N Kirillov A A Mailybaev and A P Seyranian Abstract The paper presents a theory of unfolding of eigenvalue surfaces of real symmetric and Hermitian matrices due to an arbitrary complex perturbation near a diabolic point General asymptotic formulae d ID: 76811
Download Pdf The PPT/PDF document "On eigenvalue surfaces near a diabolic p..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
wherethederivativeistakenat,andinnerproductsofvectorsin(3)aregivenby .Inexpression(3)thehigherorderterms ±=fp ±0p=±0p fp.(5) 0-7803-9235-3/05/$20.00 © 2005 IEEEPhysCon 2005, St. Petersburg, Russia319 Fig.1.Unfoldingofadiabolicpointduetocomplexperturbation. ) ,(18)ya,b=+ ) Thesetwosolutionsdeterminethepointsinparameterspace,wheredoubleeigenvaluesappear.When,thetwosolutionscoincide.For,theequationshavenorealsolutions.Inthelattercase,theeigenvaluesseparateforallNotethatthequantitiesareexpressedbymeansoftheanti-Hermitianpart=( ofthematrix 2i,=N(p0)u1,u2N(p0)u2,u1) dependsontheHermitianpart=( AT)/2as=H(p0)u1,u2)H(p0)u2,u1) ,onecansaythattheinuenceoftheanti-HermitianpartoftheperturbationisstrongerthanthatoftheHermitianpart.IftheHermitianpartprevailsintheperturbation,wehave.Inparticular,forapurelyHermitianperturbationLetusassumethatthevectorconsistsofonlytwo,andconsiderthesurfaces(13)and(14)fordifferentkindsoftheperturbation.Considerrstthecase.Then,theeigensheetsareseparate,seeFigure1a.Equationdenesalineinparameterplane.Thesheetsoftheeigensurface(14)intersectalongtheline=Imgivenbyconditions(16).Inthecasethelineandtheellipsedenedbythequationhavecommonpointswheretheeigenvaluescouple.Coordinatesofthesepointscanbefoundfromtheequations(11),wherearedenedbyexpressions(18)and(19).Herewehaveassumedthatthevectorsarelinearlyindependent.NotethatthepointscoincideinthedegeneratecaseAccordingtoconditions(15)therealeigensheetsaregluedintheintervalalpa,pb]oftheline+ReThesurfaceofrealeigenvalues(13)iscalledadoublecoffeelter[6].TheunfoldingofadiabolicpointintothedoublecoffeelterisshowninFigure1b.Notethatincrystalopticsandacousticstheintervalalpa,pb]isreferredtoasabranchcut,andthepointsarecalledsingularaxes,see[5],[7].Accordingtoequation(8)thedoubleeigenvaluesat 321 Theinversedielectrictensorisdescribedbyacomplexnon-Hermitianmatrixtranspdichroicchiral.Thesymmetricpartofconsistingoftherealmatrixtranspandimaginarymatrixdichroicconstitutetheanisotropytensor,whichdescribesthebirefringenceofthecrystal.Foratransparentcrystal,theanisotropytensorisrealandisrepresentedonlybythematrixtransp;foracrystalwithlineardichroismitiscomplex.Choosingcoordinateaxesalongtheprincipalaxesoftransp,wehavetranspThematrixdichroicdescribeslineardichroism(absorption).Thematrixchiralgivestheantisymmetricpartofdescribingchirality(opticalactivity)ofthecrystal.Itisdeterminedbytheopticalactivityvectordependinglinearlyonchiralisasymmetricopticalactivitytensor;thistensorhasanimaginarypartforamaterialwithcirculardichroism,see[7]formoredetails.First,consideratransparentnon-chiralcrystal,whendichroic.Thenthematrixtranspisrealsymmetricanddependsonavectoroftwoparameters.Thethirdcomponentofthedirectionvectorisfoundas ,wherethecasesoftwodifferentsignsshouldbeconsideredseparately.Belowweassumethatthreedielectricconstantsdifferent.Thiscorrespondstobiaxialanisotropiccrystals.Thenonzeroeigenvaluesofthematrixarefoundexplicitlyintheform[11] 2±1 2 2trace(Theeigenvaluesarethesameforoppositedirections.Byusing(33)and(36)in(37),itisstraightforwardtoshowthattwoeigenvaluescoupleat (12)/(13)=± whichdeterminefourdiabolicpoints(fortwosignsof),alsocalledopticaxes[7].Thedoubleeigenvalue Fig.3.Diabolicsingularitiesnearopticaxesandtheirlocalapproximations.ofthematrixtwoeigenvectorssatisfyingnormalizationconditions(2).Usingexpressions(36)and(39),weevaluatethevectorswithcomponents(4)foropticaxes.Substitutingthemin(9),weobtainthelocalasymptoticexpressionfortheconesingularitiesintheEquation(40)isvalidforeachofthefouropticaxes(38).Nowletusassumethatthecrystalpossessesabsorptionandchirality.Thenthematrixfamily(36)takesacomplex,wheredichroicchiralAssumethattheabsorptionandchiralityareweak,i.e.,dichroicchiralissmall.ThenwecanuseasymptoticformulaeofSections2and3todescribeunfoldingofdiabolicsingularitiesoftheeigenvaluesurfaces.Forthispurpose,weneedtoknowonlythevalueoftheperturbationattheopticaxesofthetransparentnon-chiralcrystalSubstitutingmatrix(41)evaluatedatopticaxes(38)intoexpression(7),andthenusingformulae(12),weobtainWeseethat,andarepurelyimaginarynumbersdependingonlyondichroicpropertiesofthecrystal(absorp-tion).Thequantitydependsonlyonchiralpropertiesofthe 323 [7]BerryM.V.andDennisM.R.Proc.R.Soc.Lond.A..pp.1261 1292.2003.[8]BerryM.V.Czech.J.Phys..2004.1039 1047.[9]SeyranianA.P.,KirillovO.N.,MailybaevA.A.Couplingofeigenval-uesofcomplexmatricesatdiabolicandexceptionalpoints.J.ofPhys.A:Math.Gen.(8).pp.1723 1740.2005.[10]LandauL.D.,LifshitzE.M.,PitaevskiiL.P.Electrodynamicsofcon-tinuousmedia.Oxford:Pergamon,1984.[11]LewinM.DiscreteMathematics.(1 3).pp.255 262.1994. 325