RESURGENCE in Quasiclassical Scattering

1. RESURGENCEinQuasiclassical Scattering • Richard E. Prange • Department of Physics, University of Maryland • [Work done at MPIPKS, Dresden] • Thanks to Peter Fuldeand 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 chaos Typical form for Quasiclassical Approximations. Gutzwiller Trace Formula, density of states p labels classicalperiodic orbits MillerFormula for scattering S- matrix p labels classicalscatteringorbits from channel j to channel i

4. The Miller series is oftena good approximationto the S-matrixThe trace formula is nevera good approximation tothe energy levels. [It is good for other things, like energy level correlations, but that’s another story.]

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

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

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

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 s0 = 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 Please direct questions to the many members of the audience more knowledgeable than me. We now want to apply resurgence ideas to scattering. Simple scattering doesn’t need it, but resonant scattering does.

11. Resonant scattering Surfaces of Section, Bogolmony External [SSE] and Internal [SSI] Georgeot and Prange Miller Formula NOT convergent 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. Many choices of SS: Many operator expressions for S: Same Miller

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

13. The unitary Umatrix • 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 Sab 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. UEE has the rank of S U is unitary, UEE and UII 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. triality A remarkable result The zeroes of DU(E) coincide with the zeroes of DS(E) = det(1-S(E)) Doron and Smilansky found this without U. A version of inside-outside duality and also The zeroes of DU(E) coincide with the zeroes of DW(E) = det(1-W(E)) Note also, W is unitary

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

19. Wigner and Impedance Let so makes S manifestly unitary 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

20. Resurgence for L and K Using 1/(1-λW) = Σ λnXn/DW(E), taking the Hermitean conjugate, using the recursion relation Xn = 1wn + WXn-1, etc, etc,… One obtains where Manifestly Hermitean In terms of orbits, L1 is composed of one scattering orbit or scattering pseudo-orbit and some number, possibly zero, periodic orbit or periodic pseudo-orbit.

21. A simple scattering orbit A A simple periodic pseudo-orbit B A complicated scattering orbit C 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, ANDit is manifestly Hermitean. Some remarks: Resurgence, in bad cases, is not the most efficient numerical method. It is always possible to eliminate direct scattering by choice of SS’s. Unfortunately, a matrix inversion is still needed to get S. So it is still tricky to address questions like weak localization. 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.

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 H0 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. is an NIxNI idempotent matrix with trace NE, i. e. it has NE unit eigenvalues, with the rest zero. It is `geometrical’ and less energy dependent than UU. A formula similar to VWZ’s. R determines the resonance positionand thescaleof 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 [after Barry Simon] “No satisfactory definition of a resonance can depend only on the structure of a single operator on an abstract Hilbert space.”

27. Estimation of typical resonance width of weakly open scattering systems. W(E) has NI eigenphases θa(E) Near an energy level, one of the phases has the form θa(E) (E-Ea)/Γ where Γ NIδ/2πand δ is the mean level spacing. Then, when W is diagonal, Raa (E-Ea)/2Γ. The width matrix will not generally be diagonal when R is. On average, the effective size of will be NE/NI 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. In expansion for det(1-W), keep only w0 = 1 Discard all Xn’s except X0 = 1. That is, set An extreme approximation Sinai/4 scatterer SSI, SSE CHAOTIC Orbits Special orbits 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 {¡Chaotic orbits are summarized by Φ !} 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.

30. Graph Scattering, (Smilansky, again.)

31. Graph Scattering c2+s2=1 Note similarity with extreme approximation!

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

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. Avoided crossing for these para- meters 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. Resonance Trapping Opening the system more does NOT imply ALL resonances broaden. The bouncing ball takes up lots of width, leaving little for the others bouncing ball Resonance widths `chaotic’ states Energies of closed, (t = 0) system Resonance energies [just a few]

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.