2 [24]). Higher partial wave scatterers were analyzed in the Ref. [25]. monic potentials (see review [5] for references). The unperturbed Hamiltonian of the relative motion is given by the sum of the longitudinal and transverse kinetic energies and the transverse trapping energy ˆ U = µω 2 $_{⊥}$ρ $^{2}$/ 2, At rational values of λ/π $^{2}$, the unperturbed energy spectrum shows degeneracies that are not fully lifted in the full [deduced from (4)] spectrum. To minimize the effect of the degeneracies, we, following Ref. [11], fix the length of the guide to ( L/a ⊥ ) 2 ≡ λ = 2 φ$_{gr}$π 7 ≈ 9774, where φ$_{gr}$ $_{= (1 +}$√$_{5)}$$_{/}$$_{2 is the golden ratio.}$ Standard quantum-chaotic tests .– The results of the study of the level spacing distribution in our system are fully consistent with the analogous results for the ˇ Seba billiard [12, 13]. For the energy range E > ∼ 100¯ hω ⊥ and the aspect ratio ( L/a ⊥ ) 2 = 2 φ$_{gr}$π $^{7}$, the distribu- tion quickly converges to the ˇ Seba distribution [12] at a$_{s}$ > ∼ 10 − $^{1}$a ⊥ . This distribution does show a gap at small level spacings but fails to reproduce the Gaussian tail predicted by the Gaussian Orthogonal Ensemble. At small a$_{s}$ the level spacing distribution tends to the Pois- son one. We have also verified that in the unitary regime, a$_{s}$ ≫ a ⊥ , the statistics of the point-by-point variations of the eigenstate wavefunction in the spatial representa- tion is close to the Gaussian one, according to the general prediction [26] and a particular observation in the case of the ˇ Seba billiard [11]. where ω ⊥ is the transverse frequency, Δ$_{ρ}$ is the transverse two-dimensional Laplacian, and µ is the reduced mass. In our model, the transverse and longitudinal degrees of freedom are coupled by a potential of a Fermi-Huang type [5]: ˆ H$_{0}$ = − ¯ h 2 2 µ ∂ 2 ∂z 2 − ¯ h 2 2 µ Δ$_{ρ}$ + ˆ U − ¯ hω ⊥ , (1) ˆ V = 2 π ¯ h $^{2}$a$_{s}$ µ δ$_{3}$ ( r )( ∂/∂r )( r · ) , (2) were, a$_{s}$ is the three-dimensional s -wave scattering length. This approximation is valid at low collision en- ergies (e.g., below 300 mK for Li, see a recent review [19], while the ultracold atom energies lie well below 1 µ K). The model (2) stands in a excellent agreement with the experimentally measured equation of state of the waveguide-trapped Bose gas [20] and with numerical calculations with finite-range potentials [17, 21]. The ring geometry of the waveguide imposes period- L boundary conditions along z -direction. In what fol- lows, we will restrict the Hilbert space to the states that have zero z -component of the angular momentum and that are even under the z ↔ − z reflection; the inter- action has no effect on the rest of the Hilbert space. The unperturbed spectrum is given by (see [22]) E$_{nl}$ = 2¯ hω ⊥ n + ¯ h $^{2}$(2 πl/L ) $^{2}$/ (2 µ ), where n ≥ 0 and l ≥ 0 are the transverse and longitudial quantum numbers, respec- tively. Eigenstate thermalization .– According to the eigen- state thermalization hypothesis [7–9], the ability of an isolated quantum system (with few or many degrees of freedom regardless) to thermalize follows from the sup- pression of the eigenstate-by-eigenstate fluctuations of the quantum expectation values of relevant observables. Scattered dots at the Fig. 1 show the quantum expec- tation values of the transverse trapping energy, 〈 α | ˆ U | α 〉 as a function of the eigenstate energy. The variation of 〈 α | ˆ U | α 〉 remains (in contrast to the quantum-chaotic billiards [27]) of the order of the mean, even for the en- ergies much larger than any conceivable energy scale of the system. Here and below, we work in the regime of the infinitely strong interactions, a$_{s}$ = 10 $^{6}$a ⊥ . The interacting eigenfunctions with the eigenenergy E$_{α}$ can be expressed as (see [22]) For a given eigenstate, quantum expectation values of the occupation probabilities for the transverse (Fig. 2a) and longitudinal (Fig. 2b) modes show no conver- gence to their microcanonical expectation values, P ( n ) ∝ ∫ dl δ ( E$_{nl}$ − E ) and P ( l ) ∝ ∫ dn δ ( E$_{nl}$ − E ) respectively. Both distributions are heavily dominated by only a few peaks (cf. scars in the momentum space for ˇ Seba billiards [15]). where the rescaled energy ϵ$_{α}$ is given by ϵ$_{α}$ ≡ λE$_{α}$/ (2¯ hω ⊥ ), λ ≡ ( L/a ⊥ ) 2 is the aspect ratio, ζ z/L − 1 / 2, ξ ≡ ( ρ/a ⊥ ) $^{2}$, L (0) n ( ξ ) are the Legendre poly- nomials, a ⊥ = (¯ h/µω ⊥ ) 1 / 2 is the size of the transverse ground state, and C$_{α}$ is the normalization constant. The eigenenergies are solutions of the following transcenden- tal equation (see [22]): 〈 ρ, z | α 〉 = C$_{α}$ ∞ ∑ n =0 cos ( $_{2}$√$_{ϵ}$$_{α}$ − λnζ ) √$_{ϵ}$$_{α}$ − λn $_{sin}$√$_{ϵ}$$_{α}$ − λn e − ξ 2 L (0) n ( ξ ) , (3) √$_{λ}$ ∞ ∑ n =0 $_{cot}$√$_{ϵ}$$_{α}$ − λn + i √$_{ϵ}$$_{α}$ λn − ζ ( 1 2 , − ϵ$_{α}$ λ ) = a ⊥ a$_{s}$ , (4) Relaxation from an initial state .– Now we are going to address directly the ability of our system to ther- malize from an initial state | ψ ( t = 0) 〉 . In this case, the infinite time average of the quantum expectation value of an observable ˆ A will be given by A$_{relax.}$ ≡ lim$_{t}$$_{max}$ →∞ (1 /t$_{max}$ ) ∫ t$_{max}$ 0 〈 ψ ( t ) | ˆ A | ψ ( t ) 〉 = ∑ α |〈 α | ψ ( t = 0) | $^{2}$〈 α | ˆ A | α 〉 , where | α 〉 are the eigenstates of the system (see [8]). For a quantum-chaotic system, it is expected that the above infinite time average coincides with the where ζ ( ν, x ) is the Hurwitz ζ -function (see [5]). Similar solutions were obtained for two atoms with a zero-range interaction in a cylindrically-symmetric harmonic poten- tial [23] (that system was analysed numerically in Ref.