RESURGENCEin

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).

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

( )exp p

pp

Ea

iS

The most important formula

in quantum chaos

Gutzwiller Trace Formula, density of states

p labels classical periodic orbits

Miller Formula for scattering S- matrixp labels classical scattering orbits from channel j to channel i

( )OSC

D E =

( )ijS E =

Typical form

forQuasiclassical

Approximations.

The Miller series is oftena 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.]

One scattering orbit only in these directionsso Miller sum for Sij is one term`In’ direction same as `out’ direction

Reflection gives new direction

If internal lines are ona periodic orbit, scatteringorbits align.

The periodic orbit hasthe same action as thesum of the scatteringorbits.

Scattering from a stadium billiard.

Keep on going

j

i

Conclusion: Trace of Sn gives sum over period n orbits. Inside-outside duality

Sij

Ski

Sjk

i

k

k j

We will use pieces ofinternal orbits later (Bogomolny)

Uab

Large angular momentum orbitdoes not scatter

Inside quasiclassical energy levels

( ) det(1 ( ))SD E S E are the zeroes of

This gives the inside spectrum in terms of outside S matrix

( )S E has finite rank, N, so0

( ) det(1 ( ))N

nS n

n

D E S E s

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

Smilansky, Doron and Dietz found the

ALSO

a finite sum for the zeroes instead of an infinite sum for infinities

The nontrivial part of

ResurgenceResurgence

* †

*

0

det( )

exp( 2 ( ))

( ) ( , ) 2R

det(1 ) ( ,(1

, )

[

)1

e ( )]

N

N n n

i iS S n

n

SS S

s i E s

S D E

E e D E e s E

D E

N/2

( ) ( )E E N

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, Bogolmonyand Smilansky et al. ]

¡Manifestly real!

N

A

[cumulative smoothed DOS]

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.

•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

cos( ( ))S E N The mean level spacingis correct!

To study the influence of orbits on wavefunctions, however

we need to study scattering!

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.

Resonant scatteringSurfaces of Section,Bogolmony

External [SSE] and Internal [SSI]Georgeot and Prange

Miller Formula0

( ) ( )nEE E I II IE

n

S E U U U U

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

1

1 (( )

)EEII

E I IESU E

E U U U

Many choices of SS: Many operator expressions for S: Same Miller

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?

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

EE EI

IE II

U UU

U U

The S matrix is unitary if U is.

( ) exp( ( ) / )ab ab abU E a iS E Structure of U matrix elements

Sab is the action of the classical orbit from point b on SSB to a first encounterat 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]

The meaning of U• It can be shown that the zeroes of

( ) det(1 ( ))UD E U E are the energy levels of the closed system obtained by

REFLECTINGorbits at the external surfaces of section instead of entering or leaving.

Remark: there is a considerable degree of arbitrariness in thedefinition of the `closed system’.

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 weaktunnelling 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

A remarkable resultThe zeroes of DU(E) coincide with the zeroes of DS(E) = det(1-S(E))

A version of inside-outside duality

The zeroes of DU(E) coincide with the zeroes of DW(E) = det(1-W(E))

and also

1

1II IE EIEE

W U U UU

triality

Note also, W is unitary

Doron and Smilansky found this without U.

Still no resurgence

II IE EIW U U U 1

1EI IEII

S U UU

1

1EI IEL U

WU

Take the special case that UEE = 0,

no direct scattering

CAN RESURGETHIS!1

LS

L

Orbits in

volving this

term ar

e pseu

do orbits.

1

1EI IEIE EI

U UW U U

Wigner and Impedance1

2 2

iL K 1

1

iKS

iK

Let so

where

†K KK is Wigner’s R-matrix, or alternatively the impedance matrix

K(E) has poles at the energies of `the’ closed system, thus S = 1at these energies, which we already knew. The residues at thesepoles are related to the widths (and shifts) of the resonances.

We can thus apply resurgence to the impedance matrix

makes S manifestly unitary

Resurgence for L and K

2( )

( )W

iE

†

1 1L - LK

Using 1/(1-λW) = Σ λn Xn/DW(E), taking the Hermitean conjugate,using the recursion relation Xn = 1wn + WXn-1, etc, etc,… One obtains

where

2

1

/

0

Ii

EI

N

IEn

e

nL U X U

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.

Manifestly Hermitean

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 actionas orbit C, and also the prefactors are almost the same.

A+B appears in the expansion with opposite sign from Cso their total contribution is small.

This physics has been understood since Cvitanovic and Eckhardt.The Fredholm determinant expansion makes it systematic.

This is the main resultThe impedance matrix can be found by resurgence,The impedance matrix can be found by resurgence,ANDAND it is manifestly Hermitean.it is manifestly Hermitean.

Unfortunately, a matrix inversion is still needed to get S. So it isstill 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 correspondingto random matrix theory for the energy levels is independent of RMTsymmetry. 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.

Some additional results and examples

An important scattering formula

Verbaashot, Weidenmüller and Zirnbauer, (also assuming nodirect scattering) obtain for –S (different convention for S)

††

0

11 2S iV V

E H iVV

Here H0 is a presumed Hamiltonian for a closed system, (gives resonance positions) and V is a rectangular matrix connecting scattering channelsto the closed system, giving the resonance widths. This formulation is convenient for RMT. It is not so suitable for quasiclassics.

A formula similar to VWZ’s.

††

11 2

( )IE IEIE IE

S iU UR E iU U

1 ( )( )

1 ( )

W ER E i

W E

†IE IEU U 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.

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.

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]

Estimation of typical resonance width of weakly open

scattering systems.

†IE IEU U

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/2π and 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 isis related to the resonance trapping phenomenon

/EN

An extreme approximation

cos( ( ))W E N

Sinai/4 scatterer SSI, SSE

Special orbitsCHAOTIC Orbits

In expansion for det(1-W), keep only w0 = 1Discard all Xn’s except X0 = 1. That is, set

22 2

1 4i i ikb i ikb

EI IE

kaL e U U e e e e

ib

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.

Resonances of the scattering matrix are given by the zeroes of

2 * 2 ( ) 2( ) 1 ikb i E ikbD E e e Since 22 k / 2A if kb >> 1, Φ >> kb. To first

approximation, the zeroes are where * 2 ( ) 1i Ee

The widths aren

1 1ln( )

k | |n

A

The remaining terms in the above expression give corrections tothis result.

In particular, no resonances are associated with the term

2ikbe That is, the bouncing ball just comes inand immediately goes out.

{ ¡Chaotic orbits aresummarized by Φ !}

Graph Scattering, (Smilansky, again.)

Graph Scattering

2

2

0

s

c c

s

c

s

c

c D D c D

c

s

s s

s D sD D

U

c

0

0 s

cWD

D

22

1 1c

c

s

s

c D

D

s D

DL

2 2( ) 1 scD s DE Dc

2ikbcD e

2 ( )i ksD e

(1 )(1 )1

( )csS

E

DD

D

Note similarity with extreme approximation!

c2 +s2 =1

Graph Scattering with a tunnelling barrier

Tunnel barrier,reflection probabilityamplitude r, transmissionamplitude t.

' ( ...)

1

1

S S S S S S Sr r r r

r

t t

t S tSr

S

1

( )r

S E

Resonances for 0 < r < 1

(1 )(( (1 )1

(

))

)csS

D

E

E ED

D

Complex E that solve

At r = 1, S = 1, E on closed spectrum. At r close to 1, E will acquire small imaginary part.

Resonance Trapping

Resonance energies [just a few]

Res

onan

ce w

idth

s

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

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

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.