On Spectrum Generating Algebra of the Heun Operator

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by  Priyasri Kar

On Spectrum Generating Algebra of the Heun Operator
PriyasriKara,RiteshK.Singhb,AnandaDasguptac,PrasantaK.Panigrahid
DepartmentofPhysicalSciences,
IndianInstituteofScienceEducationandResearchKolkata,Mohanpur741246,India
a [email protected],b [email protected],
c [email protected],d [email protected]
April6,2016
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Abstract
r
p
A
TheHeunoperatorhasbeencast,intermsoftheelementsofanunderlyingsu(1,1)algebra,
5 underspecificparametricconditions,forthepurposeofspectrumgeneration. Theseelements
aredifferentialoperatorsofdegrees±1/2and0.Itisfoundthattheregularsingularitiesat0and

] ∞ofthegeneralHeunequationmustbeelementaryundertherequiredparametricconditions.
h
Thespectrumgenerationhasbeendemonstratedthroughasetofexamples.
p
-
t
n
Differential equations, in general, do not admit exact solutions. Several techniques have been de-
a
u vised, to obtain partial, or in certain cases, the complete eigenspace of the differential operator
q
under study [1]. In this regard, the well known factorization method, which at a fundamental
[
levelreducestheorderofthedifferentialoperatorinvolved,hasfoundextensiveapplications[2].

2
The harmonic oscillator spectrum generating Heisenberg algebra, the Darboux transformation,
v
1 factorization of the hypergeometric equation by Schro¨dinger [3] and subsequent study of a large
0 classofequationsbyInfeldandHull[2]serveasclassicexamples. Supersymmetricquantumme-
6
chanics(SUSYQM)usesfactorization,torevealunexpectedinterconnectionsamongstapparently
3
0 differentquantalproblems[4,5,6].
.
1
0 Grouptheoreticalstructuresalsohaveplayedsignificantroleinunravellingthesymmetryofdif-
6
ferentialoperators[1]. Apartialalgebraizationofthewholesolutionspacehasbeenachieved[7,
1
8,9]forawideclassofquantalquasi-exactlysolvable(QES)Hamiltoniansofoneormoredimen-
:
v
sions, which are essentially hermitian differential operators of the second order. These Hamil-
i
X tonians, or their appropriately similarity transformed versions, often take the form of some well
r known differential operators, e.g., Heun, Lame´ etc. Hence, the study of the underlying symme-
a
tries of these operators is of immense importance and several attempts have been made to this
end. For example, the Lame´ equation has been expressed as a bilinear in both su(2) [10] and
su(1,1) [11] generators. The latter case relates the non-unitary representations of su(1,1), to the
eigenfunction of the periodic Lame´ potential. Closed form solutions for confluent hypergeomet-
ricandhypergeometricequationshavebeenfound[12],usingaconnectionbetweenthespaceof
monomials and the solution space of the above equations [13, 14], which reveals the underlying
su(1,1)anddeformedsl(2)structures,respectively[15].
Recently, the Heun operator has been cast [16] in terms of the elements of cubic deformations
of sl(2) algebra. This has led to two known solutions of a Heun equation, encountered earlier
byChristandLee[17]. However,non-lineardeformationsofsl(2),areassociatedwithnon-trivial
1
representationtheory[18]. Castingthedifferentialoperatorintermsoflinearsl(2)algebraenables
one to exploit available representation theory. Linear su(1,1) symmetry of the Heun operator
is present in the literature [7]. In the present work, we use a different representation of su(1,1)
generators. Under certain parametric conditions, this reveals a new symmtery of the equation
and leads to a number of solutions, that are unavailable from the methods in [7]. It is found that
this new symmetry exists if the singularities at z = 0 and z = ∞ are elementary1. Under these
conditions, an exact map is established between the Heun solution space and the unitary and
non-unitaryrepresentationsofsu(1,1)algebra.
TheHeunequationisgivenby[20]
d2y(z) (cid:18)γ δ ε (cid:19) dy(z) αβz−q
Hy(z) = + + + + y(z) = 0, (1)
dz2 z z−1 z−a dz z(z−1)(z−a)
with regular singularities at z = 0, 1, a((cid:54)= 0,1)and ∞; the exponents at these singularities being
(0, 1−γ),(0, 1−δ),(0, 1−ε)and(α, β),respectively. Here,qplaystheroleoftheeigenparameter.
In this work, we study Heun equations with only real parameters. The equation, being a second
orderlinearhomogeneousequationwith4regularsingularities,satisfiestheFuchsiancondition:
γ +δ+ε = α+β +1, (2)
allowingeliminationofεinfavoroftheothers. Eq.(1)canbewrittenas:
(cid:20) d2 d (cid:21)
Hy(z) = f (z) +f (z) +f (z) y(z) = 0, (3)
1 dz2 2 dz 3
where,f (z) = a z3+a z2+a z, f (z) = a z2+a z+a and f (z) = a z+a . Theparameters
1 0 1 2 2 3 4 5 3 6 7
a ∈ R, fori = 0,...,7aregivenby,
i
a =1 a = 1+α+β a = αβ (4a)
0 3 6
a =−(a+1) a = −[aγ +aδ−δ+α+β +1] a = −q (4b)
1 4 7
a =a a = aγ. (4c)
2 5
Evidently,theHeunequationconsistsofdifferentialoperatorsofdegrees2 +1, 0and−1,whichare
denotedbyP ,F(P )andP ,respectively. Eq.(3)canthenberewrittenas[16],
+ 0 −
Hy(z) = [P +F(P )+P ]y(z) = 0 (5)
+ 0 −
where,
d2 d d d2 d
P = a z3 +a z2 +a z, P = z −j, P = a z +a (6)
+ 0 dz2 3 dz 6 0 dz − 2 dz2 5dz
and F(P ) = a P2+((2j −1)a +a )P +(j(j −1)a +ja +a ). (7)
0 1 0 1 4 0 1 4 7
Theaboveoperatorssatisfycubicdeformationofsl(2)algebraamongthemselves,whichhasbeen
exploited[16]totractapartoftheeigenspaceofaHeunoperatoranalytically,asmentionedearlier.
WiththeaimtocasttheHeunoperatorintermsoflinearsu(1,1)algebra,wedefinetheconstituent
operatorsas,
d √ d d 2ν
E = 2z3/2 +2µ z, H = 2z +µ+ν and E = 2z1/2 + √ ; (8)
+ −
dz dz dz z
1Anyregularsingularitycanbecharacterizedbytwoexponentsρ1 andρ2,whicharethetworootsoftheindicial
equation. The cases with |ρ1 −ρ2| = 1/2 are known as elementary singularity and are of special significance. All
regularandirregularsingularitiescanbeobtainedbythecoalescenceoftwoandthreeormoreelementarysingularities,
respectively[19].
2Thedegreed,ofanoperatorOd,isdefinedasthechangeinthepowerofamonomial,whenacteduponbyit,i.e.,
Odzp ∝zp+d.
where µ, ν and λ are parameters. The operators E , H and E , have degrees +1/2, 0 and −1/2,
+ −
respectively. Theyarefoundtosatisfysu(1,1)algebra,withcommutators,
[H,E ] = ±E , [E ,E ] = −2H (9)
± ± + −
andtheCasimir:
1
C(µ,ν) = (E E +E E )−H2 = −(µ−ν)(µ−ν −1). (10)
+ − − +
2
The problem in question, determines the parameters of the constituent operators, which deter-
mine the Casimir value. The Casimir value, in turn, determines the representations available for
thealgebraassociatedwiththeproblem.
We now proceed to construct the differential operators P ,P and F(P ), from the constituent
+ − 0
su(1,1) generators and identify the conditions, under which this is possible. The constituent op-
erators are of degrees ±1/2 and 0, whereas, the degrees of operators P and F(P ) are ±1 and
± 0
0, respectively. Hence, it is clear that the complete Heun operator will comprise of linear and
quadraticformsoftheconstituentoperators3. Assumingtheforms,
P = c E E and P = c E E (11)
+ + + + − − − −
wherec areconstants,weobtain,
±
a a
0 0
a = 4c , a = 2c (3+4µ) = (3+4µ), a = 2c µ(1+2µ) = µ(1+2µ), (12)
0 + 3 + 6 +
2 2
a
2
a = 4c , a = 2c (1+4ν) = (1+4ν), 2c ν(2ν −1) = 0. (13)
2 − 5 − −
2
UsingEqs.(12)and(4a),onecansolveforµ,α,β:
1
|α−β| = , (14)
2
implyingthesingularityatz = ∞iselementary. Similarly,usingEqs.(13)and(4c),oneisleftwith
justtwovaluesofν:
ν = 0 =⇒ a /a = γ = 1/2 or, ν = 1/2 =⇒ a /a = γ = 3/2. (15)
5 2 5 2
Above solutions of γ imply that the singularity at z = 0 is also elementary. Finally, F(P ) can be
0
writtenintheform
F(P ) = c HH +c H +c , (16)
0 2 1 0
where,
a = 4c , a = 2(c +2c (1+µ+ν)) and a = −q = c +(µ+ν)(c +c (µ+ν)). (17)
1 2 4 1 2 7 0 1 2
Thus, a Heun operator with elementary singularities at z = 0 and z = ∞, i.e., with parametric
conditionsEqs.(14)and(15),becomes,
H = c E E +c E E +c HH +c H +c . (18)
+ + + − − − 2 1 0
The choices of E and E fix the values of µ and ν, respectively, which in turn determines the
+ −
Casimir, C = C(µ,ν). Finding the solution to Heun equation, then boils down to identifying a
suitable representation space V of su(1,1) and finding a y ∈ V , such that, Hy = 0. For su(1,1)
C C
algebra, there are five different classes of representation spaces [21] determined by the values of
C andh,theeigenvalueofoperatorH. TheseareshowninFig.1andarelistedbelow.
3Cubicandquarticcombinations,suchasE+E+H,E+E+HH,E+E+E+E− (andsimilartermswithE+ andE−
interchanged),arealsoofthedesireddegreesandthereforecouldbeusedintheconstructionoftheHeunoperators.
However,theycontaindifferentialoperatorsofhigherorders,notpresentintheHeunequation.Thus,thecoefficientsof
thoseundesiredtermsmustbeputtozero,whichrevealednomoreparametricfreedomthanthechosencombinations.
(Quadruplet) (Triplet) (Doublet) (Singlet) (PD/ND/PS)
−15/4 −2 −3/4 0 1/4
(PD/ND) (PS)
Casimir (Real line) (PD/ND/CS)
Figure 1: Casimir value for su(1,1) and corresponding representations are listed: positive discrete (PD),
negativediscrete(ND),principalseries(PS),complementaryseries(CS)andsomefinitedimensionalones
(opencircles).
Principalseries(PS):Infinitedimensionalspace,C ≥ 1/4,h ∈ (−∞,∞)and(C,h) (cid:54)= (1/4,1/2),
Complementaryseries(CS):Infinitedimensionalspace,C ∈ (0,1/4),h ∈ (−∞,∞),h (cid:54)= 1/2,
Positivediscrete(PD):Infinitedimensionalspace,C ≤ 1/4,h ≥ 0,
Negativediscrete(ND):Infinitedimensionalspace,C ≤ 1/4,h ≤ 0,
(cid:104) (cid:105)
Finitedimensions: Finitedimensionalspace,C = −(n2−1),h ∈ −(n−1), (n−1) withn ∈ N.
4 2 2
For a given value of Casimir C, one of the available representation space V can be chosen. The
C
elements of this space are |C,h(cid:105) ∝ zp, with p = (h−µ−ν)/2. We represent the solution y as a
linearcombinationofzp fromV andfindtheco-efficients. DuetothequadraticdependenceofH
C
onE andE ,wenoticethat,thesolutionspacesplitsasV = V ⊕V —theevenandoddstates
+ − C e o
oftherepresentationspace. TheprocessoffindingsolutionsinV andV isdemonstratedbelow,
e o
throughasetofexamples.
Example 1: We choose γ = 1/2, δ = ε = −1/2, αβ = 1/2. This leads to |α−β| = 1/2, ensuring
thesingularitiesat0and∞tobeelementary. TheHeunequationtakestheform
d2y 1 (cid:18)1 1 1 (cid:19) dy z−2q
+ − − + y = 0, (19)
dz2 2 z z−1 z−a dz 2z(z−1)(z−a)
withµ = −1,ν = 0i.e.,C = −2. Thisallowsthepositivediscreteandnegativediscreterepresen-
tationspacesalongwithatripletrepresentationwithh ∈ {−1,0,+1}orp ∈ {0,1/2,1}. Choosing
thetripletspace,V correspondstop ∈ {0,1}andV top ∈ {1/2},leadingtothesolutions
e o
√ √
y ∈ V = z+ a, witheigenvalue q = + a/2, (20)
1 e
√ √
y ∈ V = z− a, witheigenvalue q = − a/2 (21)
2 e
√
and y ∈ V = z, witheigenvalue q = (a+1)/4. (22)
3 o
The positive discrete and negative discrete representations also yield solutions, not explicitly
shownforthisexample.
Example 2: We choose γ = 3/2, δ = ε = −1/2, αβ = 0, leading to |α − β| = 1/2. The Heun
equationisgivenby,
d2y 1 (cid:18)3 1 1 (cid:19) dy q
+ − − + y = 0 (23)
dz2 2 z z−1 z−a dz z(z−1)(z−a)
andcorrespondstoµ = −1/2,ν = 1/2withC = −2, asinthepreviouscase. Here, forthetriplet
space,oneobtainsp ∈ {−1/2,0,1/2},withp ∈ {0}beingV . Thesolutionsare:
o
√ (cid:112) √
y ∈ V = z+ a/z, witheigenvalue q = −(a+1)/4+ a/2, (24)
1 e
√ (cid:112) √
y ∈ V = z− a/z, witheigenvalue q = −(a+1)/4− a/2 (25)
2 e
and y ∈ V = const., witheigenvalue q = 0. (26)
3 o
Heretoo,wehavepositivediscreteandnegativediscreterepresentations,likethepreviousexam-
ple.
Example3: Next,weanalyzetheLame´ equation,whichisaspecialcaseoftheHeunequation:
d2y 1 (cid:18)1 1 1 (cid:19) dy q+ρ(ρ+1)z/4
+ + + − y(z) = 0. (27)
dz2 2 z z−1 z−a dz z(z−1)(z−a)
Here, the singularities at z = 0, 1 and a are elementary and choosing ρ(ρ + 1) = 0, makes the
singularity at z = ∞ also elementary. With this choice, we have µ = 0 = ν, i.e., C = 0, giving us
thepossibilitytochoosefromsinglet,positivediscreteandnegativediscreterepresentations. The
singlet solution is, y = const., with q = 0. The positive discrete space splits into the even (V+)
0 e
andtheodd(V+)subspacesandthecorrespondingsolutions,forachoseneigenvalueq ∈ R,are:
o
(cid:34)z q+a+1 (cid:16)z(cid:17)2 2q2+10q(a+1)+8(cid:0)a2+1(cid:1)+7a (cid:16)z(cid:17)3 (cid:35)
y ∈ V+ = 1+2q + + +... (28)
1 e a 3 a 45 a
√ (cid:34) 4q+a+1 (cid:16)z(cid:17) 16q2+40q(a+1)+9(cid:0)a2+1(cid:1)+6a (cid:16)z(cid:17)2 (cid:35)
y ∈ V+ = z 1+ + +... . (29)
2 o 6 a 120 a
Similarly,forthenegativediscretespacetoo,wehavetheeven(V−)andtheodd(V−)subspaces
e o
andthecorrespondingsolutions,forachoseneigenvalueq ∈ R,are:
(cid:34)1 q+a+1 (cid:18) 1 (cid:19) 2q2+10q(a+1)+8(cid:0)a2+1(cid:1)+7a (cid:18) 1 (cid:19) (cid:35)
y ∈ V− = 1+2q + + +...(30)
3 e z 3 z2 45 z3
1 (cid:34) 4q+a+1 (cid:18)1(cid:19) 16q2+40q(a+1)+9(cid:0)a2+1(cid:1)+6a (cid:18) 1 (cid:19) (cid:35)
y ∈ V− = √ 1+ + +... .(31)
4 o z 6 z 120 z2
Thedomainofconvergenceforthepositivediscretespacesolutionsis0 < z < min{1,a},whereas,
that for the negative discrete space solutions is max{1,a} < z < ∞. The solutions in infinite
dimensional spaces can always be written as a power series in z or 1/z, with an overall factor of
zp,p ∈ R,liketheabovefoursolutions.
For all the above examples, we have obtained solutions, which are not available from the previ-
ously known su(1,1) symmetry of the Heun equation [7]. These solutions are presented here by
√
Eqs. (22), (24), (25), (29) and (31). It may be noted that all of them involve powers of z, which
reflects the advantage of using a different representation of the su(1,1) generators in the present
work.
In conclusion, we have found that, the Heun operator can be exactly cast as a quadratic polyno-
mial of elements of su(1,1) algebra, if the singularities at z = 0 and z = ∞ are elementary. This
allows one to use the representations of su(1,1) to find explicit solutions of Heun equation. The
√
finite dimensional representations yield polynomial solutions in z, while positive and negative
discreterepresentationsgivepowerseriessolutionsinz and1/z,respectively,moduloanoverall
factor of zp. Heun equation is connected with a host of physical problems, related to quantum
mechanical and non-linear systems. Hence, the relevance of the algebraically generated solution
spacetotheseproblemsneedscarefulconsiderations. Weintendtoreturntosomeoftheseprob-
lemsinnearfuture.
Acknowledgements: WethankAruneshRoyforusefuldiscussions.
References
[1] WillardMillerJr.,SymmetryGroupsandtheirApplications,AcademicPress(1972).
[2] L.InfeldandT.E.Hull,Rev.Mod.Phys.23,1(1951).
[3] E.Schro¨dinger,Proc.Roy.IrishAcad.A47,53(1941).
[4] F.Cooper,A.KhareandU.Sukhatme,Phys.Rep.251,267(1995).
[5] F. Cooper, A. Khare and U. Sukhatme, Supersymmetry in Quantum Mechanics, World Scientific
(2001).
[6] A. Gangopadhyaya, J. V. Mallow, C. Rasinariu, Supersymmetic Quantum Mechanics: An Intro-
duction,WorldScientific(2010).
[7] A.V.Turbiner,Commun.Math.Phys.118,467(1988).
[8] M.A.ShifmanandA.V.Turbiner,Commun.Math.Phys.126,347(1989).
[9] A. G. Ushveridze, Quasi-exactly Solvable Models in Quantum Mechanics, Inst. of Physics Publ.
(1994).
[10] Y.Alhassid,F.GurseyandF.Iachello,Phys.Rev.Lett.50,873(1983).
[11] H.LiandD.Kusnezov,Phys.Rev.Lett.83,1283(1999).
[12] N.Gurappa,P.K.JhaandP.K.Panigrahi,SIGMA3,057(2007).
[13] N.GurappaandP.K.Panigrahi,Phys.Rev.B62,1943(2000).
[14] N.GurappaandP.K.Panigrahi,J.Phys.A37,L605(2004).
[15] T.Shreecharan,P.K.PanigrahiandJ.Banerji,Phys.Rev.A69,012102(2004).
[16] A.Roy,A.SenandP.K.Panigrahi,Preprint[arXiv:1304.2225].
[17] N.H.ChristandT.D.Lee,Phys.Rev.D12,1606(1975).
[18] M.Rocek,Phys.Lett.B255,554(1991).
[19] E.L.Ince,OrdinaryDifferentialEquations,DoverBooksonMathematics(1926).
[20] A.Ronveaux(ed.),Heun’sDifferentialEquations,OxfordUniversityPress(1995).
[21] Seeforexample,W.Groenevelt,Indag.Math.(N.S.)14,329(2003)andreferencestherein.