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Unexpected features of quantum degeneracies in the many-body continuum M. Ploszajczak (GANIL) with J. Okolowicz (INP-Krakow), J. Dukelsky (CSIC-Madrid) closed quantum system open quantum system continuum coupling effects in binding: 0.1-3 MeV Instability of SM eigenstates at channel threshold - segregation of time scales in the continuum - alignment of near-threshold states with decay channels - instability of SM eigenstates at channel threshold - level degeneracies Level degeneracy is the subtlest quantal effect that appears at the highest energy resolution How accidental is a degeneracy? J.von Neumann, E.Wigner (1929) Hermitian problems Generic situation: avoided level crossings "2 Level crossing if "1 # " 2 ! ! "1 ! but... consider H ( R* ) " {H ( R)} , ( R " X,Y,...) "E ( R) # E 2 ( R) $ E1 ( R) = { 2 { [H11'(R) $ H 22' (R)] + 4 H12' (R) 2 2 1/ 2 } " #E ( X,Y ) = A( X $ X * ) + 2B( X $ X * )(Y $ Y * ) + C (Y $ Y * ) ! ! ! H (R * H ( R) ) M.V. Berry (1984) "n (C ) = "n e ±i#(C ) / 2 ! ! 2 1/ 2 } =0 Molecular physics: M.Vaz Pires et al., J. Chem Phys. 69 (1978) 3242 Nuclear physics: D.L.Hill, J.A.Wheeler, Phys. Rev. 89 (1952) 1102 R.S. Nikam, P. Ring, Phys. Rev. Lett. 58 (1987) 980 Non-Hermitian problems VH Level repulsion Width clustering ! VNH vs Level clustering Width repulsion ! VH VNH "2 " ! "!1 ! ! E " = v in + iw ex ! ! ! * - Exceptional point "1 = " 2 ("1 = "1 ) is a generic situation in NH problems - Defect of the vector space: DH = n "1 - Topological phase: "1 # i"2 , "2 # $i"1 ! ! ! C. Dembowski et al., PRL 86 (2001) 787(microwave cavity) ! ! Exceptional points in the scattering continuum - Do they exist in the low-energy scattering continuum? - Do they appear for realistic values of continuum coupling parameters? Shell Model Embedded in the Continuum K. Bennaceur et al. Nucl. Phys. A671 (2000) 203 J. Rotureau et al., PRL 95 (2005) 042503; NP A767 (2006) 13 Q = [ A] SM SMEC P = [ A "1] # [1] T = [ A " 2] # [2] HTT H PT ... = ... HQP HQQ ! ! H PP HQQ " HQQ ( E ) ! ! ! HQQ ( E ) = HQQ + HQP GP ( E ) H PQ ! ! ( +) ! HQQ H PP closed quantum ! system ! ! HQT HTT ! open quantum system Open QS solution in Q : ! HQQ "# = E# "# biorthogonal basis * # "#˜ HQQ = E "#˜ "# = $ a#i %Ai ! * ; "#˜ = "# i " ! The scattering function (in I=Q+P+...) ˜$ ; " = # + % c$ " c E c E $ ! c" = ! #" HQP $ Ec E % E" ˜ # $ 1+ GP( +) H PQ "# " ( ) ! ! Inside of the interaction region : ! "Ec ~ $ c# "# # eigenfunctions of HQQ ( E ) ! ! O #$ = "#˜ O "$ ! For bound states: E" ( E ) is real and E" ( E ) = E For unbound states: physical resonances poles of S-matrix ! ! ! "#˜ "$ = %#$ E " = Re(E" ( E )) E = E" #" = $2Im(E" ( E )) E = E" Exceptional points in the scattering continuum appear due to an interplay between Hermitian and non-Hermitian parts of HQQ ( E ) : ' + VQQ ( E;V0 ) HQQ ( E ) = HQQ + HQP GP( +) ( E ) H PQ = HQQ ! scattering energy ! E=0: 1st N-emission threshold ! continuum coupling constant Single-root solutions of a discriminant: " (# ) det [HQQ ( E;V0 ) $ EI] = 0 "E ( ! ! ! ) (" = 0,1) give positions ( E;V0 ) µ of singularities, where µ = 1,2,..., M " n ( n #1) ! vector space and n is a dimension of the M = n ( n "1) in non-integrable quantum models ! ! J. Dukelsky et al., arXiv:0811.4105 Conclusions - EPs (double-poles of the S-matrix) can appear in the low-energy continuum for realistic continuum Shell Model Hamiltonians (SMEC, Gamow Shell Model) - Number of EPs (M=n(n-1)) is typical for non-integrable systems - Several EPs ‘escape’ from a physical region of (E;V0 )-parameters - The non-resonant continuum is important for the spectroscopy of weakly bound nuclei (energy shifts and degeneracy, ! modification of effective NN correlations (spectroscopic factors), binding systematics, manifestation of clustering, exceptional points, segregation of times scales,...) Experimental observation of EPs would provide a stringent test of configuration mixing involving nucleons in discrete and continuum states