Quantum pseudo-integrable Hamiltonian impact systems. Omer Yaniv and Vered Rom-Kedar Department of Computer Science and Applied Mathematics

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Quantum pseudo-integrable Hamiltonian impact systems.

Omer Yaniv and Vered Rom-Kedar

Department of Computer Science and Applied Mathematics,

Weizmann Institute of Science, Rehovot 7610001, Israel

(Dated: October 7, 2022)

Quantization of a toy model of a pseudointegrable Hamiltonian impact system is introduced,

including EBK quantization conditions, a veriﬁcation of Weyl’s law, the study of their wavefunctions

and a study of their energy levels properties. It is demonstrated that the energy levels statistics

are similar to those of pseudointegrable billiards. Yet, here, the density of wavefunctions which

concentrate on projections of classical level sets to the conﬁguration space does not disappear at large

energies, suggesting that there is no equidistribution in the conﬁguration space in the large energy

limit; this is shown analytically for some limit symmetric cases and is demonstrated numerically for

some nonsymmetric cases.

Quantum chaos studies how classical dynamics (inte-

grable and non-integrable) are reﬂected in the proper-

ties (e.g. eigenvalues and eigenfunctions) of the corre-

spondent quantum system. It is accepted that in inte-

grable systems, the distribution of the level spacing is

provided by the Poisson distribution e−s[1], while that

in chaotic systems (hereafter, meaning mixing system on

energy surfaces, studied by simulating chaotic billiards)

they distribute as eigenvalues of random matrix ensem-

bles (GOE) [2]. When a system has a mixed phase space,

which is the common behavior of smooth Hamiltonian

systems, it is found that a Berry-Robink distribution, a

convex hall of the Poisson and the GOE distributions,

describes the level spacing [3, 4]. This distribution re-

ﬂects the existence of eigenfunctions supported on the

islands of stability and of eigenfunctions supported on

the chaotic components of the classical phase-space [5].

Pseudointegrable dynamics, correspond to systems

with intermediate complexity: the phase space trajecto-

ries are not ergodic on the full energy surface, yet, they

are not always periodic or quasi-periodic. Such systems

arise in the study of plane polygonal rational billiards

(polygonal tables with all corners being rational frac-

tions of π), where trajectories move on invariant two-

dimensional surfaces of genus g > 1[6, 7]. The level

spacing in such quantum systems appears to have in-

termediate statistics: the nearest-neighbor distribution

displays repulsion at small distances and an exponential

decay at large distances [8].

Another important characteristic of quantum systems

is the asymptotic distribution of their wavefunctions. For

systems with classical ergodic dynamics, in the semi-

classical limit, the eigenfunctions which are equidis-

tributed form a density 1 sequence [9]. In particular,

such wavefunctions are equidistributed in both conﬁgu-

ration space and momenta space. The other wavefunc-

tions, which are not equidistributed, have scars - they

concentrate along invariant phase space sets or on sin-

gular sets of the classical dynamics [10, 11]. For chaotic

billiards, the most visible scars are associated with low

period unstable periodic orbits and orbits at corners of

the billiard table [10, 12].

Since plane rational polygonal billiards are ergodic

only in the conﬁguration space (and not in the momenta

space), equidistribution of the wavefunctions can be ex-

pected only in their conﬁguration representation. Fol-

lowing [9], it was established that also here, in the semi-

classical limit, scars in conﬁguration space can only ap-

pear for a vanishing density of eigenfunctions [13]. Yet,

it was observed, for ﬁnite energies, that some of the ex-

ceptional wavefunctions here have superscars; these con-

centrate on invariant sets associated with families of clas-

sical periodic orbits [14]. Such structures were observed

experimentally [15, 16].

In this letter we investigate eigenvalues statistics and

eigenfunctions properties of a class of systems that be-

longs to the recently discovered family of classical pseu-

dointegrable Hamiltonian systems with impacts. Such

systems combine motion under a smooth potential ﬁeld

with continuous symmetries and reﬂections from a cor-

responding family of billiards that keeps the continuous

symmetries only locally and not globally. For example,

trajectories of a separable Hamiltonian

H=H1+H2, Hi(qi, pi) = p2

2m+Vi(qi), i = 1,2(1)

in a right-angled polygonal billiard with at least one con-

cave corner are pseudointegrable [17, 18].

Here, we study the quantum step oscillators: we take

Vito be conﬁning potentials which are even smooth func-

tions with a single minimum at the origin and are mono-

tone elsewhere, and take the right angled polygon to be

R2\S, where

Sqwall ={(q1, q2)|q1< qwall

1≤0and q2< qwall

2≤0}.

(2)

The trajectories are conﬁned by the potential and reﬂect

from the step Sqwall [17], see Figure 1a. Since the step

boundaries are parallel to the axes, the vertical and hor-

izontal momenta are conserved at reﬂections, so the mo-

tion occurs along the level sets Hi(qi, pi) = Ei, i = 1,2.

Passing to the action angel coordinates of the smooth

separable system, provided Ei> Vi(qwall

i), i = 1,2,

the motion on each level set is conjugated to the di-

rected motion on the ﬂat cross-shaped surface, see Figure

1b. The direction of motion on this surface is given by

arXiv:2210.02854v1 [math-ph] 6 Oct 2022

ω2(E2)/ω1(E1)and the cross shaped concave corners are

at {±θwall

1(E1),±θwall

2(E2)}, where ωi(Ei)denotes the

frequency of the smooth periodic motion under Hiand

θwall

i(Ei)denotes the angle of an impacting trajectory

(with the convention that θi= 0 at the maximum of

qi). So, the direction of motion and the surface dimen-

sions depend continuously on (E1, E2). For the case of

harmonic oscillators, i.e. when Vi(qi) = 1

2ωiq2

i, the fre-

quencies are ﬁxed at ωiand the values of θwall

i(Ei)can

be explicitly computed. Equivalently, by folding the sur-

face, the motion on such level sets is conjugated to the

directed billiard motion on an L-shaped billiard, see Fig-

ure (1)c. Thus, this system is pseudointegrable [17]. In

general, the dynamics on such surfaces has non-trivial

ergodic properties. It was proven that if qwall

i<0for

i= 1,2, the motion is typically uniquely ergodic, and,

for the case of resonant harmonic oscillators, there are

level sets with co-existing periodic ribbons and dense or-

bits on some parts of the cross-shaped surface [18].

(a)

(b)

(c)

FIG. 1: A trajectory of a separable Hamiltonian

reﬂecting from a step. (a) Projection to the

conﬁguration space. (b) The corresponding directed

motion on the cross-shaped surface in the angles space.

to the corresponding billiard motion on an L-shaped

billiard. Here, Eq. (1) are integrated with elastic

reﬂections from the step of Eq. (2), with

Vi(qi) = 1

2ωiq2

i, ω1= 1, ω2=√2, qwall

1=qwall

−1, E1= 5.625, E2= 5.50.

As we are interested in quantization, and, in particular,

in studying the role of superscars in the system, we look

ﬁrst for families of periodic orbits. Given a family of

periodic orbits on a given level set (E1, E2=E−E1),

with µ= (µ1, µ2)turning points (µ1in the horizontal

direction and µ2in the vertical one), and b= (b1, b2)

impacts (b1with the right side of the step and b2with

the upper part of the step), and an action I(E;µ, b), we

can quantize it by using the EBK quantization conditions

[19, 20]:

I(E;µ, b) = ~(n+µ1+µ2

4+b1+b2

2).(3)

Moreover, denoting by Ii(Ei)the action of the smooth

Hisystem and by Iwall

i(Ei) = Rqi≥qwall

ipi(qi;Ei)dqi=

Ii2θwall

2πthe action of the impact Hisystem, we obtain:

I(E1, E2;µ, b) =

i=1

biIwall

i+ (µi−bi

2)Ii,(4)

namely, given µ,b, Ii(Ei)and θwall

i(Ei), we expect that

the EBK quantization rule will predict the energy levels.

Yet, in general, it is non-trivial to ﬁnd µand b(see e.g.

section 7 in [18]) nor to invert I(E1, E2;µ, b)on the given

family of periodic orbits.

We consider ﬁrst some simple limit cases in which pe-

riodic motion can be easily identiﬁed. When the step

is at the origin (S0=Sqwall

1=qwall

2=0), the corner angles

are ﬁxed at θwall

i(Ei)|qwall

1=qwall

2=0 =π

2, so the dimen-

sions of the cross-shaped surface are independent of the

energy. When the potentials are harmonic, the direc-

tion of motion, ω2/ω1is independent of the energy as

well and Ii=Ei/ωi. Thus, by choosing resonant har-

monic potentials and a step at the origin, we conclude

that for all partial energies the motion is periodic and

of the same type and that Iwall

i=Ii/2. In particular,

setting : ω1= 1, ω2=n

m(with gcd(n, m) = 1), it can

be shown that there are exactly 2 options for dynamics;

When mis odd there is a single family of periodic orbits,

whereas an even mleads to 2 distinct families of periodic

orbits. In this latter case, one of the families has half

of the action of the other one. Taking the simplest case

of n= 1, we can compute the number of impacts and

turning points for each of these families, and then, using

Eqs. (3) and (4) provide a prediction for the eigenvalues,

Ek. For odd m, we obtain that the periodic trajectory

has 3(m+1) turning points (µ1= 3m, µ2= 3) and m+1

impacts (b1=m, b2= 1), hence

Ek=k

1.5m+5(1 + m)

6m.(5)

For even mwe obtain that the ﬁrst family of periodic

orbits has 2(m+ 1) turning points (µ1= 2m, µ2= 2)

and m(b1=m, b2= 0) impacts, whereas the second one

has m+ 1 (µ1=m, µ2= 1) turning points and 1impact

(b1= 0, b2= 1), hence

k1=k1

m+4m+ 2

4m(6)

EII

k2=2k2

m+m+ 3

2m,(7)

In ﬁgure 2, we validate the above results. Notice that for

even mthere are inﬁnite number of energy levels at which

k1=EII

k2(marked with green lines), and in particular,

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Quantumpseudo-integrableHamiltonianimpactsystems.OmerYanivandVeredRom-KedarDepartmentofComputerScienceandAppliedMathematics,WeizmannInstituteofScience,Rehovot7610001,Israel(Dated:October7,2022)QuantizationofatoymodelofapseudointegrableHamiltonianimpactsystemisintroduced,includingEBKquantizationcondi...

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Quantum pseudo-integrable Hamiltonian impact systems. Omer Yaniv and Vered Rom-Kedar Department of Computer Science and Applied Mathematics

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