A PT Symmetric QES Partner to the Khare Mandal Potential With Real Eigen Values

a r X i v :q u a n t -p h /0107095v 1 19 J u l 2001A PT-symmetric QES partner to the Khare-Mandalpotential with real eigenvaluesB.Bagchi a ∗,S.Mallik a ,C.Quesne b †&R.Roychoudhury c ‡aDepartment of Applied Mathematics,University of Calcutta,92Acharya Prafulla Chandra Road,Kolkata 700009,IndiabPhysique Nucl´e aire Th´e orique et Physique Math´e matique,Universit´e Libre de Bruxelles,Campus de la Plaine CP229,Boulevard du Triomphe,B-1050Brussels,BelgiumcPhysics &Applied Mathematics Unit,Indian Statistical Institute,203B.T.Road,Kolkata 700035,IndiaABSTRACTWe consider a PT-symmetric partner to Khare-Mandal’s recently proposed non-Hermitian potential with complex eigenvalues.Our potential,which is quasi-exactly solvable,is shown to possess only real eigenvalues.PACS :03.65.Bz,03.65.GeKeywords :quantum mechanics,PT-symmetry,quasi-solvableCorresponding author : B.Bagchi,Department of Applied Mathematics,University of Calcutta,92Acharya Prafulla Chandra Road,Kolkata 700009Fax :(0091)(033)5545741E-mail :bbagchi @cucc.ernet.inExploration of non-Hermitian Hamiltonians,in particular the PT-symmetric ones,is currently a topic of active research interest(see for example,[1]-[14]).As is well known, PT-symmetric Hamiltonians are conjectured[15]to preserve the reality of their bound state eigenvalues except possibly for situations when PT may be spontaneously broken. It should be noted that PT-invariance in itself is not a sufficient condition for the Hamil-tonian to possess an entirely real spectrum[1,2].Recently,Khare and Mandal(KM)have enquired[9]into the invariance of a non-Hermitian Hamiltonian under the combined operations of a complex shift(x→a−x, a=iπ−(ζcosh2x−iM)2(3)dx22H(−)=−d 2dx 2−ζ2log z ,H (±)becomeH (±)=−4z 2d 2dz−ζz−iM2.(6)As such if we setµ(±)(z )=z(1−M )/2ei ζz)(7)the Hamiltonians (6)can be mapped to their gauge-transformed formsH (±)g=µ(±)(z )−1H(±)µ(±)(z ).(8)From the relationsµ−1ddz +µ′dz2µ=d 2µd µ(10)where primes denote differentiations with respect to z ,we easily obtainH (±)g=−4z2d2µz +1dµ−4zµ′2(z ±1µ=1−M41∓1µ=µ′2z 2±iζz 3(13)3we arrive at the following representations of H(±)g:H(±)g =−4z2d2dz+2iζ(M−1)z+2M−1∓ζ2.(14)We thusfind that the Schr¨o dinger equationH(±)ψ(±)(x)=E(±)ψ(±)(x)(15)is equivalent toH(±)gφ(±)(z)=E(±)φ(±)(z)(16) whereψ(±)(x)=µ(±)(z)φ(±)(z).It should be noted here that in terms of the sℓ(2,R)generators[16]J+=z2ddz−j,J−=d2)in the true spirit of quasi-solvability[16].We now turn to some specific cases ofφ(±)(z)by focussing on the following choices: (i)φ(±)(z)=c(±)0(ii)φ(±)(z)=c(±)0+c(±)1z c(±)1=0(iii)φ(±)(z)=c(±)0+c(±)1z+c(±)2z2 c(±)2=0(iv)φ(±)(z)=c(±)0+c(±)1z+c(±)2z2+c(±)3z3 c(±)3=0where c(±)i(i=0,1,2,3)are constants.It is obvious that we can generalizeφ(±)(z)to higher degrees of z apart from the ones chosen here.First considerφ(±)(z)=c(±)0.For this case,Eq.(16)becomes2iζ(M−1)z+2M−1∓ζ2−E(±)=0(19) leading to2ζ(M−1)=0(20)42M−1∓ζ2−E(±)=0.(21) Hence M=1(j=0)and as a resultE(±)=1∓ζ2.(22) The accompanying wave functions readψ(+)∝e iζsinh2x.(24)2We therefore see that for M=1the energy eigenvalues corresponding to(1)as well as its modified PT-symmetric version(2)are real.Next considerφ(±)(z)=c(±)0+c(±)1z.Eq.(16)gives2iζ(M−2)c(±)1=0(25) 2iζ(M−1)c(±)0+ 6M−9∓ζ2−E(±) c(±)1=0(26)±2iζc(±)1+ 2M−1∓ζ2−E(±) c(±)0=0.(27) Eq.(25)implies M=2while Eqs.(26)and(27)give2iζc(±)0−ǫ(±)c(±)1=0(28)−ǫ(±)c(±)0±2iζc(±)1=0(29) whereǫ(±)=E(±)−3±ζ2.Solving(28)and(29)we getǫ(+)±=±2iζ(complex)andǫ(−)±=±2ζ(real).We thus 1find for M=2(j=cosh2x(e−x±e x)(31)2for the KM potential and2(32)E(−)±=3±2ζ+ζ5iζψ(−)±∝e1−4ζ2(41)∝e iζψ(+)cosh2x 2cosh2x−i1−4ζ2 (43) 26for the KM potential andE(−) 0=5+ζ2,E(−)±=7+ζ2±22sinh2x cosh2x(45)ψ(−)±∝e iζζ 1±2,all three eigenvalues of the PT-symmetric potential(2)remain real for all values ofζ. We now take up the case(iv)namelyφ(±)(z)=c(±)0+c(±)1z+c(±)2z2+c(±)3z3for which we obtain from Eq.(16)the relations2iζ(M−4)c(±)3=0(47) 2iζ(M−3)c(±)2+ 14M−49∓ζ2−E(±) c(±)3=0(48)2iζ(M−2)c(±)1+ 10M−25∓ζ2−E(±) c(±)2±6iζc(±)3=0(49)2iζ(M−1)c(±)0+ 6M−9∓ζ2−E(±) c(±)1±4iζc(±)2=0(50)±2iζc(±)1+ 2M−1∓ζ2−E(±) c(±)0=0.(51) We are thus led to M=4and definingǫ(±)=E(±)−15±ζ2,we get2iζc(±)2− ǫ(±)+8 c(±)3=0(52)4iζc(±)1−ǫ(±)c(±)2±6iζc(±)3=0(53)6iζc(±)0−ǫ(±)c(±)1±4iζc(±)2=0(54)7− ǫ(±)+8 c(±)0±2iζc(±)1=0.(55) An analysis of Eqs.(52)-(55)reveals that for consistency of the equationsǫ(±)have to fulfil the conditionǫ(±)+8 ǫ(±)+8 ǫ(±)2±16ζ2 ±24ζ2ǫ(±) +144ζ4=0.(56) This fourth-degree equation can be factorized into two quadratic ones namelyǫ(+)+8 ǫ(+)±4iζ +12ζ2=0(57)ǫ(−)+8 ǫ(−)±4ζ −12ζ2=0.(58) Let us now consider the solutions of(57)and(58).It is obvious that in the former case,all the four solutions are complex.This corresponds to the KM scenario.However, in the latter case(which corresponds to the PT-symmetric model(2)),we get quadratic equations inǫ(−)with real coefficients:ǫ(−)2+4(2±ζ)ǫ(−)+4ζ(±8−3ζ)=0.(59) For the upper signs we obtain two real solutions for anyζ:=−2(2+ζ)±4ǫ(−)+,±1+ζ+ζ2.(61) Note that we can combine the four real solutions given by(60)and(61)in the manner=−2(2+σζ)+4τǫ(−)σ,τ)2 are=11−2σζ+ζ2+4τE(−)σ,τψ(−)σ,τ∝e iζζ 1+τn!t n,t=±zSM thanks the Council of Scientific and Industrial Research,New Delhi for the award of a fellowship.CQ is a Research Director of the National Fund for Scientific Research (FNRS),Belgium.References[1]C.M.Bender,S.Boettcher:Phys.Rev.Lett80(1998)5243.[2]C.M.Bender,S.Boettcher:J.Phys A31(1998)L273.[3]E.Delabaere,F.Pham:Phys.Lett A250(1998)25,29.[4]M.Znojil:Phys.Lett A259(1999)220.[5]B.Bagchi,R.Roychoudhury:J.Phys A33(2000)L1.[6]M.Znojil:J.Phys A33(2000)L61.[7]M.Znojil:J.Phys A33(2000)4561.[8]B.Bagchi,F.Cannata,C.Quesne:Phys.Lett A269(2000)79.[9]A.Khare,B.P.Mandal:Phys.Lett A272(2000)53.[10]B.Bagchi,C.Quesne:Phys.Lett A273(2000)285.[11]F.Cannata,M.Ioffe,R.Roychoudhury,P.Roy:Phys.Lett A281(2001)305.[12]G.L´e vai,F.Cannata,A.Ventura:J.Phys A34(2001)839.[13]Z.Ahmed:Phys.Lett A282(2001)343.[14]P.Dorey,C.Dunning,R.Tateo:J.Phys.A34(2001)L391.[15]D.Bessis:Unpublished(1992).[16]hveridze:Quasi-Exactly Solvable Models in Quantum Mechanics,Instituteof Physics Publishing,Bristol(1994).10。