The expansion o he single-pa icle wa e unc ions
in a ha monic-oscilla o basis 关22兴.
This las me hod has become e y popula since i p o-
ides a simple comple e disc e e basis. Howe e , o weakly
bound sys ems he Gaussian asymp o ic beha io o he
ha monic-oscilla o wa e unc ions is a poo ep esen a ion
o he con inuum. Thus, me hods based in a gene al local-
scaling poin ans o ma ion o he ha monic-oscilla o unc-
ions 关23–27兴ha e d awn conside able a en ion ecen ly
关1–3兴. The so-called ans o med ha monic-oscilla o basis
共THO兲 e ains he simplici y o he ha monic-oscilla o ex-
pansion and includes he co ec asymp o ic beha io .
In his pape we discuss a way o de ining a THO basis
designed o ake in o accoun he con inuum by an app op i-
a e disc e iza ion. We p esen he me hod and apply i o wo
analy ic one-dimensional po en ials o in e es in molecula
physics: he Mo se po en ial and he Poeschl-Telle po en ial.
The me hod can be equally applied o h ee-dimensional po-
en ials. The pape is s uc u ed as ollows. Fi s , in Sec. II,
he o malism is p esen ed and he ans o ma ion o in o-
duce a THO basis is p oposed. In Sec. III, he applica ion o
he o malism o he Poeschl-Telle and Mo se po en ials is
wo ked ou . Finally, in Sec. IV he ou look and conclusions
o his pape a e p esen ed.
II. FORMALISM OF THE TRANSFORMED HARMONIC-
OSCILLATOR STATES „THO…IN ONE-DIMENSIONAL
HAMILTONIANS
In his sec ion we will apply he o malism o ans o med
ha monic-oscilla o s a es o weakly bound sys ems. Such
sys ems as he deu e on, halo nuclei, o Van de Waals mol-
ecules a e o cu en in e es . We conside he one-
dimensional Hamil onian gi en by
h⫽⫺ ប2
2
␮
d2
d 2⫹ 共 兲,共1兲
whe e is he ela i e coo dina e o wo pa icles,
␮
is he
educed mass, and ( ) is he in e ac ion be ween bo h pa -
PHYSICAL REVIEW A, VOLUME 63, 052111
1050-2947/2001/63共5兲/052111共9兲/$20.00 ©2001 The Ame ican Physical Socie y63 052111-1
icles. I dis ances a e measu ed in uni s o
␣
⫺1such ha x
⫽
␣
is dimensionless and ene gies a e gi en in uni s o
ប2
␣
2/
␮
, he p eceding Hamil onian can be w i en as
h⫽⫺ 1
2
d2
dx2⫹ 共x兲,共2兲
which will be he Hamil onian used he ea e . No e ha h
and (x) a e hen dimensionless quan i ies.
In o de o maximize he con inuum con ibu ion we as-
sume ha he sys em has jus one bound s a e,
␺
B(x), hough
he p esen o malism can be easily ex ended o sys ems wi h
se e al bound s a es as well as o h ee-dimensional sys ems:
h
␺
B共x兲⫽eB
␺
B共x兲.共3兲
The Hamil onian hhas also an in ini e numbe o eigens a es
in he con inuum ha , howe e , a e no no malizable. Ou
objec i e is o de elop a p ocedu e ha allows a con enien
desc ip ion o he s a es in he con inuum by means o a
ini e numbe o no malizable s a es.
The gene al o malism p esen ed he ewi h is applied in
he nex sec ion o wo cases o in e es in molecula physics,
he Poeschl-Telle 关28兴, and Mo se 关29兴po en ials.
A. Coo dina e ans o ma ion
Le us conside he one dimensional ha monic-oscilla o
basis
␾
n
HO共s兲⫽NnHn共s兲exp共⫺s2/2兲,共4兲
whe e Hn(s) a e he usual He mi e polynomials and Nn
⫽(
冑
␲
2nn!)⫺1/2 he co esponding no maliza ion cons an s.
This basis is o hogonal and o ms a comple e se o all
unc ions ha a e squa e in eg able in s. Besides, i we make
an a bi a y change o coo dina es, gi en by he mono o-
nously inc easing unc ion x⫽x(s), and i s in e se s⫽s(x),
he unc ions
␸
n
THO共x兲⫽
冑
ds
dx
␾
n
HO关s共x兲兴共5兲
a e o hogonal and o m a comple e se o all he unc ions
ha a e squa e in eg able in x. These unc ions a e called
THO s a es.
The ans o ma ion x(s) is a bi a y in p inciple. How-
e e , i can be chosen in o de o desc ibe p ope ly he p op-
e ies o bound s a es in ini e po en ials. So, o small alues
o s, he ha monic-oscilla o may be a easonable app oxima-
ion o he po en ial (x), and hus xshould depend linea ly
on s. Howe e , o la ge alues o s, he ha monic-oscilla o
wa e unc ions beha e as exp(⫺s2/2), while he bound wa e
unc ion in (x) beha es as exp(⫺qx), whe e q2/2⫽eB. So,
o la ge s,qx has o be p opo ional o s2/2.
I he bound s a e wa e unc ion
␺
B(x) is known, he
ans o ma ion x(s) can be comple ely de e mined by equi -
ing ha
␸
0
THO共x兲⫽
␺
B共x兲.共6兲
This condi ion oge he wi h Eq. 共5兲p o ide a basis se wi h
he asymp o ic beha io desc ibed abo e. Equa ion 共6兲is
equi alen o he nonlinea equa ion
冕
⫺⬁
x
兩
␺
B共x⬘兲
兩
2dx⬘⫽
冕
⫺⬁
s
兩
␾
0
HO共s⬘兲
兩
2ds⬘⫽1⫹e 共s兲
2,
共7兲
which de ines implici ly he unc ion x⫽x(s) as well as i s
in e se. I should be no iced ha he de i a i e can be w i -
en as
dx共s兲
ds ⫽
冉
␾
0
HO共s兲
␺
B关x共s兲兴
冊
2
.共8兲
Once he wa e unc ion o he g ound s a e p o ides he
unc ion x⫽x(s), we can employ he THO basis, Eq. 共5兲, o
desc ibe he con inuum o ou sys em. No e ha , as he THO
a e o hogonal, and he n⫽0 s a e is he only bound s a e o
he sys em, he s a es wi h n⭓1 desc ibe he con inuum. We
expec ha , as he dimension o he THO basis inc eases, he
wa e unc ions explo e dis ances beyond he ange o he
po en ial and, a he same ime, hey ha e oscilla ions inside
he po en ial. Thus, he THO basis allows o an app op ia e
desc ip ion o long ange phenomena, and a he same ime i
pe mi s o desc ibe accu a ely sho - ange e ec s.
B. Diagonaliza ion o he Hamil onian. Wid h o he s a es.
We e alua e he ma ix elemen s o he Hamil onian hin
he THO basis. I should be no iced ha he s a e
␸
0
THO(x)
⫽
␺
B(x) is an eigens a e o h, bu his is no he case o he
s a es wi h n⭓1. Le us conside he ma ix elemen
具
THO,n
兩
共h⫺eB兲
兩
THO,m
典
⫽
冕
dx
␸
n
THO共x兲共h⫺eB兲
␸
m
THO共x兲.共9兲
We can ake in o accoun ha
␸
m
THO(x)
⫽
␲
1/4NmHm关s(x)兴
␸
0
THO(x), and ha (h⫺eB)
␸
0
THO(x)⫽0,
o w i e
具
THO,n
兩
共h⫺eB兲
兩
THO,m
典
⫽
冑
␲
NnNm
2
冕
dx
␸
0
THO共x兲关Hn关s共x兲兴,
†共h⫺eB兲,Hm关s共x兲兴‡兴
␸
0
THO共x兲.共10兲
The double commu a o is independen o he po en ial, and
gi es
关Hn关s共x兲兴,†共h⫺eB兲,Hm关s共x兲兴‡兴⫽dHn关s共x兲兴
dx
dHm关s共x兲兴
dx ;
共11兲
w i ing he in eg al in e ms o s, one ge s
F. PE
´REZ-BERNAL e al. PHYSICAL REVIEW A 63 052111
052111-2
具
THO,n
兩
共h⫺eB兲
兩
THO,m
典
⫽2nNnmNm
冕
dsexp共⫺s2兲Hn⫺1共s兲Hm⫺1共s兲
冉
ds
dx
冊
2
.
共12兲
This exp ession can be easily e alua ed using Gaussian
quad a u es. No e ha he only in o ma ion equi ed is he
de i a i e o he unc ion x(s), e alua ed a he poin s sn,
which de ine he quad a u e.
The ma ix elemen s wi h n⫽0o m⫽0 anish. This is
due o he ac ha he s a e o n⫽0 is an eigens a e o he
Hamil onian. Le us conside ha we diagonalize he Hamil-
onian in a Ndimensional basis o THO s a es, om i⫽0 o
i⫽N⫺1. The eigens a es o he Hamil onian, in his e-
s ic ed basis, a e gi en by
兩
N,0
典
⫽
兩
THO,0
典
共13兲
兩
N,i
典
⫽兺
j⫽1
N⫺1
兩
THO,j
典具
THO,j
兩
N,i
典
,共14兲
whe e he s a es
兩
N,i
典
(i⫽1,...,N⫺1) ep esen he con-
inuum s a es in he unca ed Ndimensional THO basis.
They can be exp essed in he x ep esen a ion as
具
x
兩
N,i
典
⫽
␺
i
N共x兲⫽
␲
1/4Pi
N⫺1关s共x兲兴
␸
0
THO共x兲,共15兲
whe e Pi
N⫺1(s) is a polynomial gi en by
Pi
N⫺1共s兲⫽兺
j⫽1
N⫺1
NjHj共s兲
具
THO,j
兩
N,i
典
.共16兲
The eigen alues o he Hamil onian, in he es ic ed basis,
a e ela ed o he wa e unc ion h ough
共ei
N⫺eB兲⫽1
2
␦
ij
冕
dsexp共⫺s2兲
⫻dPi
N⫺1共s兲
ds
dPj
N⫺1共s兲
ds
冉
ds
dx
冊
2
.共17兲
The use o he THO basis also allows o calcula e he
wid h o he s a es. In o de o do so, we e alua e he ma ix
elemen s o he ope a o (h⫺eB)2in he basis
兩
N,i
典
. No e
ha i he s a es
兩
N,i
典
we e he ue eigens a es o he Hamil-
onian, in a comple e basis, hen his ma ix elemen would
jus be (Ei
N⫺eB)2. Howe e , as
兩
N,i
典
a e only eigens a es o
he Hamil onian in a es ic ed basis, hey will show a sp ead
o ene gies when expanded in e ms o he ue con inuum
eigens a es o h. A measu e o ha sp ead is gi en by
⌫i
N⫽
冑
具
N,i
兩
共h⫺eB兲2
兩
N,i
典
⫺共Ei
N⫺eB兲2.共18兲
Le us use he ac ha he THO s a es o m a comple e basis.
Then, we ha e
具
N,i
兩
共h⫺eB兲2
兩
N,i
典
⫽兺
n⫽0
⬁
具
N,i
兩
h⫺eB
兩
THO,n
典具
THO,n
兩
h⫺eB
兩
N,i
典
.
共19兲
Using Eqs. 共12兲and 共16兲
具
THO,n
兩
共h⫺eB兲
兩
N,i
典
⫽1
2Nn
冕
dsexp共⫺s2兲dHn共s兲
ds
dPi
N⫺1共s兲
ds
冉
ds
dx
冊
2
.
共20兲
This exp ession can be in eg a ed by pa s o gi e
具
THO,n
兩
共h⫺eB兲
兩
N,i
典
⫽1
2Nn
冕
dsHn共s兲exp共⫺s2兲
冉
2s⫺d
ds
冊
⫻
冠
dPi
N⫺1共s兲
ds
冉
ds
dx
冊
2
冡
.共21兲
Now, we can use he closu e p ope ies o he He mi e poly-
nomials, o ob ain
具
N,i
兩
共h⫺eB兲2
兩
N,i
典
⫽
冉
1
2
冊
2
冕
dsexp共⫺s2兲
冋
冉
2s⫺d
ds
冊
⫻
冠
dPi
N⫺1共s兲
ds
冉
ds
dx
冊
2
冡
册
2
.共22兲
I is ema kable ha he knowledge o he unc ion x(s) is all
we need o ob ain wa e unc ions, ene gies, and wid hs o
he Hamil onian eigens a es in he THO basis.
C. Ma ix elemen s o ope a o s. Sum ules
Le us conside he ma ix elemen s o an a bi a y local
ope a o O(x), which is a unc ion o he coo dina e x. The
ma ix elemen ha connec s he g ound s a e
兩
N,0
典
o he
con inuum s a e
兩
N,i
典
is jus
具
N,i
兩
O
兩
N,0
典
⫽
␲
1/4
冕
dx Pi
N⫺1关s共x兲兴O共x兲
兩
␸
0
THO共x兲
兩
2.
共23兲
This in eg al is mo e con enien ly w i en in e ms o he
a iable s
具
N,i
兩
O
兩
N,0
典
⫽
冑
␲
冕
ds Pi
N⫺1共s兲O„x共s兲…exp共⫺s2兲,
共24兲
an exp ession ha can be e y simply e alua ed using Gauss-
ian quad a u es. F om his las o mula we can de ine he
ollowing global magni udes.
共i兲To al s eng h: I is gi en by
CONTINUUM DISCRETIZATION IN A BASIS OF... PHYSICAL REVIEW A 63 052111
052111-3
ST共O;N兲⫽兺
i
兩
具
N,i
兩
O
兩
N,0
典
兩
2.共25兲
In he limi o e y la ge numbe o s a es N, he se o s a es
becomes a comple e se , and one can use closu e, so ha
ST共O兲⫽lim
N→⬁
ST共O;N兲⫽
具
N,0
兩
O2
兩
N,0
典
.共26兲
So, one should ob ain, in he la ge Nlimi ,
ST共O兲⫽
␲
⫺1/2
冕
dsO„x共s兲…2exp共⫺s2兲
⫽
冕
dxO共x兲2
␺
B共x兲2.共27兲
共ii兲Ene gy weigh ed sum ule: I is gi en by
EW共O;N兲⫽兺
i共ei
N⫺eB兲
兩
具
N,i
兩
O
兩
N,0
典
兩
2.共28兲
In he limi o e y la ge numbe o s a es N, we can use he
basis closu e o exp ess EW(O)⫽limN→⬁EW(O;N) in e ms
o a double commu a o
EW共O兲⫽1
2
具
N,0
兩
†O共x兲,关h⫺eB,O共x兲兴‡
兩
N,0
典
.共29兲
The double commu a o can be explici ly e alua ed, o gi e
†O共x兲,关h⫺eB,O共x兲兴‡⫽
冉
dO共x兲
dx
冊
2
.共30兲
Thus, one ob ains
EW共O兲⫽
␲
⫺1/2 1
2
冕
ds
冉
dO共x兲
dx
冏
x⫽x(s)
冊
2
exp共⫺s2兲共31兲
⫽1
2
冕
dx关dO共x兲/dx兴2
␺
B共x兲2.共32兲
共iii兲Pola izabili y: I is gi en by
P共O;N兲⫽兺
i⫽0共ei
N⫺eB兲⫺1
兩
具
N,i
兩
O
兩
N,0
典
兩
2.共33兲
In he limi o a la ge numbe o s a es, P(O)
⫽limN→⬁P(O;N) con e ges o a cons an alue, ha can be
e alua ed om he a ia ion o ma ix elemen o he Hamil-
onian h⫺eBwi h he g ound s a e
兩
gs( )
典
o a pe u bed
Hamil onian h⫹ O(x)关30兴:
P共O兲⫽1
2lim
→0
d2
d 2
具
gs共 兲
兩
h⫺eB
兩
gs共 兲
典
.共34兲
We can use he alues o hese global magni udes o
e alua e he con e gence as he numbe o THO s a es in-
cluded in he calcula ion is inc eased. Wi h ega d o he
o m o he ope a o s O(x), we will conside wo di e en
cases. Fi s , we ake O(x)⫽x, a long- ange ope a o , o de-
sc ibe e ec s o ex e nal ields, such as he Coulomb ield. In
his case, x ep esen s he elec ic dipole ope a o . In second
place, we conside a sho - ange ope a o O(x)⫽ (x), o
desc ibe possible e ec s o in e nal co ela ions, which
would ha e a ange simila o he po en ial.
III. APPLICATION TO ANALYTIC ONE-DIMENSIONAL
HAMILTONIANS
A. The Poeschl-Telle Hamil onian
The Poeschl-Telle po en ial 关28兴is widely used in mo-
lecula physics, o example, o model bending ib a ions,
and con eys a conside able a en ion in o he ields 关6兴.I is
w i en as
共x兲⫽⫺D1
cosh2共x兲,共35兲
whe e ⫺Dis he alue o he po en ial in i s minimum. The
a iable x⫽
␣
, whe e is he ela i e coo dina e and
␣
is
he in e se o he ange o he po en ial. The dep h o he
po en ial Dcan be w i en as
D⫽1
2j共j⫹1兲,共36兲
in e ms o a new pa ame e j关31兴which is a posi i e eal
numbe . The bound eigens a es o he Poeschl-Telle Hamil-
onian can be w i en in e ms o jas
⌿j 共x兲⫽Nj Pj
(j⫺ )共z兲,共37兲
whe e is an in ege aking alues om 0 o he in ege pa
o j,Nj ⫽
冑
(j⫺ ) !/(2j⫺ )! is a no maliza ion cons an ,
z⫽ anh(x), and Ps
(p)(z) a e he associa ed Legend e unc-
ions when sis in ege . In he p esen pape we conside j
⫽1, he only ue bound s a e ⫽0, has an ene gy o eB⫽
⫺1
2and i s wa e unc ion is w i en as
␺
B
PT共x兲⫽1
冑
2 cosh共x兲.共38兲
In his case he e is ano he s a e o ⫽1, which is no
no malizable, co esponding o a esonance in he con inuum
a ze o ene gy. The use o an in ege alue o jis assumed, in
he p esen pape o simplici y, bu i is no manda o y. The
ela ionship be ween xand ss ems om he Eq. 共5兲
1
冑
2 cosh共x兲
⫽
冑
ds
dx
␲
⫺1/4 exp共⫺s2/2兲.共39兲
Di ec in eg a ion gi es bo h he dependence o son xand
ice e sa: e (s)⫽ anh(x). The s(x) unc ion is p esen ed in
Fig. 1.
In his case he de i a i e o he unc ion s(x) can be
w i en in e ms o he s a iable as
F. PE
´REZ-BERNAL e al. PHYSICAL REVIEW A 63 052111
052111-4
ds
dx⫽
冑
␲
2exp共s2兲关1⫺e 2共s兲兴,共40兲
acili a ing he calcula ions. This esul allows us o w i e he
THO basis in he scoo dina e space as
␸
n
THO共x兲⫽1
冑
2n⫹1n!Hn关s共x兲兴
冑
1⫺e 2关s共x兲兴.共41兲
In Fig. 2 we plo he i s i e s a es o he THO basis o j
⫽1. In his case he Hamil onian ma ix can be easily com-
pu ed om Eq. 共12兲and i s diagonaliza ion p o ides us wi h
eigen alues and eigen unc ions. Acco ding o Eq. 共6兲,we
ob ain one nega i e eigen alue a he p ecise ene gy eB⫽
⫺1
2, and in addi ion, N⫺1 posi i e eigen alues co espond-
ing o he con inuum disc e iza ion. The esul ing ene gies
o inc easing alues o he Npa ame e a e depic ed in Fig.
3, wi h N anging om 2 o 20. The appea ance o symme y
double s is due o he symme ic o m o he po en ial, which
p o ides wa e unc ions wi h well-de ined pa i y. As he di-
mension o he THO basis inc eases, new ene gy le els ap-
pea . On he one hand, many o hem lie close o he ze o
ene gy, inc easing he le el densi y in he egion. On he
o he hand, new le els explo e highe ene gies.
The wa e unc ions ob ained a e o hono mal 共see Fig. 4
whe e we p esen he N⫽5 case兲and, as expec ed, wi h
inc easing ene gy hey ex end o highe x anges while he
nodes accumula e in he icini y o he o igin. They ha e he
desi ed asymp o ic beha io and hei na u e allows o a
s aigh o wa d use in calcula ions.
Once he eigen alues and eigen unc ions o he Hamil-
onian a e ob ained, we p oceed o check con e gence and
closu e o he unca ed basis, calcula ing he o al s eng h,
ene gy weigh ed sum ule, and pola izabili y o a ypical
long- ange ope a o (x) and a sho - ange one 关 he po en ial
(x)]. The esul s ob ained a e p esen ed in Tables I and II,
espec i ely.
In Table I we include only e en N alues. The odd N
⫹1 cases gi e iden ical esul since he nega i e pa i y s a es
a e he only ones connec ed o he g ound s a e due o he
an isymme ic na u e o he xope a o . The ope a o (x)is
FIG. 1. Func ion s(x) o he Poeschl-Telle Hamil onian cha -
ac e ized by j⫽1.
FIG. 2. THO basis o he Poeschl-Telle Hamil onian, om n
⫽0 o4.
FIG. 3. Eigen alues o he Poeschl-Telle Hamil onian in he
THO basis wi h dimensions anging om N⫽2 o 20.
FIG. 4. Eigens a es (n⫽0 o 4) o he Poeschl-Telle Hamil-
onian diagonalized in he N⫽5 THO basis.
CONTINUUM DISCRETIZATION IN A BASIS OF... PHYSICAL REVIEW A 63 052111
052111-5

symme ic and hus he si ua ion is he opposi e. Conse-
quen ly, only odd N alues a e shown in Table II.
In he long- ange ope a o case he con e gence is e y
as o he h ee obse ables compu ed. Fo N⫽4共 h ee
s a es in he con inuum, only wo wi h he igh pa i y兲we
ob ain he exac alues wi hin a 2/1000 ela i e e o . Fo he
po en ial ope a o , he con e gence is also as al hough we
need N⫽17 共eigh s a es wi h he igh pa i y兲in he wo s o
he cases, o each he same ela i e e o as be o e. I is
ema kable ha he pola izabili y associa ed o he po en ial
ope a o con e ges e y apidly. This indica es ha pe u -
ba i e co ec ions o he ene gy o he bound s a e, due o
changes o he po en ials, will be ob ained accu a ely in his
basis. In bo h cases we should s ess he as con e gence
ob ained, which poin s ou ha he disc e iza ion pe o med
is able o simula e co ec ly he con inuum e ec s wi h he
inclusion o ew s a es in he THO basis.
B. The Mo se Hamil onian
The Mo se po en ial 关29兴is a commonplace o model an-
ha monic ib a ions in dia omic molecules 关32兴and i is be-
coming o wide use in polya omic s uc u e calcula ions
h ough he local mode pic u e 关33,34兴. The o m o he po-
en ial is
共x兲⫽D
兵
关1⫺exp共⫺x兲兴2⫺1
其
,共42兲
whe e x⫽
␣
, wi h he ela i e coo dina e and
␣
he in-
e se o he po en ial ange, and Dis he po en ial dep h in
he minimum (x⫽0). Dcan be w i en in e ms o a pa am-
e e j关35兴, which is a posi i e eal numbe , as
D⫽1
2
冉
j⫹1
2
冊
2
.共43兲
The bound wa e unc ions o he Mo se po en ial a e w i -
en as
⌿j 共x兲⫽Nj exp共⫺z/2兲zj⫺ L
2j⫺2 共z兲,共44兲
whe e is an in ege numbe aking alues om 0 up o he
in ege pa o j,Nj ⫽
冑
(2j⫺2 ) !/(2j⫺ )! is a no mal-
iza ion cons an , z⫽(2j⫹1)exp(⫺x) is he Mo se a iable,
and Ls
(p)(z) a e he gene alized Lague e polynomials o de-
g ee sand o de p. As in he p e ious case we ake j⫽1. The
only ue bound s a e in his case, ⫽0, has ene gy eB⫽
⫺1
2and i s wa e unc ion is
␺
B
M共x兲⫽3 exp共⫺x兲exp关⫺3 exp共⫺x兲/2兴.共45兲
Fo ⫽1 he e is ano he s a e ha is no no malizable and
co esponds o a esonance in he con inuum a ze o ene gy.
Di ec in eg a ion in Eq. 共7兲p o ides he ela ion be ween
xand s:
关1⫹e 共s兲兴/2⫽关1⫹3 exp共⫺x兲兴exp关⫺3 exp共⫺x兲兴.
共46兲
Nume ically sol ing his equa ion we ge he s(x) unc ion
plo ed in Fig. 5. The beha io o s(x) e lec s he po en ial
TABLE I. Con e gence o he o al s eng h (ST), ene gy
weigh ed sum ule (EW), and pola izabili y 共P兲o he ope a o xas
a unc ion o he THO basis dimension o he Poeschl-Telle
Hamil onian. Nis he o al numbe o basis s a es. In his case,
because o he pa i y selec ion ule, only odd pa i y s a es a e con-
nec ed o he g ound s a e h ough he xope a o .
NST(x,N)EW(x,N)P(x,N)
2 0.815 77 0.527 09 1.262 54
4 0.822 45 0.500 34 1.420 50
6 0.822 467 0.499 99 1.423 44
8 0.500 00 1.423 49
10 0.500 00 1.423 50
Exac Value 0.8224 67 0.500 00 1.423 50
TABLE II. Con e gence o he ST,EWsum ule, and Po he
ope a o (x) as a unc ion o he THO basis dimension o he
Poeschl-Telle Hamil onian. Nis he o al numbe o basis s a es. In
his case, because o he pa i y selec ion ule, only e en pa i y s a es
a e connec ed o he g ound s a e h ough he (x) ope a o .
NST( ,N)EW( ,N)P( ,N)
3 0.511 992 0.061 675 0.073 9799
5 0.527 628 0.108 383 0.074 0682
7 0.531 714 0.132 586 0.074 0737
9 0.532 854 0.143 771 0.074 0741
11 0.533 187 0.148 694
13 0.533 287 0.150 812
15 0.533 318 0.151 713
17 0.533 328 0.152 096
Exac Value 0.533 333 0.152 381 0.074 0741
FIG. 5. Func ion s(x) o he Mo se Hamil onian cha ac e ized
by j⫽1.
F. PE
´REZ-BERNAL e al. PHYSICAL REVIEW A 63 052111
052111-6
asymme y. Once he s(x) unc ion is compu ed we can de-
ine he THO basis ollowing Eq. 共5兲. The esul o N⫽5is
depic ed in Fig. 6.
The Hamil onian diagonaliza ion in he THO basis p o-
ides wi h eigen alues and eigen unc ions. We plo in Fig. 7
he ene gies ob ained inc easing he dimension o he basis
om N⫽2 o 20. The bound-s a e ene gy lies a i s exac
alue, eB⫽⫺ 1
2, while he beha io o he posi i e eigen al-
ues is simila o he p eceding case, excluding he appea ance
o pa i y double s. No e he di e en scaling in Figs. 3 and 7,
which shows he di e en beha io o he Poeschl-Telle and
Mo se po en ials.
The eigen unc ions 共see Fig. 8兲 o m an o hono mal se .
They a e no symme ic, as expec ed, bu as in he p e ious
case, hey bo h inc ease he numbe o nodes in he egion
a ound he o igin and explo e highe
兩
x
兩
alues as nin-
c eases. Posi i e alues o xa e explo ed much mo e apidly
as a unc ion o n han he nega i e ones.
Wi h he ob ained eigen alues and eigen unc ions, we
again check he con e gence and closu e o he unca ed
basis calcula ing he o al s eng h, ene gy weigh ed sum
ule, and pola izabili y o he xope a o and he po en ial .
The esul s ob ained a e p esen ed in Tables III and IV. In
his case we canno make any symme y simpli ica ion.
In Table III he esul s o he xope a o a e p esen ed,
showing a e y as con e gence o all he compu ed ob-
se ables. Fo N⫽5共 ou s a es in he con inuum兲we ob ain
a ound 1/1000 maximum ela i e e o . Fo he po en ial op-
e a o 共see Table IV兲wi h N⫽6 he maximum ela i e e o
is a ound 1/1000. Also, in his case, he con e gence o he
pola izabili y associa ed o he po en ial ope a o is e y as .
As in he Poeschl-Telle po en ial we should s ess he
as con e gence ob ained, e en as e in his case. Tha sup-
po s he e idence o conside ing he unca ed THO basis
as a sui able ool o con inuum disc e iza ion.
IV. SUMMARY, CONCLUSIONS, AND OUTLOOK
In his pape a THO basis has been in oduced o p oduce
app op ia e no malizable s a es o disc e izing he con-
inuum. This is a undamen al p oblem in quan um mechan-
ics and is especially ele an when ea ing weakly bound
sys ems. The THO basis used in his pape is ob ained by a
local scale ans o ma ion 共LST兲 ha con e s he g ound
s a e o he sys em in o he ha monic-oscilla o 共HO兲g ound
s a e. Thus he only p e ious equi emen o apply his o -
FIG. 6. Basis o THO o he Mo se Hamil onian, om n⫽0
o 4.
FIG. 7. Eigen alues o he Mo se Hamil onian in he THO basis
wi h dimensions anging om N⫽2 o 20.
FIG. 8. Eigens a es (n⫽0 o 4) o he Mo se Hamil onian di-
agonalized in he N⫽5 THO basis.
TABLE III. Con e gence o he ST,EWsum ule, and Po he
ope a o xas a unc ion o he THO basis dimension o he Mo se
Hamil onian. Nis he o al numbe o basis s a es.
NST(x,N)EW(x,N)P(x,N)
2 1.082 07 0.593 44 0.658 937
3 1.100 99 0.500 68 0.894 608
4 1.101 67 0.500 04 0.950 650
5 1.101 68 0.500 00 0.960 174
6 0.961 193
7 0.961 235
8 0.961 235
Exac Value 1.101 68 0.500 00 0.961 237
CONTINUUM DISCRETIZATION IN A BASIS OF... PHYSICAL REVIEW A 63 052111
052111-7
malism is o know 共ei he analy ically o nume ically兲 he
g ound s a e o he sys em. This de ines he LST and allows
o gene a e all he s a es in he THO basis by ans o ming
he HO wa e unc ions. The s a es in he THO basis a e
disc e e, no malizable, and ha e exponen ially dec easing
asymp o ic beha io . Al hough he basis is in ini e, i is pos-
sible o ge good app oxima ions o he exac esul s when
calcula ing obse ables o in e es by unca ing he basis o
ew s a es in he con inuum egion. In he calcula ions p e-
sen ed in his pape , unca ing jus o 7 o 8 s a es in he
con inuum gi es he exac esul s wi hin a ound one pe mil
ela i e e o in he wo s o he cases.
In his pape we ha e p esen ed he o malism o one-
dimensional po en ials and we ha e chosen he case o jus
one bound s a e. Howe e we ha e pe o med calcula ions
o se e al bound s a es and he same kind o esul s a e
ob ained. The THO basis con e ge e y apidly o he exac
ene gies o he bound eigens a es while s a es in he con-
inuum lie close o ze o ene gy, inc easing he le el densi y
in ha egion, and ew o hem explo e highe -ene gy e-
gions.
The o malism p esen ed he e can be o use whene e
bound s a es close o he dissocia ion limi a e conce ned o
in he cases in which he coupling be ween bound and con-
inuum s a es a e impo an . I can be used o s uc u e cal-
cula ion o e alua e s eng h unc ions in o he con inuum
and o pe o m sca e ing calcula ions aking in o accoun he
b eakup e ec s.
The use o THO wa e unc ions in p ac ical calcula ions
in ol es inc easing he numbe No s a es conside ed in he
calcula ion un il con e gence is achie ed. In his sense, he
THO basis has an ad an age o e he use o a box o calcu-
la e con inuum e ec s, because, in his case, one has o deal
wi h wo con inuum pa ame e s, which a e he adius o he
box, and he maximum ene gy o he s a es conside ed. Thus,
i is much ha de o demons a e con e gence when he e a e
wo pa ame e s o a y, ins ead o jus one disc e e pa am-
e e .
The CDCC calcula ions ha e simila con e gence p ob-
lems. The bins desc ibing con inuum disc e iza ion a e such
ha when exp essed in e ms o he coo dina es, hey anish
a dis ances o he o de o 1/⌬k. A p ac ical CDCC calcu-
la ion equi es o ix he maximum k alue conside ed and
he in e al ⌬ko he bins, so ha he numbe o bins is
gi en by N⫽kmax /⌬k. He e, also o demons a e con e -
gence, one has o deal wi h wo pa ame e s. In addi ion, he
CDCC me hod equi es o sol e he Sch o
¨dinge equa ion o
all he ene gies in he con inuum. In ou case, he e is only
one disc e e pa ame e o check con e gence and i is only
equi ed o sol e he Sch o
¨dinge equa ion o he g ound
s a e.
The THO basis also has some simila i ies wi h he S u -
mian basis. In bo h cases he basis is disc e e and no maliz-
able, and he wa e unc ions ha e he same asymp o ic be-
ha io as he g ound s a e. Howe e , he S u mian basis
equi es sol ing he Sch o
¨dinge equa ion o inc easing al-
ues o he po en ial dep h, ob aining in his way wa e unc-
ions wi h he same ene gy, bu mo e nodes. These wa e
unc ions a e no o hogonal, as hey co espond o di e en
Hamil onians. Besides, he S u mian basis gi es an accu a e
desc ip ion o he in e io o he po en ial, bu i con e ges
e y slowly o desc ibe la ge sepa a ions. Thus, he THO
basis has he ad an age ha one has o sol e only he Sch o
¨-
dinge equa ion o he g ound s a e, he wa e unc ions a e
o hogonal, and he desc ip ion o dis ances beyond he po-
en ial ange seems o be sa is ac o y.
ACKNOWLEDGMENTS
This wo k was suppo ed in pa by he Spanish DGICYT
unde P ojec Nos. PB98-1111 and FPA2000-1592-C03-02.
We acknowledge use ul discussions wi h F. Iachello, P. H.
Vacca o, A. F ank, R. Lemus, and R. C. Johnson.
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TABLE IV. Con e gence o he ST,EWsum ule, and Po he
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Mo se Hamil onian. Nis he o al numbe o basis s a es.
NST( ,N)EW( ,N)P( ,N)
2 0.575 235 0.120 856 0.013 4195
3 0.669 899 0.178 806 0.093 6744
4 0.740 342 0.689 425 0.093 7489
5 0.749 840 0.919 480 0.093 7500
6 0.749 996 0.938 555
7 0.750 000 0.937 510
8 0.937 500
Exac Value 0.750 000 0.937 500 0.093 7500
F. PE
´REZ-BERNAL e al. PHYSICAL REVIEW A 63 052111
052111-8
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