statistics of zeros of Husimi representations of quantum chaotic eigenstates and random polynomials

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1、February5,2008ParametricstatisticsofzerosofHusimirepresentationsofquantumchaoticeigenstatesandrandompolynomialsTomaˇzProsen1PhysicsDepartment,FacultyofMathematicsandPhysics,UniversityofLjubljana,Jadranska19,1000Ljubljana,SloveniaAbstractLocalparametricstatisticsofzerosofHusimireprese

2、ntationsofquantumeigen-statesareintroduced.ItisconjecturedthatforaclassicallyfullychaoticsystemsoneshouldusethemodelofparametricstatisticsofcomplexrootsofGaussianrandompoly-nomialswhichisexactlysolvableasdemonstratedbelow.Forexample,thevelocities(derivativesofzerosofHusimifunctionwit

3、hrespecttoanexternalparameter)arepre-dictedtoobeyauniversal(non-Maxwellian)distributiondP(v)222−3=(1+

4、v

5、/σ),dv2πσ2whereσ2isthemeansquarevelocity.Theconjectureisdemonstratednumericallyinagenericchaoticsystemwithtwodegreesoffreedom.Dynamicalformulationofthe“zero–flow”intermsofanintegrab

6、lemany-bodydynamicalsystemisgivenaswell.arXiv:chao-dyn/9612006v13Dec19961IntroductionTheintenseresearchintheso-calledquantumchaologyhasproducedmanydifferentsig-naturesofclassicalchaosinthecorrespondingquantumHamiltoniansystems.Ithasbeenfoundthatenergyspectraandeigenstatesofclassically

7、fullychaoticquantumsys-temshaveuniversalstatisticalpropertieswhichcanbedescribedbystochasticmodelswithnofreeparameters,likerandommatrixtheory(RMT).IntheBargmann(orHusimi)representation,eigenstatesofquantumsystemscanbeuniquelyrepresentedintermsofcomplexanalyticfunctionsinphasespacevar

8、iables.Quiterecently,ithasbeenproposed[7]thatincaseof1-dimsystems(orPoincar´esurface1e-mail:prosen@fiz.uni-lj.si1ofsectionreductionsof2-dimsystems[10]),wherephasespaceistwodimensional,andonehasonlyonecomplexvariablez=q+ip,anyeigenstatecanbeuniquelyrepresentedbythecollectionof(complex)

9、zerosofitsBargmannorHusimirepresentation.Thishasbeencalledthestellarrepresentation.Ithasbeenfound[10]thatthestructureofzerosofHusimirepresentationofaneigenstateisreminiscentofthestructureofclassicalphasespace:(i)inclassicallyfullychaoticsystemsthezerostendtospreaduniformlyoverthewhol

10、eclassically