SEMICLASSICAL DESCRIPTION OF THE

DENSITY OF STATES IN THE CONTINUUM REGION

OF THE HENON-HEILES SYSTEM

M. Brack, University of Regensburg

“Critical Stability of Few-Body Quantum Systems”

MPI-PKS Dresden, October 17 - 21, 2005

The Henon-Heiles system:

– periodic orbits and bifurcation cascade– classical escape over the barriers– Gutzwiller’s semiclassical trace formula– density of states above barrier

Work done in collaboration with:

P. Winkler (University of Nevada, Reno, USA)J. Kaidel (University of Regensburg, Germany)S. Fedotkin (INR Kiev, Ukraine) (for partial results)

Motivation

• Simple few-body reactions with energy thresholdsmay be parameterized by the motion of a parti-cle in a (multidimensional) potential landscapeV (x, y, . . . ) with one or more saddles.

• Classical approach: look for classical trajectories(including complex paths for tunnelling); calculatephase-space flow through PODS; . . .

• Quantum-mechanical approach: density of states,scattering matrix, . . .

• Here: We consider the well-known 2-dimensionalmodel potential V (x, y) of Henon and Heiles.Look for classical trajectories for escape; calculatedensity of states above the barriers in termsof periodic orbits using the semiclassical traceformula of Gutzwiller (generalized for bifurcationsand symmetry breaking)

The Henon-Heiles potential

V (x, y) = 12 (x2 + y2) + α (x2y − 1

3 y3)

• Originally: gravitational potential of a galaxy, withpossibility of escape for E > Eth = 1/6α2

[M. Henon and C. Heiles, Astron. J. 69, 73 (1964)]

• Model for the deformation (vibrational) energy oftriatomic molecules such as H+

3 , with possibility ofdissociation

• Model for an electron in a triangular quantum dotwith 3 external leads

• Formally: 3-particle cyclic Toda chain (integrable,bound!), Taylor expanded around x=y=0 andtruncated at 3rd order (becomes open and non-integrable!)

• Has become a textbook example of a non-integrablemixed system with transition from regular (E → 0or α → 0) to chaotic motion (E À Eth)

The shortest orbits

Equipotential lines and the three shortest periodicorbits A, B and C (at scaled energy e = 6Eα2 = 1, i.e.at threshold energy E = Eth) – note D3 symmetry!

-1.0 -0.5 0.0 0.5 1.0

-0.5

0.0

0.5

1.0

y

x

A

C

B

3 orbits B, always unstable, exist at all energies

2 orbits C, stable up to e ' 0.892, exist at all energies

3 orbits A, stable up to e ' 0.97, but then .....

A, B, C correspond to vibrational modes in a triatomicmolecule! [N. Fulton et al., J. Chem. Phys. 99, 906 (1993)]

Bifurcation cascade of the A orbit:

0.97 0.98 0.99 1.0e

-4

-2

0

2

4

trM

A6

R5

L6

R7

A7

A5

Trace of stability matrix of A orbit versus energy e:bifurcation of orbits R5 and L6

Orbit A undergoes an infinite sequence of pitchforkbifurcations before it reaches the barrier at e = 1 withperiod TA = ∞!

The orbits Rn and Lm born at these bifurcations existup to infinite energy (all unstable for e ≥ 1)!

Bifurcation cascade of the A orbit:

0.99925 0.9995 0.99975 1.0e

-4

-2

0

2

4

trM

0.97 0.98 0.99 1.0e

-4

-2

0

2

4

trM

A6

A8

R5

R7

R5

L6

L8

R7

R9

A7

A9

A5

A7

<zoom>

Zoom into the last 0.1% of the energy scale:bifurcation of orbits R7 and L8

Zoom again: orbits R9 and L10, and so on .....

0.99998 0.99999 1.0e

-4

-2

0

2

4

trM

0.99925 0.9995 0.99975 1.0e

-4

-2

0

2

4

trM

0.97 0.98 0.99 1.0e

-4

-2

0

2

4

trM

A6

A8

A10

R5

R7

R9

R5

R5, R7

L6

L8

L10

R7

R9

R11

A7

A9

A11

A5

A7

A9

<zoom>

<zoom>

“HH fan”: TrM'−4.183 for all R orbits; similar for L orbits!

Bifurcation cascades in a double-well potential

V (u, v) =1

2

(u2 − v2

)+ λ

(v4 − 1

2u2v2

)+

1

16λ

equipotential lines and projection along u = 0:

-0.75 -0.5 -0.25 0 0.25 0.5 0.75

v

-1

-0.5

0

0.5

1

u

-0.75 -0.5 -0.25 0 0.25 0.5 0.75

v

0

0.5

1

1.5

2

2.5

V(0

,v)

A

B

barrier at scaled energy e = 1; system is (quasi-)boundup to e = 9

A orbit in v direction has bifurcation cascades bothbelow barrier (for e ↑ 1) and above barrier (for e ↓ 1)!(TA → ∞ in both cases)

Self-similarity of R and L orbits

Shapes or R and L orbits in (x, y) plane at e = 1,successively zoomed in the x direction by factor 0.1631:

-0.1 0.0 0.1

-0.5

0.0

0.5

1.0

R5, L6

x*0.163

R7, L8

x*0.1632

R9, L10

x*0.1633

R11, L12

x*0.1634

R13, L14

x*0.1635

R15, L16

y

x

1more precisely: exp(−π/√

3) = 0.1630335 . . .(an analytical ’Feigenbaum constant’ !)

[M. Brack, Foundations of Physics 31, 209 (2991)]

Self-similarity of R and L orbits

Shapes or R and L orbits in (x, y) plane at e = 1,successively zoomed in the x direction by factor 0.163:

-0.1 0.0 0.1

-0.5

0.0

0.5

1.0

R5, L6

x*0.163

R7, L8

x*0.1632

R9, L10

x*0.1633

R11, L12

x*0.1634

R13, L14

x*0.1635

R15, L16

y

x

Now zoom also in y direction by factor 0.163 (frombarrier position at y = 1):

-0.1 0.0 0.1

-0.5

0.0

0.5

1.0R5, L6

(x,y)*0.163

R7, L8

(x,y)*0.1632

R9, L10

(x,y)*0.1633

R11, L12

(x,y)*0.1634

R13, L14

(x,y)*0.1635

R15, L16

y

x

Analytical form of these shapes near bifurcations givenby Lame functions[M. Brack, M. Mehta, K. Tanaka, J. Phys. A 34, 8199 (2001)]

The “Henon-Heiles fan”

Numerical finding:[M. Brack, Foundations of Physics 31, 209 (2991)]

for n, m → ∞ (i.e., for e → 1), the values of trM ofthe Rn orbits intersect linearly at trMRn = −4.183,those of the Lm orbits at trMLm = +8.183, thus

trMR,L(e) −→ 2 ∓ 6.183

(e − e∗

1 − e∗

)

where e∗ are their respective bifurcation energies.

Explanation by first-order perturbation theory, follow-ing Creagh [Ann. Phys. 248, 60 (1996)]:

using the term x2y as a perturbation for the rationaltori bifurcating in the separable HH potential (α = 1)V (x, y) = 1

2 (x2 + y2)− 13 y3 from the A orbit, we get:

trMR,L(e) −→ 2 ∓ 5.069

(e − e∗

1 − e∗

)

[S. Fedotkin, and M. Brack, work in progress]

Other orbits above the barrier (e ≥ 1)

-1.0 -0.5 0.0 0.5 1.0

-0.5

0.0

0.5

1.0

y

x

D

2 orbits D born at a bifurcation from C2 at e ' 0.892,

stable until e ' 1.238, then always unstable

3 orbits τ exist for all e > 1, always unstable

(play the roles of PODS!)

Escape from the Henon-Heiles potential

Unbound non-periodic orbits starting from (x, y) =(0, 0) at e = 1.2:

-1.0 -0.5 0.0 0.5 1.0

-0.5

0.0

0.5

1.0

y

x

Note the sensitive dependence of the exit barrier onthe initial direction!

Escape from the Henon-Heiles potential

Where does the particle exit, when it starts from thepoint (x,y) = (0,0) ?

We vary the initial velocity (components vx,vy)

-1.5 -1 -0.5 0 0.5 1 1.5x

-1.5

-1

-0.5

0

0.5

1

1.5

y vx

vy

blue

green red

and insert coloured points corresponding to the exitbarrier (blue, green, red) into a diagram (vx,vy)

Escape from the Henon-Heiles potential

-1.21 -0.61 0 0.61 1.21vx

-1.21

-0.61

0

0.61

1.21

vy

Colour gives exit barrier:blue: upwards

red: down right

green: down left

black: no escape (circle: e ≤ 1)

Escape from the Henon-Heiles potential

-1.21 -0.61 0 0.61 1.21vx

-1.21

-0.61

0

0.61

1.21

vy

Colour gives exit barrier:blue: upwards

red: down right

green: down left

black: no escape (circle: e ≤ 1)

zoom −→

0.39 0.7 1.0.55 0.85

-0.21

0.1

0.39

-0.05

0.25

Appearance of a fractal structure!

Escape from the Henon-Heiles potential

-1.21 -0.61 0 0.61 1.21vx

-1.21

-0.61

0

0.61

1.21

vy

Colour gives exit barrier:blue: upwards

red: down right

green: down left

black: no escape (circle: e ≤ 1)

zoom −→

0.39 0.7 1.0.55 0.85

-0.21

0.1

0.39

-0.05

0.25

further zoom −→

0.57 0.605 0.64

-0.03

0.03

0

black island for e > 1: capture by stable orbit D above barrier!

Bifurcation of the D orbit

trace of stability matrix:

1.165 1.17 1.175 1.18 1.185 1.19

e1 ee2

- 0.5

0

0.5

1

1.5

2

2.5

3

Tr

M~

G7

D9

D7

E8

E8

~)Im(Tr M

E’

~Im(Tr M )

G’

~Re(Tr M

E’,G’)

dashed lines show “ghost orbits” (needed for trace formula)

shapes of orbits D, E and G: (at e = 1.17, 1.18 and 1.19)

-0.5 0.0 0.5x

-0.5

0.0

0.5

1.0

y

-0.5 0.0 0.5x

-0.5

0.0

0.5

1.0

-0.5 0.0 0.5x

-0.5

0.0

0.5

1.0

solid: D, dashed: E, dash-dotted: G

A closer look at the captures by the D orbit:

-1.21 -0.61 0 0.61 1.21vx

-1.21

-0.61

0

0.61

1.21

vy

note the “capture islands” at angles n · 2π/6(n = 0, 1, 2, 3, 4, 5)!

Poincare surface of section (x = 0)

at e = 1.14, zoom of central stability island of D orbit

The Gutzwiller trace formula

The quantum-mechanical density of states:

g(E) =∑

n δ(E − En) = g(E) + δg(E)

Smooth part g(E): from extended Thomas-Fermi(ETF) model or Strutinsky smoothing of {En}

Oscillating part δg(E): semiclassical trace formula[M. Gutzwiller, J. Math. Phys. 12, 343 (1971)]

δg(E) ' ∑poApo(E) cos

[1~Spo(E) − π

2 σpo

]

Sum over periodic orbits (po) of the classical system!

Spo =∮

pop · dq = action integral along po

Apo = amplitude (related to stability and degeneracy)

σpo = Maslov index ( = Conley-Zehnder index forsymplectic path on a Lagrangian manifold)

In systems with mixed classical dynamics:no convergence of po sum due to bifurcations!

Influence of periodic orbit bifurcations

Amplitude Apo(E) for isolated orbit:

Apo(E) =Tpo(E)

rpo

√|det (Mpo(E) − 1)|

Mpo = stability matrix, rpo = repetition number ofprimitive periodic orbit

Apo diverges at po bifurcations, where detMpo = 1!(Reason: break-down of stationary-phase approximationin doing trace integrals)

Remedy: go beyond saddle-point approximation!This yields so-called uniform approximations

[Ozorio de Almeida & Hannay, Tomsovic et al., Creagh, Sieber,

Schomerus & Sieber, Brack et al., ...]

But: this becomes increasingly complicated with in-creasing orbit lengths. (Number of bifurcations growsexponentially!)

Coarse graining

For finite resolution of energy spectrum, convolutelevel density over energy range γ

gγ(E) =1√πγ

∑

n

e−(E−Enγ )

2

⇒ get exponential damping factor in trace formula,suppressing orbits with longer periods Tpo:

δgγ(E) ∼∑

po

Apo(E) e−(γTpo/2~)2cos

[1

~Spo(E) − π

2σpo

]

⇒ Only shortest orbits relevant for gross-shell effects!

Solves problem of convergence of po sum!

The Henon-Heiles level density below barrier

with semiclassical trace formula (γ = 0.4 ~ω):solid line: quantum-mechanical (α=0.03)

dotted line: semiclassical, Gutzwiller trace formula1

dashed line: semiclassical, Gutzwiller + uniform approximations2

0.0 0.2 0.4 0.6 0.8 1.0e

-8-4

04

8

g(e)

0.9 0.925 0.95 0.975 1.0e

-2-1012

g(e)

D7 bif.

R5 bif. L6 bif.

only 11 periodic orbits contribute: (up to e ∼ 0.85 only A,B,C!)

A5/6/7, C3, B4, R5 and L6 (k = 1, 2), D7 (k = 1)

1including uniform approximation for U(2) symmetry breaking

[M. Brack, P. Meier, K. Tanaka, J. Phys. A 32, 331 (1999)]2[J. Kaidel, M. Brack, Phys. Rev. E 70, 016206 (2004)]

Level density in continuum

The trace formula is mathematically justifiable also incontinuum regions. There, it describes the density ofresonances:

En −→ En − iΓn

Γn = width of resonance, ~/Γn ∝ escape time frompotential

Then, the level density g(E) is given by

∑

n

δ(E − En) −→ −1

πIm

∑

n

1

E − En + iΓn/2

after subtraction of non-resonating continuum density(∝

√E in 3d, constant in 2d)

Role of finite widths Γn:

• affect smooth level density g(E) [cf. H. Schomerus and

J. Tworzydlo, Phys. Rev. Lett. 93, 154102 (2004)]

• give “natural” coarse-graining (selection of shortestorbits) in semiclassical trace formula for δg(E)

The Henon-Heiles level density above barrier

using semiclassical trace formula with uniform ap-proximation of orbit D bifurcation (codimension two)(smooth part g(E) obtained by complex Strutinskysmoothing; has small uncertainties at e ∼> 1!)

Coarse-graining with γ = 0.5 (energy unit ~ω = 1):dashed line: semiclassical

solid line: quantum-mechanical (with complex spectrum, α=0.1)

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0e

-0.4

-0.2

0.0

0.2

0.4

g(e)

only 7 periodic orbits contribute (all unstable):C3, B4, R5 and L6 (k = 1), τ (k = 1, 2, 3)

Note the pronounced regular shell structure in a 99.9%chaotic system!

[J. Kaidel, P. Winkler, M. Brack, Phys. Rev. E 70, 066208 (2004)]

The Henon-Heiles level density above barrier

using semiclassical trace formula with uniform ap-proximation of orbit D bifurcation (codimension two)(smooth part g(E) obtained by complex Strutinskysmoothing; has small uncertainties at e ∼> 1!)

Coarse-graining with γ = 0.25 (energy unit ~ω = 1):dashed line: semiclassical

solid line: quantum-mechanical (with complex spectrum, α=0.1)

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0e

-3.0

-1.5

0.0

1.5

3.0

g(e)

18 periodic orbits contribute:C3, B4 (k = 1, 2); R5, L6, R7 and L8 (k = 1) (all unstable)

D7/9, G7 (k = 1, stable); E8 (k = 1, unstable) + 2 “ghosts”

τ (k = 1, 2, 3, 4, 5, all unstable)

[J. Kaidel, P. Winkler, M. Brack, Phys. Rev. E 70, 066208 (2004)]

Summary and Conclusions

1. Semiclassical trace formula (with uniform approxi-mations for bifurcating orbits) reproduces detailedshell structure in the quantum-mechanical densityof states in the continuum region

2. Only real orbits needed1 for reproducing density ofcomplex resonances

3. Pronounced shell structure in an (almost) chaoticsystem

4. Good prospectives for semiclassical calculations ofmolecular (or other few-body) reactions includingquantum effects

1apart from “ghosts” required for uniform approximations near

bifurcations

Thanks

1. To my collaborators and former students:– R. K. Bhaduri (McMaster, Canada)

– M. V. N. Murthy (I.M.Sc. Madras, India)

– S. Creagh (Nottingham, UK)

– K. Tanaka (Saskatoon, Canada)

– P. Winkler (Reno, USA)

– S. Fedotkin, A. Magner (I.N.R. Kiev, Ukraine)

– P. Meier, Ch. Amann, J. Kaidel (Regensburg)

2. To my sponsors:– Deutsche Forschungsgemeinschaft

– NSERC (Canada)

3. To a patient audience!