1. RESURGENCE in Quasiclassical Scattering Richard E. Prange Department of Physics, University of Maryland [Work done at MPIPKS, Dresden] Thanks to Peter Fulde and many others Supported by the BSF (with S. Fishman). Phys. Rev. Lett. 90, 070401-1-4 (2003).

2. Outline Review of resurgence in closed systems Review of the U-matrix formulation Resurgence and impedance ( main result ) Another U-matrix formulation of S-matrix Howland’s razor, resonance trapping, extreme approximations and graph scattering Summary

3. The most important formula in quantum c h a o s Gutzwiller Trace Formula, density of states p labels classical periodic orbits Miller Formula for scattering S- matrix p labels classical scattering orbits from channel j to channel i = = Typical form for Quasiclassical Approximations.

4. The Miller series is often a good approximation to the S-matrix The trace formula is never a good approximation to the energy levels. [It is good for other things, like energy level correlations, but that’s another story.]

5. One scattering orbit only in these directions so Miller sum for S ij is one term `In’ direction same as `out’ direction Reflection gives new direction If internal lines are on a periodic orbit, scattering orbits align. The periodic orbit has the same action as the sum of the scattering orbits. Scattering from a stadium billiard. Keep on going j i Conclusion: Trace of S n gives sum over period n orbits. Inside-outside duality S ij S ki S jk i k k j We will use pieces of internal orbits later (Bogomolny) U ab Large angular momentum orbit does not scatter

6. Inside quasiclassical energy levels are the zeroes of This gives the inside spectrum in terms of outside S matrix has finite rank, N, so s n can be expressed as sums of products of traces of powers of S These have the form a p exp(i S p /ħ) where p is a composite orbit Smilansky, Doron and Dietz found the ALSO a finite sum for the zeroes instead of an infinite sum for infinities The nontrivial part of

7. Resurgence Although finite, number of periodic orbits can increase exponentially with Use the unitarity of S! [The name resurgence was used by Berry and Keating. The results above were obtained independently by BK, Bogolmony and Smilansky et al. ] ¡Manifestly real! [cumulative smoothed DOS]

8. Benefits of Resurgence Number of orbits needed is greatly reduced, e.g. 10,000 before resurgence - > 100 after resurgence Zeroes of a manifestly real function Resurgence works in numerical applications, but usually there are better ways to do numerics. The real benefits are possible insights into a system, not so much for `universal’ phenomena, best studied via RMT, but non-universal, special phenomena, e. g. scars, superscars, special states, etc. which are reflected in the classical mechanics.

9. With resurgence, even gross approximations can have merit Example: Keeping only zeroth term s 0 = 1 gives nonsense before resurgence. After resurgence it is found that The mean level spacing is correct! To study the influence of orbits on wavefunctions, however we need to study scattering !

10. End of Review of Resurgence We now want to apply resurgence ideas to scattering. Simple scattering doesn’t need it, but resonant scattering does. Please direct questions to the many members of the audience more knowledgeable than me.

11. Resonant scattering Surfaces of Section, Bogolmony External [SSE] and Internal [SSI] Georgeot and Prange Miller Formula The U’s are (continuous) energy dependent matrices from one point on an SS to the next encounter with the SS. Doing integrals by stationary phase recovers Miller. NOT convergent Many choices of SS: Many operator expressions for S: Same Miller

12. Resonances Resonance energies, E = E a – iΓ a, the zeroes of D II (E) = det(1-U II (E)) are complex because U II is subunitary. Because U II has finite rank, D II can be expanded to a finite series. Because U II is NOT unitary, the resurgence arguments fail ! Question: Can a way be found to resurge?

13. The unitary U matrix Introduced by Ozorio de Almeida and Vallejos [In this context, U is really Bogolmony’s transfer operator] Livsic, Arov, Helton `anticipated’ O de A by 25 years! (we physicists really should keep up with the work of the Siberian mathematical engineers) Discovered for physicists by Fyodorov and Sommers The S matrix is unitary if U is.

14. Structure of U matrix elements S ab is the action of the classical orbit from point b on SSB to a first encounter at point a on SSA The U’s have finite rank. U EE has the rank of S U is unitary, U EE and U II are subunitary [Eigenvalues inside the unit circle]

15. The meaning of U It can be shown that the zeroes of are the energy levels of the closed system obtained by REFLECTING orbits at the external surfaces of section instead of entering or leaving. Remark: there is a considerable degree of arbitrariness in the definition of the `closed system’.

16. Weakly open is the challenging case : The widths Γ are comparable to the level spacing, and the number of orbits needed is large. The nearly closed case is relatively easy. The weak tunnelling in and out can be treated as a perturbation. Γ << level spacing. To do Miller, need to take ray splitting into account. The Miller series is rapidly convergent for the open case, Γ >> level spacing

17. A remarkable result The zeroes of D U (E) coincide with the zeroes of D S (E) = det(1-S(E)) A version of inside-outside duality The zeroes of D U (E) coincide with the zeroes of D W (E) = det(1-W(E)) and also triality Note also, W is unitary Doron and Smilansky found this without U.

18. Still no resurgence Take the special case that U EE = 0, no direct scattering CAN RESURGE THIS! Orbits involving this term are pseudo orbits.

19. Wigner and Impedance Let so where K is Wigner’s R-matrix, or alternatively the impedance matrix K(E) has poles at the energies of `the’ closed system, thus S = 1 at these energies, which we already knew. The residues at these poles are related to the widths (and shifts) of the resonances. We can thus apply resurgence to the impedance matrix makes S manifestly unitary

20. Resurgence for L and K Using 1/(1-λW) = Σ λ n X n /D W (E), taking the Hermitean conjugate, using the recursion relation X n = 1w n + WX n-1, etc, etc,… One obtains where In terms of orbits, L 1 is composed of one scattering orbit or scattering pseudo-orbit and some number, possibly zero, periodic orbit or periodic pseudo-orbit. Manifestly Hermitean

21. A complicated scattering orbit C A simple scattering orbit A A simple periodic pseudo-orbit B Orbit A composed with orbit B has almost the same total action as orbit C, and also the prefactors are almost the same. A+B appears in the expansion with opposite sign from C so their total contribution is small. This physics has been understood since Cvitanovic and Eckhardt. The Fredholm determinant expansion makes it systematic.

22. This is the main result The impedance matrix can be found by resurgence, AND it is manifestly Hermitean. Unfortunately, a matrix inversion is still needed to get S. So it is still tricky to address questions like weak localization. Some remarks: There is no unique closed system, so its energy levels are not unique. The distribution of an ensemble of impedance matrices corresponding to random matrix theory for the energy levels is independent of RMT symmetry. This is good in view of the previous remark. It is always possible to eliminate direct scattering by choice of SS’s. Resurgence, in bad cases, is not the most efficient numerical method.

23. Some additional results and examples

24. An important scattering formula Verbaashot, Weidenmüller and Zirnbauer, (also assuming no direct scattering) obtain for –S (different convention for S) Here H 0 is a presumed Hamiltonian for a closed system, (gives resonance positions) and V is a rectangular matrix connecting scattering channels to the closed system, giving the resonance widths. This formulation is convenient for RMT. It is not so suitable for quasiclassics.

25. A formula similar to VWZ’s. is an N I xN I idempotent matrix with trace N E, i. e. it has N E unit eigenvalues, with the rest zero. It is `geometrical’ and less energy dependent than UU. R determines the resonance position and the scale of total width, UU † the distribution of width over the channels.`Howland’s razor.’ This `width sum rule’ structure makes it possible to understand resonance trapping in this formulation.

26. Howland’s razor “No satisfactory definition of a resonance can depend only on the structure of a single operator on an abstract Hilbert space.” [after Barry Simon]

27. Estimation of typical resonance width of weakly open scattering systems. W(E) has N I eigenphases θ a (E) Near an energy level, one of the phases has the form θ a (E)  (E-E a )/Γ where Γ  N I δ/2π and δ is the mean level spacing. Then, when W is diagonal, R aa  (E-E a )/2Γ. The width matrix will not generally be diagonal when R is. On average, the effective size of will be N E /N I So the typical width is The case that the width matrix is almost diagonal when W is is related to the resonance trapping phenomenon

28. An extreme approximation Sinai/4 scatterer SSI, SSE Special orbits C H AO T I C Orbits In expansion for det(1-W), keep only w 0 = 1 Discard all X n ’s except X 0 = 1. That is, set a is the width of the lead, b the square size, k is of order π/a. So β is rather small, of order (a/b) 1/2, but not TOO small.

29. Resonances of the scattering matrix are given by the zeroes of Since if kb >> 1, Φ >> kb. To first approximation, the zeroes are where The widths are The remaining terms in the above expression give corrections to this result. In particular, no resonances are associated with the term That is, the bouncing ball just comes in and immediately goes out. { ¡Chaotic orbits are summarized by Φ !}

30. Graph Scattering, (Smilansky, again.)

31. Graph Scattering Note similarity with extreme approximation! c 2 +s 2 =1

32. Graph Scattering with a tunnelling barrier Tunnel barrier, reflection probability amplitude r, transmission amplitude t. S

33. Resonances for 0 < r < 1 Complex E that solve At r = 1, S = 1, E on closed spectrum. At r close to 1, E will acquire small imaginary part.

34. Resonance Trapping Resonance energies [just a few] Resonance widths Energies of closed, (t = 0) system The evolution of some resonance positions and widths of a toy graph scattering model as it goes from closed (t = 0) to open (t = 1) system. bouncing ball `chaotic’ states Opening the system more does NOT imply ALL resonances broaden. The bouncing ball takes up lots of width, leaving little for the others Avoided crossing for these para- meters

35. Summary Resurgence can be used to approximate the impedance matrix quasiclassically Because this approximation is robust, it can be used to obtain toy models which capture some features of a complex system These toy models are similar to but not identical to graph scattering toy models

36. More summary The method of unitary matrices employed here has many applications. Even within this method, there are alternative expressions for things like the S- matrix which are useful for different things.