The Hierarhy of Hamiltonians for a Restrited
Class of Natanzon Potentials
ElsoDrigo Filho a
and Regina MariaRiotta b
a
InstitutodeBioi^enias,LetraseCi^eniasExatas,IBILCE-UNESP
RuaCristov~ao Colombo,2265, 15054-000S~aoJosedoRioPreto,SP,Brazil
b
FauldadedeTenologiade S~ao Paulo,FATEC/SP-CEETPS-UNESP
PraaFernandoPrestes,30,01124-060, S~aoPaulo,SP, Brazil
Reeivedon10November,2000. Revisedversionreeivedon10January,2001
TherestritedlassofNatanzonpotentialswithtwofreeparametersisstudiedwithintheontext
ofSupersymmetriQuantumMehanis. ThehierarhyofHamiltoniansandageneralformforthe
superpotentialispresented. Therstmembersofthesuperfamilyareexpliitlyevaluated.
I Introdution
ThelassesofNatanzonpotentials,namely,the
hyper-geometriandtheonuent,reertopotentialswhose
Shrodinger equation is analytially and exatly
solv-ablebymeansofhypergeometrifuntions. Theyhave
motivated several works onerning the mathematial
and algebrai aspets of their struture and solutions
andhavenumerousappliations inseveralbranhesof
physis,[1℄-[7℄.
Inpartiular,therehavebeenstudieswithin
Super-symmetriQuantum Mehanisformalism. Cooperet
al, [6℄, for instane, investigated the relationship
be-tweenshapeinvarianeandexatlyanalytialsolvable
potentialsand showed that the Natanzon potential is
notshapeinvariantalthoughithasanalytialsolutions
for the assoiated Shrodinger equation. Levai et al,
[7℄, havedetermined phase-equivalentpotentials for a
lass of Natanzon potentialsemploying the formalism
ofsupersymmetry.
However,the hierarhy of the Hamiltonians
orre-sponding to Natanzon potentials has not been
deter-minedyet. Inref. [6℄,Cooperet alhaveskethedthe
rstfewpotentialsofthehierarhyfromtheknowledge
of their asymptoti behaviour from the series
approx-imation. In this paper we onstrut the hierarhy of
Hamiltonians of the restrited lass of Natanzon
po-tentials, (Ginohio lass), with two free parameters.
Therstfewmembersofthesuperfamilyareexpliitly
evaluatedandageneralform forthesuperportentialis
proposedbyindution.
II Supersymmetri Quantum
Mehanis Formalism
IntheformalismofSupersymmetriQuantum
Mehan-istherearetwooperatorsQandQ +
,that satisfythe
algebra
fQ;Qg=fQ +
;Q +
g=0; fQ;Q +
g=H
SS (1)
where H
SS
is the supersymmetri Hamiltonian. The
usualrealisationoftheoperatorsQandQ +
is
Q=a
1
=
0 0
A 0
Q +
=a +
1
+
=
0 A +
0 0
(2)
where
are written in terms of the Pauli matries
and A
are bosoni operators. With this realisation
thesupersymmetriHamiltonianH
SS
isgivenby
H
SS =
A +
A 0
0 A A
+
=
H +
0
0 H
: (3)
where H
are supersymmetri partner Hamiltonians
andshare thesamespetra,apartfrom the
nondegen-erate ground state. Using the super-algebra a given
Hamiltonian anbefatorizedin termsof thebosoni
operators.In~==1units,itis givenby
H
1 =
d 2
dr 2
+V
1
(r)=A +
1 A
1 +E
(1)
0
(4)
where E (1)
0
isthelowesteigenvalue. Thebosoni
oper-atorsaredenedby
A
1 =
d
dr +W
1 (r)
(5)
where the superpotential W
1
(r) satises the Riati
equation
W 2
1 W
0
1 =V
1 (r) E
(1)
0
: (6)
Theeigenfuntionfortheloweststateisrelatedtothe
superpotentialW as
(1)
0
(r)=Nexp( Z
r
0 W
1
(r)dr) (7)
oronversely
W
1 (r)=
d
dr ln(
(1)
0
): (8)
Now it is possible to onstrut the supersymmetri
partnerHamiltonian,
H
2 =A
1 A
+
1 +E
(1)
0 =
d 2
dr 2
+(W 2
1 +W
0
1 )+E
(1)
0
: (9)
IfonefatorizesH
2
intermsofanewpairofbosonioperators,A
2
one gets,
H
2 =A
+
2 A
2 +E
(2)
0 =
d 2
dr 2
+(W 2
2 W
0
2 )+E
(2)
0
(10)
d
whereE (2)
0
isthelowesteigenvalueofH
2 andW
2 satisfy
theRiatiequation,
W 2
2 W
0
2 =V
2 (r) E
(2)
0
: (11)
Thus a whole hierarhy of Hamiltonians an be
on-struted , with simple relations onneting the
eigen-valuesandeigenfuntionsofthen-members,[8℄-[13℄
H
n =A
+
n A
n +E
(n)
0
(12)
A
n =
d
dr +W
n (r)
(13)
(1)
n =A
+
1 A
+
2 :::
(n+1)
0
E (1)
n =E
(n+1)
0
(14)
(1)
0
(r)=Nexp( Z
r
0 W
1
(r)dr): (15)
III Natanzon Potential and the
Hierarhy of Hamiltonians
TherestritedlassofNatanzonpotentialshavingtwo
parametersandgivenintermsofthevariabley(r)is,
V(r)=f 2
v(v+1)+1=4(1 2
)[5(1 2
)y 4
(7
2
)y 2
+2℄g(1 y 2
) ; (16)
wherethevariablefuntiony(r)satisesdy=dr=(1 y 2
)[1 (1 2
)y 2
℄. Thedimensionlessfreeparametersv and
measurethedepthandtheshapeofthepotential,respetively.
TheShrodingerequationforthispotential,[2℄,[3℄,in dimensionlessunits,isgivenby
[ d 2
=dr 2
+V(r)℄
n (r)=
n n
(r) (17)
where V(x)=v
0 V(r),
n =E
n =v
0
andr=bx=(2mv
0 =~
2
) 1=2
x.
Theanalytisolutionsfortheenergyeigenfuntionsaregivenby,
n
=(1 2
)
n =2
[g(y)℄ (2
n +1)=4
C
n +1=2
(y=[g(y)℄ 1=2
where g(y)= 1 (1 2
)y 2
. The fatorC (a)
n
(x) is a Gegenbauer polynomial when n is a non-negative integer,
whihisourase. Theorrespondingenergyeigenvaluesaregivenby
n
=
2
n
4
;
n
>0,where
n
2
=[ 2
(v+1=2) 2
+(1 2
)(n+1=2) 2
℄ 1=2
(n+1=2): (19)
Notietherelationshipbetweentheenergylevelswhihwillbeextensivelyusedin whatfollows,
( 2
n
2
n 1 )
2
=( 2n (2n+1)
n
+(2n 1)
n 1
): (20)
d
Inordertoonstrutthesuperfamilywerstly
fa-torizethe Natanzonpotential,alling V(r)=V
1 (r) =
V (r)+ (1)
0
, [6℄. Thefatorized Shrodinger equation
isgivenby
H
1
(1)
0 =a
+
1 a
1 ; a
1
=d=dr+W
1
(r) (21)
where (1)
n =
n
. ThesuperpotentialW
1
(r)isevaluated
from the knowledge of the ground stateeigenfuntion
ofV(r)byusing(8)with (1)
n =
n
,givenby(18). It
satisestheRiatiequationanditisgivenby
W
1
(r)=(1 2
)y(y 2
1)=2+y
0
2
: (22)
ThesuperpartnerHamiltonian satisestheequation
H
2
(1)
0 =a
1 a
+
1
(23)
whihiswrittenin termsofV
2 (r)as
W 2
1 +W
0
1 =V
2 (r)
(1)
0
(24)
whereV
2
(r),thepotentialfortheseondmemberofthe
hierarhy,isgivenby
V
2
(r)=f 2
0
4
+
0
2
+1=4(1 2
)[ 7(1 2
)y 4
+(9 3 2
8
0
2
)y 2
2℄g(1 y 2
): (25)
d
Toonstrut the nextmember of the superfamily, we
fatorizetheShrodingerequationforV
2
. Itgives
H
2
(2)
0 =a
+
2 a
2 ; a
2
=d=dr+W
2
(r) (26)
whereW
2
(r)satisestheassoiatedRiatiequation,
W 2
2 W
0
2 =V
2 (r)
(2)
0
: (27)
(2)
0
is the energy ground state of the potential V
2 (r)
and itis suh that (2)
0 =
(1)
1
. ThesuperpotentialW
2
anbeomputedfrom thegroundstatewavefuntion
(2)
0
. It is given by W
2 (r) =
d
dr log(
(2)
0
), where
(2)
0 =a
1 (1)
1 ,i.e.,
W
2
(r)=3=2(1 2
)y(y 2
1)+y
1
2 d
dr log(f
1 ) (28)
where
f
1
(y)=1+a
11 y
2
; (29)
andtheoeÆienta
11
isgivenby
a
11 =(
0
1 )
2
1: (30)
ThenewsuperpartnerofH
2
isgivenby
W 2
2 +W
0
2 =V
3 (r)
(2)
0
(31)
where V
3
(r),thepotentialforthethirdmemberof the
hierarhy,isgivenby
V
3
(r) = f 2
1
4
+
1
2
+1=4(1 2
) 27(1 2
)y 4
+(33 15 2
)y 2
6
+ (32)
+ 2a
11 g(y)
f (y)
f1+( 9+6 2
2
1
2
)y 2
+8(1 2
)y 4
g+8a 2
11 y
2
(1 y 2
)( g(y)
f (y) )
2
g(1 y 2
andg(y)=1 (1 2
)y 2
. Thus,fatorizingtheHamiltonianforthis potentialwehave
H 3 (3) 0 =a + 3 a 3 ; a 3
=d=dr+W
3
(r) (33)
where W
3
(r)satisestheRiatiequation,
W 2 3 W 0 3 =V 3 (r) (3) 0 : (34) (3) 0
istheenergygroundstateof thepotentialV
3 (r),with (3) 0 = (2) 1 = (1) 2
. Again,thesuperpotentialW
3 anbe
omputed fromthegroundstatewavefuntion (3)
0
, dened byW
3 (r)= d dr log( (3) 0 ), with (3) 0 =a 2 a 1 (1) 2 . It
isgivenby
W 3 =5=2(1 2 )y(y 2
1)+y
2 2 + d dr log(f 1 ) d dr log(f 2 ) (35) where f 2
(y)=1+a
21 y 2 +a 22 y 4 (36)
withoeÆientsaregivenby
a 21 =2( 1 2 ) 2 2 a 22
=1+ 2 (2 0 3 1 +6 2 )=3+ 4 ( 4 0 +6 1 2 2 2 0 5 0 2 +3 0 1 +3 1 2 )=3:
For the next member of the superfamily, we show the result of the evaluation of the superpotential, W
4 (r) = d dr log( (4) 0 )with (4) 0 =a 3 a 2 a 1 (1) 3
. Itis givenby
W 4 =7=2(1 2 )y(y 2
1)+y
3 2 + d dr log(f 1 )+ d dr log(f 2 ) d dr log(f 3 ) (37) where f 1 andf 2
areevaluated in(29)and(36)andf
3
issetto
f
3
(y)=1+a
31 y 2 +a 32 y 4 +a 33 y 6 ; (38)
withtheoeÆientsgivenby
a 31 =3( 2 3 ) 2 +2( 0 1 ) 2 5 a 32
=10 + 2 (6 25 0 +15 1 6 2 +22 3 )=3+ + 4 (6 0 +2 2 0 18 1 6 0 1 +18 2 +13 0 2 3 1 2 6 3 11 0 3 3 1 3 +8 2 3 )=3 a 33
= 10 +
2
( 6+13
0 3 1 12 2 4 3 )+ 4 148 15 + 484 45 0 206 45 2 0 84 5 1 + + 42 45 0 1 +2 2 + 1 3 0 2 1 2 394 45 3 + 497 45 0 3 59 5 1 3 16 3 2 3 + + 6 8 3 0 + 328 45 2 0 116 45 3 0 24 5 1 48 5 0 1 + 12 5 2 0 1 + 8 3 2 + 82 9 0 2 14 9 2 0 2 6 1 2 +2 0 1 2 8 15 3 58 45 0 3 + 182 45 2 0 3 2 5 1 3 26 5 0 1 3 + 8 9 2 3 + 44 9 0 2 3 4 0 2 3
Castingalltheresultswehavesofarforthehierarhy,thefollowingnth-termforthesuperpotentialisindued
W
n+1
=(n+1=2)(1 2
)y(y 2
1)+y
n 2 + d dr log( Q n i=0 f i 1 f n
) ; f
0 =f
1
=1 (39)
where f
n
(y)isa2n-orderpolynomialoftheform
f n (y)= n X a ni y 2i ; a n0
WestressthatsineW
n+1
isasuperpotentialithekstheRiatiequation,
W 2
n+1 W
0
n+1 =V
n+1 (r)
(n+1)
0
(41)
whereV
n+1
(r)isthesuperpartnerpotentialof V
n
whihsatises
W 2
n +W
0
n =V
n+1 (r)
n
0
: (42)
WehavethereforeareursiverelationshipbetweenW
n+1 andW
n
givenby
W 2
n+1 W
0
n+1 =W
2
n +W
0
n +
n
0
(n+1)
0
(43)
where n
0
=
2
n 1
4
and (n+1)
0
=
2
n
4
. Afterthesubstitutionsweend upwiththeondition
2n(1 2
) 2
y 2
(y 2
1) 2
+(y 2
1) 2
((2n 1)
n 1
(2n+1)
n
)(1 (1 2
)y 2
) 2n
+
+ n 1
X
i=0 f
0
i 1
f
i 1
4 f
0
n 1
f
n 1 2
f 0
n
f
n
+2(1 2
)y(y 2
1)+2y 2
(
n
n 1 )
+
+f 0
n 1 =f
n 1
4n(1 2
)y(y 2
1) 2f 0
n =f
n +2y
2
(
n +
n 1 )
(44)
f 0
n =f
n
(1
2
)y(y 2
1)(2n+1)+2y 2
n
+f 00
n =f
n
2 n 1
X
i=0 f
00
i 1
f
i 1 +2
n 1
X
i=0 (
f 0
i 1
f
i 1 )
2
+ 2n(1 2
)(1 3y 2
)
2
(
n +
n 1 )
dy=dr=0:
wheref 0
=df=drandf 00
=d 2
f=dr 2
.
d
Therefore,f
n+1
anbedeterminedfromthe
knowl-edge off
n
. In thisway,thepartiular asesof n=1,
n=2andn=3anbehekedbyinspetionandthe
resulting funtions f
1 , f
2 and f
3
perfetly agree with
equations(29),(36)and(38)respetively. Notiethat
thepartiularasewhen=1triviallyreduesthe
n-thtermsuperpotential,equation (39)to W
n+1 =y
n ,
with
n
= v n, sine all the f's beome 1 one all
the a
in
's are heked to redue to zero. The related
potentialsofthehierarhyarethengivenby
V
n+1
(r)= (v n)(v n+1)=osh 2
r n=0;1;:::
(45)
ThisisknownastheshapeinvariantPoshl-Teller(PT)
potential.
IV Conlusions
The hierarhy of Hamiltonians is studied for the
re-stritedlassofNatanzonpotentials,(Ginohiolass),
withtwoparametersandageneralformforthe
super-potentialisproposed.Thesuperalgebradrivesustothe
onlusionthatthewhole superfamilyisaolletionof
exatlysolvableHamiltonians. Thease=1served
asahekofourformulaeandwasshowntoreduethe
original potential to the Poshl-Teller(PT) potential,
knownto beshapeinvariant.
Asanal remark,theshapeinvarianeonept
in-disussions about the exatly solvable potentials. In
ref. [8℄ there is a disussion aboutthis subjet whih
hasreentlybeenextendedin[14℄onerningpotentials
depending onn parameters. TheNatanzonpotential
is not shape invariant in the usual sense, as most of
the exatly solvable potentials are. However, for the
restritedlassanalisedhere,itwaspossibleto obtain
ageneralform forthe superpotential,asshown in the
previoussetion.
TheHulthenpotentialwithoutthepotentialbarrier
termisanotherexampleofanexatlysolvablepotential
whihisnotshapeinvariant,butforwhihitispossible
todetermineageneralexpressionforthesuperpotential
in thehierarhy,[13℄.
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