Multiple parameter structure of Mielnik's isospectrality in unbroken SUSYQM

12 Pages·0.16 MB·

by  H. C. Rosu

0 MULTIPLE PARAMETER STRUCTURE OF MIELNIK’S ISOSPECTRALITY
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IN UNBROKEN SUSYQM
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HARETC.ROSU
J
[email protected]
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2 Instituto de F´ısica, Universidad de Guanajuato, Apdo Postal E-143, Le´on, Gto, Mexico

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v Abstract. Within unbroken SUSYQM and for zero factorization energy, I present an
7 iterativegeneralizationofMielnik’sisospectralmethodbyemployingaSchro¨dingertrue
0 zeromodeinthefirst-stepgeneralRiccatisolutionandimposingthephysicalcondition
0 ofnormalizationateachiterativestep. Thisprocedureleadstoawell-definedmultiple-
4 parameterstructurewithinMielnik’sconstructionforbothzeromodesandpotentials.
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- Int. J. Theor. Phys. 39, 105-114(January 2000)
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Thesupersymmetricproceduresareaninterestingandfruitfulextensionof(one-
dimensional) quantum mechanics. For recent reviews see [1]. These techniques
are, essentially, factorizations of one-dimensional Schr¨odinger operators, first dis-
cussedin the supersymmetric contextby Witten in 1981[2], andwell knowninthe
mathematicalliteratureinthebroadersenseofDarbouxcovarianceofSchroedinger
equations [3].
In 1984, Mielnik [4] introduced a different factorization of the quantum har-
monic oscillator based on the general Riccati solution. As a result, Mielnik ob-
taineda one-parameterfamily ofpotentialswithexactlythe samespectrumas that
of the harmonic oscillator. However,even though in the same year Nieto discussed
the connection of such a factorization with the inverse scattering approach, and
Ferna´ndez applied it to the hydrogen atom case, Mielnik’s result remained a cu-
riosity for a decade during which only a few authors paid attention to it. On the
other hand, constructing families of strictly isospectral potentials is an important
possibilitywith manypotentialapplicationsin physics[1]. This explainsthe recent
surge of interest in this supersymmetric issue [5]. My goal in this work is to give a
multiple-parametergeneralizationofMielnik’sprocedurebasedontheground-state
function of any soluble one-dimensional quantum mechanical problem. This is just
a form of Crum’s iterations, i.e., repeated Darboux transformations. Some work
along this line has already been done by Keung et al. [6], who performed an itera-
tive construction for the reflectionless, solitonic, sech potentials and the attractive
Coulomb potential presenting relevant plots as well. However,they first go n steps
awayfromagivengroundstateandonlyafterwardsperformthenstepsbackwards.
On the other hand, Pappademos et al. [7], working in the continuum part of the
spectrum,gotone-andtwo-parametersupersymmetricfamiliesofpotentialsstrictly
isospectral with respect to the half-line free particle and Coulomb potentials and
focused on the supersymmetric bound states in the continuum. Their procedure
is closer to the method I will present in the following. For more recent works see
BagrovandSamsonov[8],Ferna´ndezetal[9],JunkerandRoy[10],andRosas-Ortiz
[11].
In the following, I first briefly recall the mathematical background of Mielnik’s
method andnext passto a simple multiple-parametergeneralizationfor the partic-
ular but physically relevant zero-mode case.
I begin with the “fermionic” Riccati (FR) equation y′ = y2 + V (x) [the
1
“bosonic” one being y′ = y2 +V (x)] for which I suppose to k−now a particular
0
solution y . Notice also that I do not put any free constant in the Riccati equa-
0
tions, that is, I work at zero factorization energy. Let us seek the general solution
in the form y = w +y . By substituting y in the FR equation one gets the
1 1 0 1
Bernoulliequation w′ =w2+(2y )w . Furthermore,using w =1/w , we obtain
− 1 1 0 1 ′ 2 1
the simple first-order linear differential equation w (2y )w 1 = 0, which can
be solved by employing the integration factor F0(x2)−= e−0cx22y0−, leading to the so-
x R
lution w (x)=(λ+ F (z)dz)/F (x), where λ occurs as an integrationconstant.
2 c 0 0
In applications the loRwerlimit c is either or 0 depending on whether one deals
−∞
2
with full-line or half-line problems, respectively. In the latter case, λ is restricted
to be a positive number. Thus, the general FR solution reads
e− cx2y0 F0 x
y1 =y0+ λ+ xRe− cz2y0 ≡y0+ λ+ cx(F0) =y0+Dln(cid:18)λ+Zc (F0)(cid:19) , (1)
c R R
R
where D = d . It is easy to reach the conclusion that the particular FR solution
dx
y corresponds to Witten’s superpotential [2], while the general FR solution y =
0 1
y + F0 is of Mielnik type [4]. This is especially clear when one is able to
0 x
λ+ F0
c
identifyRf = F1/2 with the quantum mechanical ground-state wavefunction u of
0 0 0
the problem at hand. This requires suitable asymptotic behaviour of the Riccati
solutiony andapplyingthenormalizationconditiontof ,turningitintoatruezero
0 0
mode. Asis wellknown,this casecorrespondstothe so-calledunbrokenSUSYQM,
which will be assumed to hold henceforth. Moreover, 2y′ ( 2 d2 lnF1/2) is
− 0 ≡ − dx2 0
the Darboux transform contribution to the initial Schr¨odinger potential, i.e., V =
1
′
V 2y . Also, the modes
0− 0
F1/2 u
u (x)= 0 = 0 (2)
λ λ+ xF λ+ xu2
c 0 c 0
R R
can be normalized and therefore considered as ground-state wavefunctions of the
bosonic family of potentials corresponding to Mielnik’s parametric superpotential.
The one-parameter true zero modes read
√ΛF1/2 √Λu
v (x)= 0 = 0 , (3)
λ λ+ xF λ+ xu2
c 0 c 0
R R
where √Λ= λ(λ+1) is the normalization constant.
′
Moreover,p2y canbethoughtofasthegeneralDarbouxtransformcontribution
− 1
totheinitialpotentialgeneratingthebosonicstrictlyisospectralfamily,whichreads
d2 x 4u u′ 2u4
VM =V (x) 2 ln λ+ u2 =V (x) 0 0 + 0 . (4)
λ 0 − dx2 (cid:18) Z 0(cid:19) 0 − λ+ xu2 (λ+ xu2)2
0 0
R R
This family of potentials can be seen as a continuous deformation of the original
potential, because the latter is included in the infinite limit of the deforming pa-
rameter λ and v±∞ =u0 as well. In more intuitive terms, Mielnik’s method based
onanintialSchroedingertruezeromodemaybecalledadoubleDarbouxtechnique
of deleting followed by reinstating a nodeless ground-state wavefunction u (x) of
0
a potential V (x) by means of which one can generate a one-parameter family of
0
isospectral potentials V (x), where λ is a labeling, real parameter of each member
λ
potential in the set.
One can go on with one of the strictly isospectral bosonic zero modes u =
λ1
u0 (i.e., by choosingλ=λ ) and repeatthe strictly isospectralconstruction,
λ1+ cxu20 1
R
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getting a new two-parameter zero mode
u u
u = λ1 = 0 . (5)
λ1,λ2 λ + xu2 (λ + xu2)(λ + xu2 )
2 c λ1 1 c 0 2 c λ1
R R R
The two-parameter true zero modes read
√Λ Λ u
1 2 0
v = . (6)
λ1,λ2 (λ + xu2)(λ + xv2 )
1 0 2 λ1
R R
The resulting two-parameter family of strictly isospectral potentials will be
d2 x x
V =V 2 ln λ + u2 λ + v2 . (7)
λ1,λ2 0− dx2 h(cid:18) 1 Z 0(cid:19)(cid:18) 2 Z λ1(cid:19)i
At the ith -parameter level, one will have
d2 x x x
V =V 2 ln λ + u2 λ + v2 ... λ + v2
λ1,λ2,...λi 0− dx2 h(cid:18) 1 Z 0(cid:19)(cid:18) 2 Z λ1(cid:19) (cid:18) i Z λ1...λi−1(cid:19)i
(8)
and
√Λ Λ ...Λ u
1 2 i 0
v = . (9)
λ1...λi (λ + xu2)...(λ + xv2 )
1 0 i λ1...λi−1
R R
Explicit formulas for the parametric zero modes can be obtained if one uses a
x
notation based on the integration factor F = (x) (c)=∆ . Then
c 0 F −F xF
R
√Λ u
1 0
v (x)= . (10)
λ1 λ +∆
1 x
F
Next, one can calculate
x u2 x ′dx F(x) dz
0 = F = =
Zc (λ+ cxu20)2 Zc (λ−F(c)+F(x))2 ZF(c) (λ−F(c)+z)2
R
1 1 ∆
x
= = F . (11)
λ − λ+∆ λ(λ+∆ )
x x
F F
Thus
√Λ Λ u
1 2 0
v (x)= . (12)
λ1,λ2 λ λ +(λ +λ +1)∆
1 2 1 2 x
F
At the next step one gets
√Λ Λ Λ u
1 2 3 0
v (x)= (13)
λ1,λ2,λ3 λ λ λ +(λ λ +λ λ +λ λ +λ +λ +λ +1)∆
1 2 3 1 2 2 3 3 1 1 2 3 x
F
and the general formula at the i level can be written down in the form
√Λ ....Λ u
1 i 0
v (x)= , (14)
λ1,λ2,...,λi C(i)+C(i)∆
1 2 xF
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where the first coefficient in the denominator is the product of all parameters,
whereasthe secondcoefficient is just the sum over allthe rest of lowerorder Viete-
type expressions in the parameters. By the same token, one can write a general
formula for the strictly isospectral potentials
4C(i)u u′ 2(C(i))2u4
V =V 2D2ln[C(i)+C(i)∆ ]=V 2 0 0 + 2 0 ,
λ1,λ2,...,λi 0− 1 2 xF 0−C(i)+C(i)∆ (C(i)+C(i)∆ )2
1 2 xF 1 2 xF
(15)
which may be considered as the generalization of the furthest right-hand side of
Eq.(4)andrepresentsasimplegeneralizationofMielnik’sone-parameterpotentials.
Since Λ ...Λ =C(i)(C(i) +C(i)), one might think that there is nothing new in
1 i 1 1 2
(14)and(15)withrespecttoacommonMielniksolutionwithaneffectiveparameter
λ(i) =C(i)/C(i). However,Iwillarguethatbyperformingsuchanequivalenceone
eff 1 2
loses a certain type of information. This information is a consequence of the sym-
metry of (14)and (15) in the space of parameters. One cansee that the subindices
ofanypairofparameterscanbeinterchangedwithoutaffectingtheformulas. Thus,
each λ parameter can be varied independently of the others, making it possible to
put questions related to the following type of situation. Suppose we construct two
Mielnikpotentialscorrespondingtoλ andλ andaskwhatisthepotentialbearing
1 2
true zero modes that for λ goes to the Mielnik case for λ , whereas for
1 2
→ ±∞
λ itgoestotheMielnikcaseforλ . Theanswerisprovidedbytheconstruc-
2 1
→±∞
tionofthisworkandcorrespondstotheparticularcaseV bearingthetruezero
λ1,λ2
modes vλ1,λ2. Indeed, as one can easily check, vλ1,±∞ = vλ1 and Vλ1,±∞ = VλM1 ,
whereasv±∞,λ2 =vλ2 andV±∞,λ2 =VλM2. Inthe generalcase,onestartswithaset
of i Mielnik potentials corresponding to i fixed values of Mielnik’s parameter and
asks the same question, this time for the set of i asymptotic limits. The answer is
given by (14) and (15) and cannot be providedif one works with only one effective
parameter unless its multiple-parameter value found above is used.
Another interesting remark is the one-to-one relationship between any polyno-
mial equation a xi +a xi−1 +...+a = 0 and the present iterative construction.
0 1 i
If we consider the λ parameters as the zeros of such arbitrary polynomials, we can
write (14) as
( 1)ia ( i( 1)ia )u
v = q − i 0 − i 0 , (16)
λ1,λ2,...,λi ( 1)ia +( Pi−1( 1)ia )∆
− i 0 − i xF
P
and in (15) one can substitute the same type of denominator as in (16). There is
onlyoneconstraintontheemployedpolynomials,whichoneshouldimposeinorder
to avoid possible singularities. Usually the integral ∆ in the denominators of
x
F
(15) and (16) is of the kink type, i.e., it may be written in the form α+βK(x),
where the function K(x) has the kink behavior, taking values between -1 and +1
andαandβ aresomeconstants,ofwhichαmaybezero. Thentheallowedintervals
for the effective parameter are λ(i) > β α and λ(i) < (β +α). When α = 0
eff − eff −
one gets λ(i) >β.
| eff|
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Itisworthwhiletomentionthatthepreviousiterationprocesscanbeunderstood
most easily from the Riccati equation standpoint as follows. To get, for example,
′
the two-parameter zero mode, one should start again with the FR equation y =
y2 + V (x) and take as the known particular solution y(1) = y + y , where
−yλ1 = λ1+1 F0x(F0). TheintermediateBernoulliequationwillbpe−w1′ =0 w12+λ12yp(1)w1.
c
ThisisturRnedintoafirst-orderdifferentialequationbytheinversefunctionmethod.
(− x2y(1))
The integration factor of the latter is Fλ1 =exp Rc p andxthe solution for the
first-order differential equation can be written w = (λ + F dz)/F . From
2 2 c λ1 λ1
this presentation it is clear how one should proceed for anRarbitrary step. Also,
the logarithmic derivative notation in Eq. (1) is equally convenient to have a clear
image of the iteration process. Thus, one can generate hierarchies of parametric
Schr¨odinger zero modes of any desired order by means of the Riccati connection.
The parametric normalization deletes the interval [ 1,0] from the parameter
space of λ(i) . At the 1 limit, one can make a conn−ection with the Abraham-
eff −
Mosesisospectraltechnique [12], whereasatthe 0limit the connectioncanbe done
with another isospectral construction developed by Pursey[13]. This connection is
only from the point of view of the potentials; the zero modes as worked out here
just disappear.
In conclusion, I have shownexplicitly the way Crum’s iterationworkswhen the
general Riccati solutions (general superpotentials) at zero factorization energy are
basedonthecorrespondingSchr¨odingerground-statewavefunctions,obtaininggen-
eral formulas for this simple ‘generalization’ of Mielnik’s one-parameter SUSYQM
isospectrality. Plots of the two-parameterformulas for the harmonic oscillator case
are presented in Fig. 1-5. One may consider the results of this work as pointing to
an interesting hierarchical structure within the general Riccati solution produced
by a particular type of repeated Darboux transformations when the normalization
condition of quantum mechanics is taken care of at each iterative step.
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Acknowledgments
This work was partially supported by the CONACyT Project 458100-5-25844E. I
wish to thank D.J. Ferna´ndez and B. Mielnik for very useful correspondence.
References
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in Symmetries in Quantum Mechanics and Quantum Optics, to appear (quant-
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(Berlin: Springer, 1996).
[2] E. Witten, Nucl. Phys.B 188 (1981) 513.
[3] V.B. Matveev and M.A. Salle, Darboux Transformations and Solitons (Berlin:
Springer, 1991).
[4] B. Mielnik, J. Math. Phys. 25 (1984) 3387; M.M. Nieto, Phys.Lett. B145 (1984) 208;
D.J. Fern´andez,Lett. Math. Phys. 8 (1984) 337.
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(1997) 2269; Phys.Rev.A 54 (1996) 2571.
[6] W.-Y. Keunget al., J. Phys. A 22 (1989) L987; Q. Wang et al, Mod. Phys. Lett. A 5
(1990) 525.
[7] J. Pappademos, U. Sukhatme,and A.Pagnamenta, Phys. Rev.A 48 (1993) 3525.
[8] V.G. Bagrov and B.F. Samsonov, Phys. Part. Nucl. 28 (1997) 374, and references
therein.
[9] D.J. Fern´andez,V.Hussin andB. Mielnik, Phys.Lett.A 244 (1998) 309; see also D.J.
Fern´andez, J. Negro, and M.A. del Olmo, Ann.Phys.252 (1996) 386; D.J. Fern´andez,
Int.J. Mod. Phys. A 12 (1997) 171.
[10] G. Junkerand P.Roy,Ann.Phys. 270 (1998) 155.
[11] J.O.Rosas-Ortiz,J.Phys.A31(1998) 10163, L507;Ph.D.Thesis1997, CINVESTAV-
IPN, Mexico.
[12] P.B. Abraham and H.E. Moses, Phys.Rev. A 22 (1980) 1333.
[13] D.L. Pursey, Phys.Rev.D 36 (1987) 1103.
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8
6
4
2 5
0
4
-2
-4 3
-4 2 lambda1
-2
x0 2 1
4
Fig. 1
Two-parameter strictly isospectral harmonic oscillator potentials (¯h=m=ω=1), for
fixedλ =0.2 and λ ∈[0.1,5]. They are identical to Mielnik harmonic oscillator
2 1
potentials with λ(2) ∈[0.0154,0.1613].
eff
8
0.8
0.6
0.4
0.2
5
4
0 3
-4 2 lambda1
-2
0 1
x 2 4
Fig. 2
The corresponding truezero modes.
9
0.92
0.9
0.88
0.86
0.84
0.82
0.8
0.78
0.76
0.74
0.72
0.1 0.1
0.2 0.2
0.3 0.3
lambda2 lambda1
0.4 0.4
0.5 0.5
Fig. 3
The two-parameter true zero modes at fixed x=−1.4 as a function of thetwo
parameters.
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