MHD alpha^2-dynamo, Squire equation and PT-symmetr

arXiv:math-ph/0501069v221Mar2005MHDα2−dynamo,SquireequationandPT−symmetricinterpolationbetweensquarewellandharmonicoscillatorUweG¨unthera∗,FrankStefania†andMiloslavZnojilb‡aResearchCenterRossendorf,P.O.Box510119,D-01314Dresden,Germanyb´Ustavjadern´efyzikyAVˇCR,25068ˇReˇz,CzechRepublic27January2005AbstractItisshownthattheα2−dynamoofMagnetohydrodynamics,thehydrodynamicSquireequationaswellasaninterpolationmodelofPT−symmetricQuantumMechanicsarecloselyrelatedasspectralproblemsinKreinspaces.Fortheα2−dynamoandthePT−symmetricmodelthestrongsimilaritiesaredemonstratedwiththehelpofa2×2operatormatrixrepresentation,whereastheSquireequationisre-interpretedasarescaledandWick-rotatedPT−symmetricproblem.BasedonrecentresultsontheSquireequationthespectrumofthePT−symmetricinterpolationmodelisanalyzedindetailandtheHerbstlimitisdescribedasspectralsingularity.1IntroductionNon-HermitianPT−symmetricquantummechanicalsystems[1,2,3,4,5,6,7,8]areknowntopossessspectralsectorswithpurelyrealeigenvaluesaswellassectorswithpairsofcomplexconjugateeigenvalues.Changesofcertainsystemparameterscanleadtospectralphasetransitionsfromonesectortotheother.ThephysicsinthetwosectorshasbeenidentifiedwithphasesofunbrokenPT−symmetry(realeigenvalues)andspontaneouslybrokenPT−symmetry(pairwisecomplexconjugateeigenvalues)[1,2].Fromamathematicalpointofview,non-HermitianPT−symmetricHamiltoniansareself-adjointoperatorsinKreinspaces[9,10,11]—Hilbertspaceswithanadditionalindefinitemetricstructure—andthetwospectralsectorscorrespondtoKreinspacestatesofpositiveornegativetype(realeigenvalues)andneutral(isotropic)states(pairwisecomplexconjugateeigenvalues).ApartfromPT−symmetricQuantumMechanics(PTSQM),itisknownthatacertainclassofsphericallysymmetricmean-fielddynamomodels[12]ofMagnetohydrodynamics(MHD)canbedescribedbyself-adjointoperatorsinKreinspacesaswell[13].Thesemodelsshowsimilarspectralphasetransitionsfromrealtopairwisecomplexconjugateeigenvalues[14]—andonlythephysicalinterpretationdiffersfromthatinPTSQM.Fordynamositsimplyconsistsinatransitionfromnon-oscillatorystatestooscillatorystates.Inthepresentpaper,wearegoingtobrieflydescribetheunderlyingstructuraloperatortheoreticparallelsbetweenPTSQMmodelsandthesphericallysymmetricMHDα2−dynamo(section2).ThediscussionwillbeillustratedwiththehelpofaPT−symmetricinterpolationbetweenaharmonicoscillatorplacedinasquarewellandanemptysquarewell(section3).Thisinterpolationshowsarichstructureofspectralphasetransitionswithacoupleofunexpectedfeatures.Furthermore,wewillshowinsect.4thattheeigenvalueproblemofthePT−symmetric(intermediate)interpolationmodelwithlinearcomplexpotential(purelycomplexelectricalfield)withinthesquarewellismathematicallyidenticaltotheeigenvalueproblemoftherescaledandWick-rotatedSquireequationofhydrodynamicswhichdescribesthenormalvorticityofaplanechannelflow(Couetteflow)withlineartransversalvelocityprofile.RecentAiryfunctionbasedresultsontheSquireequationallowustoanalyticallydescribethespectralbehaviorofthePTSQMmodelinthelimitingcasewhenthewidthofthesquarewelltendstoinfinity.InthislimitwereproducetheHerbstmodel[15]withitsemptyspectrum.Thelimitingbehavioroccursasablowing-upofthespectrumtoinfinityalongthreedirectionsonthecomplex∗e-mail:[e-mail]†e-mail:[e-mail]‡e-mail:[e-mail]1plane—leavingbehindaspectrallyemptyregionatanyfixedfinitedistancefromtheoriginofthespectralplane.Insection5webrieflysketchsomelinksoftheobtainedresultstootherphysicalsetupsandanalyticaltechniques.2KreinspacepropertiesofPT−symmetricquantummodelsandofthesphericallysymmetricMHDα2−dynamo2.1PT−symmetricquantummodelsIntheirseminalletter[1]BenderandBoettcheridentifiedPT−symmetryastheessentialpropertyofthenon-HermitianquantumsystemHψ(x)=Eψ(x),H=−d2dx2+gx2(ix)ν(1)whichensurestherealityofitsspectrumforexponentsν∈[0,2)andψ(x)∈˜H=L2(−∞,∞)[16].ThisallowedthemnotonlytoextendanearlierconjectureofBessisandZinn-Justin(whosenumericalresultsindicatedthatquantumsystemswithcomplexpotentialV(x)=ix3mighthaveapurelyrealspectrum),butalsoinitiatedthestilllastingintensivestudyofgeneralizedPT−symmetricnon-Hermitiansystems[17].SuchsystemsarecharacterizedbyaPT−symmetricHamiltonianH,[PT,H]=0(2)wherePdenotesareflectionPxP=−x,Pψ(x)=ψ(−x)(3)whilethetime-reversaloperatorTperformscomplexconjugationTiT=−i,Tψ(x)=ψ(x)∗.(4)BecausebothoperatorsPandTareinvolutionoperators,P2=I,T2=I,(5)theyinducenaturalZ2−gradingsoftheHilbertspace˜H.Foroursubsequentanalysisitsufficestoconsiderthesubclassofmodelswhichcanbedefinedsolelyoverthereallinex∈R.ForsuchmodelstheT−inducedZ2−gradingcorrespondstoasplittingofthewavefunctionsψ∈˜Hintorealandimaginarycomponents(whatisofnodirectphysicalinterestinaquantummechanicalcontext;additionallyonewouldhavetoworkinarealHilbertspacewithdoubleddimensioncomparedtotheoriginalcomplexone),whereasPinducesaZ2−gradingintoparityevenandparityoddcomponentsψ(x)=ψ+(x)+ψ−(x),Pψ±(x)=ψ±(−x)=±ψ±(x).(6)ThecorrespondingZ2−gradedHilbertspacesplitsas˜H=H+⊕H−,ψ±∈H±.(7)InthecaseofasimplePT−symmetricone-particlesystemwithHamiltonianH=−∂2x+V+(x)+iV−(x),V±(−x)=±V±(x),ℑV±=