Modification of the Porter-Thomas distribution by rank-one interaction. Eugene Bogomolny

Transcription

1 Modification of the Porter-Thomas distribution by rank-one interaction Eugene Bogomolny University Paris-Sud, CNRS Laboratoire de Physique Théorique et Modèles Statistiques, Orsay France XII Brunel-Bielefeld Workshop on Random Matrix Theory, Uxbridge,

2 Outlook 1 Introduction 2 Rank-one formalism 3 Probability dist 4 κ 2 < 1 5 κ 2 > 1 6 Large-window dist 7 Numerics 8 Conclusion

3 Porter Thomas distribution = everything is Gaussian C. E. Porter and R. G. Thomas Phys. Rev ) Fluctuations of nuclear reaction widths Time-invariant systems P 1 x) = 1 exp x ) 2πlx 2l Nuclear physics : x = reduced resonance width. Random matrices : x = N Ψ 2 Time-non-invariant systems P 2 x) = 1 exp x ) l l l = mean value of x. Standard normalisation : x = 1 l = 1

4 Porter Thomas distribution = everything is Gaussian C. E. Porter and R. G. Thomas Phys. Rev ) Fluctuations of nuclear reaction widths Time-invariant systems P 1 x) = 1 exp x ) 2πlx 2l Nuclear physics : x = reduced resonance width. Random matrices : x = N Ψ 2 Time-non-invariant systems P 2 x) = 1 exp x ) l l l = mean value of x. Standard normalisation : x = 1 l = 1 Old experiments agreed well with the PT distribution

5 Porter Thomas distribution = everything is Gaussian C. E. Porter and R. G. Thomas Phys. Rev ) Fluctuations of nuclear reaction widths Time-invariant systems P 1 x) = 1 exp x ) 2πlx 2l Nuclear physics : x = reduced resonance width. Random matrices : x = N Ψ 2 Time-non-invariant systems P 2 x) = 1 exp x ) l l l = mean value of x. Standard normalisation : x = 1 l = 1 Old experiments agreed well with the PT distribution Recent experiments Reduced neutron widths in the nuclear data ensemble : experiment and theory do not agree Phys. Rev. C 84, ) Neutron resonance data exclude random matrix theory Fortschritte Phys. 61, ) P. E. Koehler et al

6 Modification of the model Within standard invariant) random matrix ensembles : PT distribution = theorem Modifications required

7 Modification of the model Within standard invariant) random matrix ensembles : PT distribution = theorem Modifications required A. Volya, H. A. Weidenmüller, and V. Zelevinsky PRL 115, ) Neutron resonance widths and the Porter-Thomas distribution Realistic model of nuclear s-wave resonances : M ij = G β) ij + Z δ i1 δ j1 G β) ij = standard random matrix, β = 1 GOE, β = 2 GUE Z δ i1 δ j1 = interaction which couples resonances to decay channels

8 Modification of the model Within standard invariant) random matrix ensembles : PT distribution = theorem Modifications required A. Volya, H. A. Weidenmüller, and V. Zelevinsky PRL 115, ) Neutron resonance widths and the Porter-Thomas distribution Realistic model of nuclear s-wave resonances : M ij = G β) ij + Z δ i1 δ j1 G β) ij = standard random matrix, β = 1 GOE, β = 2 GUE Z δ i1 δ j1 = interaction which couples resonances to decay channels Distribution GOE or GUE) : ) PG ij ) exp β 4σ 2 Tr G G ) β = 1, 2 Density = Wigner semicircle law : ρ W E) = 1 2πσ 2 4σ 2 N E 2 Z may be complex. ReZ is due to coupling to neutron channel and is immanent to the theory. ImZ is related with hypothetical non-statistical gamma decays Here only Hermitian matrix M is considered, Z = real. To get a nontrivial limit κ = Z σ N remains constant when N Numerically : distribution of x = N Ψ 1 ) 2 does deviate from the PT law

9 Known results Real Z local spectral statistics is independent on Z after unfolding) For GUE there is a direct proof using the Itzykson-Zuber integral E. Brezin and S. Hikami, Nucl. Phys. B ) For GOE? Imaginary Z = iγ σ there exists an exact solution using non-linear sigma model) N for the distribution of imaginary parts of eigenvalues, E = e + i 2 Γ, y = πγ/ρ W e), GUE, Y. V. Fyodorov and H.-J. Sommers, JETP Lett. 63, ) Py) = d e dy yg sinh y y ), g = 1 2 γ + γ 1 ) GOE, H.-J. Sommers, Y. V. Fyodorov, and M. Titov, J. Phys. A 32, L ) Py) = 1 d 2 1 4π dy 2 1 λ 2 )e 2λy g λ)f λ, y)dλ 1 e yp 1 dp g 1 e yp 2 dp 2 F λ, y) = g λ p 1 ) p )p 1 g) 1 λ p 2 ) p )g p 2)

10 Rank-one formalism M ij = G β) ij + Z δ i1 δ j1 Two N N Hermitian matrices G and M differ by a rank-one interaction M ij = G ij + v i v j, v j = Z1, 0,..., 0) Eigenvalues and eigenfunctions for Hermitian matrices assumed to be orthogonal Expansions : N G ij Φ j ) = e Φ i ), j=1 N M ij Ψ j ) = E Ψ i ) j=1 Ψ j ) = N β=1 C βφ j β), Φ j ) = N β=1 C 1 β Ψ j β)

11 Rank-one formalism M ij = G β) ij + Z δ i1 δ j1 Two N N Hermitian matrices G and M differ by a rank-one interaction M ij = G ij + v i v j, v j = Z1, 0,..., 0) Eigenvalues and eigenfunctions for Hermitian matrices assumed to be orthogonal Expansions : Consequences N G ij Φ j ) = e Φ i ), j=1 C β = Quantisation condition Complementary relations N M ij Ψ j ) = E Ψ i ) j=1 Ψ j ) = N β=1 C βφ j β), Φ j ) = N β=1 C 1 β Ψ j β) ab β E e β, b β = C 1 β = β N v j Φ j β), j=1 b β 2 E e β = 1, ba E β e β E, e a = β a β 2 e E β = 1 C β b β

12 Schematic picture 0 0 Quantisation conditions : β b β 2 E e β = 1, β a β 2 e E β = 1 New and old eigenvalues are interlacing e 1 e 2 e N, e E e +1, e N E N for Z > 0)

13 Main relations Solve for numerators b 2 γ Eγ e) = γ eγ e), a 2 γ eγ E) = γ Eγ E) Theorem N m=1 b m x m y n = 1, b m = n xm yn) s m xm xs) x m, y m are known) Proof one among many others) Consider the function r n x yr ) r x yr ) f nx) = s x xs) = x y n) s x xs) Asymptotically f nx) 1/x integral over a large contour encircling all poles equals 1 Rewriting this integral as the sum over all finite poles gives 1 = r xm yr ) x m m y n) s m xm xs) Q.E.D. Many different relations. E.g. β b β 2 = β a β 2 = β E β e β )

14 Initial probability distribution Eigenvalues e and eigenfunctions Φ 1 ) of matrix G β are distributed as in standard random matrix ensembles P{e }, {r }) e γ e ) β r β/2 1 δ r 1 exp V {e })) <γ r = Φ 1 ) 2 the same is valid for other components as well) V {e }) = confinement term. For standard Gaussian ensembles V {e }) = β 4σ 2 e 2 Mean density of matrix eigenvalues = Wigner semicircle law ρ W E) = 1 2πσ 2 4Nσ 2 E 2

15 Initial probability distribution Eigenvalues e and eigenfunctions Φ 1 ) of matrix G β are distributed as in standard random matrix ensembles P{e }, {r }) e γ e ) β r β/2 1 δ r 1 exp V {e })) <γ r = Φ 1 ) 2 the same is valid for other components as well) V {e }) = confinement term. For standard Gaussian ensembles V {e }) = β 4σ 2 e 2 Mean density of matrix eigenvalues = Wigner semicircle law ρ W E) = 1 2πσ 2 4Nσ 2 E 2 Main question How new eigenvectors Ψ 1 ) and new eigenvalues E are distributed? Two steps of changing variables : e, Φ 1 )) e, E ) Ψ 1 ), E )

16 Joint distribution of old and new eigenvalues P{e }, {r }) de dr e γ e β <γ r β/2 1 δ r 1) de dr b = N j=1 v j Φ j β), v j = Z1, 0,..., 0) b 2 = Z Φ 1 ) 2 Expression of r through old and new eigenvalues, r Φ 1 ) 2 = 1 Z b 2 = 1 Z γ Eγ e) eγ e) γ

17 Joint distribution of old and new eigenvalues P{e }, {r }) de dr e γ e β <γ r β/2 1 δ r 1) de dr b = N j=1 v j Φ j β), v j = Z1, 0,..., 0) b 2 = Z Φ 1 ) 2 Expression of r through old and new eigenvalues, Change of variables : r Φ 1 ) 2 = 1 Z b 2 = 1 Z γ Eγ e) eγ e) γ N quantities r N quantities E Derivatives : r / E γ = r /E γ e ) Cauchy determinant : Jacobian ) r det E γ 1 det ) = x m y n i<j x i x j ) i<j y j y i ) i,j x i y j ) = ) 1 r ) det = <β r E E β)e β e ) ) E γ e,β E e β) 1 E γ E ) = Z N e <γ e, r = 1 E e ) γ) Z

18 Change from old eigenvalues to new eigenfunctions I. L. Aleiner and K. A. Matveev Shifts of random energy levels by a local perturbation, PRL 80, ) ) γ> eγ e)eγ E) P{e }, {E }) γ, eγ δ E e ) Z de de E β/2 1 Ψ 1 ) = β C βφ 1 β) = 1 Z β C βb β = a Z β b β 2 = a E e β Z Expression of r through old and new eigenvalues z = Ψ 1 ) 2 = 1 Z a 2 = 1 γ E eγ) Z γ E Eγ)

19 Change from old eigenvalues to new eigenfunctions I. L. Aleiner and K. A. Matveev Shifts of random energy levels by a local perturbation, PRL 80, ) ) γ> eγ e)eγ E) P{e }, {E }) γ, eγ δ E e ) Z de de E β/2 1 Ψ 1 ) = β C βφ 1 β) = 1 Z β C βb β = a Z β b β 2 = a E e β Z Expression of r through old and new eigenvalues z = Ψ 1 ) 2 = 1 Z a 2 = 1 γ E eγ) Z γ E Eγ) z N quantities r N quantities e. Derivatives : = z e γ E e γ Jacobian ) z det = 1 e γ e ) e β Z N E <γ E γ) Identities e γ) =,γe ) 2, z E E γ) z = 1 E e ) Z <γ

20 Distribution of new eigenvalues and new eigenfunctions New joint distribution = the old one e γ e β <γ = E γ E β <γ r β/2 1 δ z β/2 1 δ r 1) de dr = z 1) de dz

21 Distribution of new eigenvalues and new eigenfunctions New joint distribution = the old one e γ e β <γ = E γ E β <γ r β/2 1 δ z β/2 1 δ r 1) de dr = z 1) de dz Symmetry : G ij = M ij v i v j e E P{e }, {E }) is symmetric

22 Distribution of new eigenvalues and new eigenfunctions New joint distribution = the old one e γ e β <γ = E γ E β <γ r β/2 1 δ z β/2 1 δ r 1) de dr = z 1) de dz Symmetry : G ij = M ij v i v j e E P{e }, {E }) is symmetric Symmetry is valid only without the confinement term : exp βtr GG /4σ 2 ) Tr GG = Tr M ij Zδ i1 δ j1 ) 2, M ij = EΨ i )Ψ j ), M 11 = E Ψ 1) 2 e2 = E 2 2Z Ez + Z 2 Full joint distribution of new eigenvalues E and new eigenvectors, z Ψ 1 ) 2 P{E }, {z }) <β E β E β z β/2 1 δ z 1) [ exp β 4σ 2 E 2 2Z E z )]

23 Mean level density L. A. Pastur, On the spectrum of random matrices, TMP, 10, ) Mean Green function The Pastur equation ḠE) = 1 N Tr E M) 1 ḠE) = 1 1 N E Z σ 2 NḠE) + N 1 1 N E σ 2 NḠE) Solution in 2 lowest orders in N 1 ) ZḠ 0 E) ḠE) = Ḡ 0 E) 1 + N1 ZḠ 0 E)) Ḡ 0 E) = the mean Green function for standard RM ensembles Ḡ 0 E) = 1 E σ 2 NḠ 0 E), Ḡ 0 E) = E E 2 4σ 2 N 2σ 2 N

24 Formation of a collective state = outlier) Mean level density ρe) = 1 π Tr GE i0), G 0E) = E E 2 4σ 2 N 2σ 2 N For perturbed problem E = 2σ N cos φ N ρφ) = 2π + 2κ2 cos φ κ) πκ 2 2κ cos φ + 1) ρ 0 E) = ) sin 2 φ 1 4Nσ 2πσ 2 2 E 2 N

25 Formation of a collective state = outlier) Mean level density ρe) = 1 π Tr GE i0), G 0E) = E E 2 4σ 2 N 2σ 2 N For perturbed problem E = 2σ N cos φ N ρφ) = 2π + 2κ2 cos φ κ) πκ 2 2κ cos φ + 1) ρ 0 E) = ) sin 2 φ 1 4Nσ 2πσ 2 2 E 2 N Total level number : π 0 ρφ)dφ = { N κ 2 < 1 N 1 + 1/κ 2 κ 2 > 1

26 Formation of a collective state = outlier) Mean level density ρe) = 1 π Tr GE i0), G 0E) = E E 2 4σ 2 N 2σ 2 N For perturbed problem E = 2σ N cos φ N ρφ) = 2π + 2κ2 cos φ κ) πκ 2 2κ cos φ + 1) ρ 0 E) = ) sin 2 φ 1 4Nσ 2πσ 2 2 E 2 N { Total level number : π N κ 0 ρφ)dφ = 2 < 1 N 1 + 1/κ 2 κ 2 > 1 Collective state outlier) appears at κ 2 > 1 E c Z σ 2 NḠ 0 E c) = 0 E c = σ Nκ + 1 κ ) Close to E c ḠE) Ḡ 0 E) + δge) a = 2σ1 κ 2 )) δge) = 1 N [ κ 2 κ 2 1 E Ec) σ2 NδGE)] 1 δρe) = 1 a 2 E E c) 2 πσa δρe)de = 1 κ 2. Total : ρe)de = N

27 Formation of a collective state = outlier) Mean level density ρe) = 1 π Tr GE i0), G 0E) = E E 2 4σ 2 N 2σ 2 N For perturbed problem E = 2σ N cos φ N ρφ) = 2π + 2κ2 cos φ κ) πκ 2 2κ cos φ + 1) ρ 0 E) = ) sin 2 φ 1 4Nσ 2πσ 2 2 E 2 N { Total level number : π N κ 0 ρφ)dφ = 2 < 1 N 1 + 1/κ 2 κ 2 > 1 Collective state outlier) appears at κ 2 > 1 E c Z σ 2 NḠ 0 E c) = 0 E c = σ Nκ + 1 κ ) Close to E c ḠE) Ḡ 0 E) + δge) a = 2σ1 κ 2 )) δge) = 1 N [ κ 2 κ 2 1 E Ec) σ2 NδGE)] 1 δρe) = 1 a 2 E E c) 2 πσa δρe)de = 1 κ 2. Total : ρe)de = N Exact result : E E c = Gaussian variable with mean= 0, variance= 2κ2 1)σ 2 βκ 2

28 Case κ 2 < 1 z = Ψ 1 ) 2 for all energies E are ON 1 ) Condition z = 1 can be taken into account by the Lagrange multiplier δ z 1) exp µ z 1)) z = independent, each z ze ) has PDF Pz, E) = µ Z ) β/2 2σ 2 E πz) β/2 1 exp µ βz ) ) 2σ 2 E z µ has to be calculated from normalisation : z = 1 1 z Pz, E ) dz = 1, = Z 0 E E σ 2, E = 2µσ2 βz The sum is mean unperturbed Green function, Ḡ 0 E) Ḡ 0 E) = 1 N Tr E G) 1 = E E 2 4σ 2 N 2σ 2 N µ = βn κ Distribution of perturbed problem = local PT distribution P β x) = 1 2πx) 1 β/2 exp le)) β/2 βx 2lE) ), le) = κ κ ) 1 σ N E

29 Case κ 2 > 1 When κ 2 > 1 there exists one collective state. z c = Ψ 1 E c) 2 = O1) Other z are ON 1 ) and their distribution is the local PT distribution Pz, E) = µ Z ) β/2 2σ 2 E πz) β/2 1 exp µ βz ) ) 2σ 2 E z The only difference is that their normalisation is N 1 =1 z = 1 zc µz c) = βn κ 2 1 z c) + 1 ) 2 1 z c

30 Case κ 2 > 1 When κ 2 > 1 there exists one collective state. z c = Ψ 1 E c) 2 = O1) Other z are ON 1 ) and their distribution is the local PT distribution Pz, E) = µ Z ) β/2 2σ 2 E πz) β/2 1 exp µ βz ) ) 2σ 2 E z The only difference is that their normalisation is N 1 =1 z = 1 zc µz c) = βn κ 2 1 z c) + 1 ) 2 1 z c F E, r) = 1 N Calculation of E c and z c Joint distribution of E E c and r z c PE, r) E E β µr) β Nκ 2σ E) β/2 [ exp β 4σ 2 E 2 + β Nκ ] 2σ Er 1 r)µr) Nβ F E,r) e κ lne E ) + 1 4σ 2 N E N 2σ N Er 1 r)νr), νr) = 1 2 ln νr) κ 2 1 r) r κ ) 2σ N E )

31 Final answer Saddle-point equations E c = σ N κ + 1 κ For both κ 2 < 1 and κ 2 > 1 F E,r) r F E,r) = 0 E ), z c = 1 1 κ 2 µ = βn κ = 0 plus mean Green function

32 Final answer Saddle-point equations E c = σ N κ + 1 κ For both κ 2 < 1 and κ 2 > 1 F E,r) r F E,r) = 0 E ), z c = 1 1 κ 2 µ = βn κ = 0 plus mean Green function Local Porter-Thomas distribution For 2σ N E 2σ N and all κ 1 P β x, E) = 2πx) 1 β/2 exp βx ) le)) β/2 2lE) le) N Ψ 1 E) 2 = κ κ ) 1 σ N E

33 Final answer Saddle-point equations E c = σ N κ + 1 κ For both κ 2 < 1 and κ 2 > 1 F E,r) r F E,r) = 0 E ), z c = 1 1 κ 2 µ = βn κ = 0 plus mean Green function Local Porter-Thomas distribution For 2σ N E 2σ N and all κ 1 P β x, E) = 2πx) 1 β/2 exp βx ) le)) β/2 2lE) le) N Ψ 1 E) 2 = κ κ ) 1 σ N E N 2σ N [ ] { Ψ 1 E ) 2 1 κ E 2σ ρ W E) z P β z, E ) dz de = 2 < 1 N 0 1/κ 2 κ 2 > 1

34 Final answer Saddle-point equations E c = σ N κ + 1 κ For both κ 2 < 1 and κ 2 > 1 F E,r) r F E,r) = 0 E ), z c = 1 1 κ 2 µ = βn κ = 0 plus mean Green function Local Porter-Thomas distribution For 2σ N E 2σ N and all κ 1 P β x, E) = 2πx) 1 β/2 exp βx ) le)) β/2 2lE) le) N Ψ 1 E) 2 = κ κ ) 1 σ N E N 2σ N [ ] { Ψ 1 E ) 2 1 κ E 2σ ρ W E) z P β z, E ) dz de = 2 < 1 N 0 1/κ 2 κ 2 > 1 Cf. H. A. Weidenmüler, PRL 105, ) : le) is associated with the existence of a virtual or weakly bound state near threshold

35 Another approach based on the averaged Green function The exact Schur complement formula ĜE) E M) 1 ) Ĝ 11 E) = E Z M 1j G jk M k1 ) 1 j,k 1 G jk = the Green function of N 1) N 1) matrix obtained from matrix M by removing line 1 and row 1 = standard N 1) N 1) random matrix, H For G β matrices M 1j M k1 σ 2 δ jk N Therefore G 11 E) E Z σ 2 G 0 E)) 1 N G 0 = Tr E H) 1. For N Eigenfunction expansion : G 0 E) E E 2 4σ 2 N 2σ 2 G ij E) = Ψ i )Ψ j ) Ψ 1 E) 2 Im G 11E i0) E E πρ W E) ) 1 After some algebra : le) N Ψ 1 E) 2 = κ κ σ E N Generalisation for finite rank-interaction is straightforward but i) outlier interaction requires more careful calculations and ii) the Gaussian distribution cannot be proved

36 Large-window distribution Local Gaussian distribution only for Ψ 1 E) in a small energy window δe σ N If E 1 < E < E 2 z q [E 1,E 2 ] = c βq) E2 ρ W E) κ κ ) q E2 δn E 1 σ N E de, δn = ρ W E)dE E 1 ρ W E) = Wigner s spectral density, c β q) = the Gaussian moments c 1 q) = 2q Γq+1/2) π, c 2 q) = Γq + 1) The full distribution = weighted integral of local PT distributions P β x) = 1 E2 ρ W E) δn E 1 2πx) 1 β/2 exp βx le)) β/2 2lE) ) de, le) = 1 κ If all states are included, E 1 = 2σ N, E 2 = 2σ N For β = 1 GOE) 2 π P 1 x) = π 3 dφ sin 2 φ κ κ cos φ e 1 2 κ2 2κ cos φ+1)x x 0 For β = 2 GUE) P 2 x) = I 12κx) κx e κ2 +1)x κ σ NE

37 Numerics : mean values of N Ψ 1 E)) 2 for different κ 1.5 <Ν Ψ 1 Ε)) 2 > κ Red circles are mean values for energies in the interval [ N/2, N/2] Blue diamonds are the same but for energies in the interval [ N/2, 3 N/2] Solid black lines = large-window theoretical predictions Dashed black line is the small-window predictions N = 1000, σ = 1, and each point is averaged over 50 random realisations

38 Numerics : distribution of x = NΨ 1 E) for β = 1 GOE) Px) x Px) x Left : states with energies in [ N/2, N/2] lower curve) and [ N/2, 3 N/2] upper curve) for κ = 0.6 Right : the same but for κ = 1.5 Solid lines are the Gaussian fits with zero mean whose variance agrees well with the theoretical prediction

39 Numerics : distribution of x = N Re Ψ 1 E) left) and x = N Im Ψ 1 E) right) for β = 2 GUE) and κ = 0.6 Px) x Px) x Energies in the interval [ N/2, N/2] Px) x Px) x Energies in the interval [ N/2, 3 N/2]

40 Numerics : distribution of x = NΨ 1 E) for all states, κ =.8, β = Px) x Blue dashed line is the PT distribution Gaussian) : Px) = e x2 /2 / 2π Black solid line is theoretical prediction 2 π Px) = π 3 dφ sin 2 φ κ κ cos φ e 1 2 κ2 2κ cos φ+1)x 2 0

41 Conclusion Standard PT distribution = eigenvectors of large random matrix are i.i.d.r.v. P 1 x) = 1 exp x ), P 2 x) = 1 exp x ), x = N Ψ 2, l = 1 2πlx 2l l l

42 Conclusion Standard PT distribution = eigenvectors of large random matrix are i.i.d.r.v. P 1 x) = 1 exp x ), P 2 x) = 1 exp x ), x = N Ψ 2, l = 1 2πlx 2l l l When ensemble of standard random matrices with Gaussian distribution is perturbed by a rank-one perturbation, the distribution of x = N Ψ 1 E) 2 has the same functional form but with le) N Ψ 1 E) 2 = κ κ σ N E ) 1

43 Conclusion Standard PT distribution = eigenvectors of large random matrix are i.i.d.r.v. P 1 x) = 1 exp x ), P 2 x) = 1 exp x ), x = N Ψ 2, l = 1 2πlx 2l l l When ensemble of standard random matrices with Gaussian distribution is perturbed by a rank-one perturbation, the distribution of x = N Ψ 1 E) 2 has the same functional form but with le) N Ψ 1 E) 2 = κ κ σ N E ) 1 When κ 2 > 1 there exists one collective state whose mean energy is E c = σ Nκ + κ 1 ) and Ψ c) 1 2 = 1 κ 2

44 Conclusion Standard PT distribution = eigenvectors of large random matrix are i.i.d.r.v. P 1 x) = 1 exp x ), P 2 x) = 1 exp x ), x = N Ψ 2, l = 1 2πlx 2l l l When ensemble of standard random matrices with Gaussian distribution is perturbed by a rank-one perturbation, the distribution of x = N Ψ 1 E) 2 has the same functional form but with le) N Ψ 1 E) 2 = κ κ σ N E ) 1 When κ 2 > 1 there exists one collective state whose mean energy is E c = σ Nκ + κ 1 ) and Ψ c) 1 2 = 1 κ 2 When all eigenfunctions in a large energy interval are considered their distribution is not Gaussian but is given by an integral over Gaussian functions

45 Conclusion Standard PT distribution = eigenvectors of large random matrix are i.i.d.r.v. P 1 x) = 1 exp x ), P 2 x) = 1 exp x ), x = N Ψ 2, l = 1 2πlx 2l l l When ensemble of standard random matrices with Gaussian distribution is perturbed by a rank-one perturbation, the distribution of x = N Ψ 1 E) 2 has the same functional form but with le) N Ψ 1 E) 2 = κ κ σ N E ) 1 When κ 2 > 1 there exists one collective state whose mean energy is E c = σ Nκ + κ 1 ) and Ψ c) 1 2 = 1 κ 2 When all eigenfunctions in a large energy interval are considered their distribution is not Gaussian but is given by an integral over Gaussian functions In the limit N all other components of eigenfunctions except Ψ 1 E)) remain distributed according to the usual PT distribution

46 Conclusion Standard PT distribution = eigenvectors of large random matrix are i.i.d.r.v. P 1 x) = 1 exp x ), P 2 x) = 1 exp x ), x = N Ψ 2, l = 1 2πlx 2l l l When ensemble of standard random matrices with Gaussian distribution is perturbed by a rank-one perturbation, the distribution of x = N Ψ 1 E) 2 has the same functional form but with le) N Ψ 1 E) 2 = κ κ σ N E ) 1 When κ 2 > 1 there exists one collective state whose mean energy is E c = σ Nκ + κ 1 ) and Ψ c) 1 2 = 1 κ 2 When all eigenfunctions in a large energy interval are considered their distribution is not Gaussian but is given by an integral over Gaussian functions In the limit N all other components of eigenfunctions except Ψ 1 E)) remain distributed according to the usual PT distribution Experimentally one has to fit a width distribution for a resonance in a small energy window to the local PT formula thus finding the le) from the fit Taking into account together different resonances with different energies is not a sensitive way to investigate this phenomenon

47 Conclusion Standard PT distribution = eigenvectors of large random matrix are i.i.d.r.v. P 1 x) = 1 exp x ), P 2 x) = 1 exp x ), x = N Ψ 2, l = 1 2πlx 2l l l When ensemble of standard random matrices with Gaussian distribution is perturbed by a rank-one perturbation, the distribution of x = N Ψ 1 E) 2 has the same functional form but with le) N Ψ 1 E) 2 = κ κ σ N E ) 1 When κ 2 > 1 there exists one collective state whose mean energy is E c = σ Nκ + κ 1 ) and Ψ c) 1 2 = 1 κ 2 When all eigenfunctions in a large energy interval are considered their distribution is not Gaussian but is given by an integral over Gaussian functions In the limit N all other components of eigenfunctions except Ψ 1 E)) remain distributed according to the usual PT distribution Experimentally one has to fit a width distribution for a resonance in a small energy window to the local PT formula thus finding the le) from the fit Taking into account together different resonances with different energies is not a sensitive way to investigate this phenomenon Introduction of rank-one interaction does not change local spectral statistics

48 Conclusion Standard PT distribution = eigenvectors of large random matrix are i.i.d.r.v. P 1 x) = 1 exp x ), P 2 x) = 1 exp x ), x = N Ψ 2, l = 1 2πlx 2l l l When ensemble of standard random matrices with Gaussian distribution is perturbed by a rank-one perturbation, the distribution of x = N Ψ 1 E) 2 has the same functional form but with le) N Ψ 1 E) 2 = κ κ σ N E ) 1 When κ 2 > 1 there exists one collective state whose mean energy is E c = σ Nκ + κ 1 ) and Ψ c) 1 2 = 1 κ 2 When all eigenfunctions in a large energy interval are considered their distribution is not Gaussian but is given by an integral over Gaussian functions In the limit N all other components of eigenfunctions except Ψ 1 E)) remain distributed according to the usual PT distribution Experimentally one has to fit a width distribution for a resonance in a small energy window to the local PT formula thus finding the le) from the fit Taking into account together different resonances with different energies is not a sensitive way to investigate this phenomenon Introduction of rank-one interaction does not change local spectral statistics Experimentally for resonances 3 L) at small distances = RM prediction but deviates from it at L d as for dynamical systems