Photodisintegration of 4He and the Supermultiplet Potential Model of Cluster-Cluster Interactions

p+³H and p+³He elastic collisions are described in terms of a supermultiplet model with [ƒ]-dependent potentials. The phase shifts δ ^[ƒ] _L,t,S (E) with [ƒ] = [4], t = S = 0, L = even are reconstructed from the observable nuclear phase shifts δ _L,S (E) of the above collisions. So, the initial-state interaction V ^[4] _L,0,0 (R) of the ⁴He +Υ→³H +p(³He +n) reaction can be found unambiguously, while the final-state interaction V ^[31] _{L, 1, 0} (R) is defined by the observablesδ _L,0 (E) of p+³He scattering. The data on the proton momentum distribution in ⁴He and on the charge-exchange reaction ³H +p→³He +n confirm the model. In calculating the above photonuclear reactions, in addition to the initial-state and final-state antisymmetrizations, preserving the corresponding symmetry [ƒ], the nucleon-nucleon correlations in the ³H (³He) subsystem were also taken into account. The results are in good agreement both with recent experimental data and theoretical investigations by sofianos, Fiedeldey, and Sandhas, who followed a rather different approach. Received August 5, 1994; revised November 30, 1994; accepted for publication December 30, 1994     

Representations of the cyclically symmetric q-deformed algebra U q(so3)

An algebra homomorphism from the nonstandard q-deformed (cyclically symmetric) algebra U _q(so₃) to the extension Û _q(sl₂) of the Hopf algebra U _q(sl₂) is constructed. Not all irreducible representations (IR) of U _q(sl₂) can be extended to representations of Û _q(sl₂). Composing the homomorphism with irreducible representations of Û _q(sl₂) we obtain representations of U _q(so₃). Not all of these representations of U _q(so₃) are irreducible. Reducible representations of U _q(so₃) are decomposed into irreducible components. In this way we obtain all IR of U _q(so₃) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so₃ when q 1.     

Unitary representations of suq(2) on the plane for q ∈ R+ or generic q ∈ S1

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra su _q(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions. In their study, the values of q were implicitly restricted to q R⁺. In the present paper, we extend their work to the case of generic values of q S ¹ (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, q R⁺ and generic q S ¹, by determining some appropriate scalar products.     

Positive-energy irreps of the quantum anti de Sitter algebra

We obtain positive-energy irreducible representations of theq-deformed anti de Sitter algebraU _q (so(3, 2)) by deformation of the classical ones. When the deformation parameterq isN-th root of unity, all these irreducible representations become unitary and finite-dimensional. Generically, their dimensions are smaller than those of the corresponding finite-dimensional non-unitary representations ofso(3, 2). We discuss in detail the singleton representations, i.e. the Di and Rac. WhenN is odd, the Di has dimension 1/2(N ²–1) and the Rac has dimension 1/2(N ²+1), while ifN is even, both the Di and Rac have dimension 1/2N ². These dimensions are classical only forN=3 when the Di and Rac are deformations of the two fundamental non-unitary representations ofso(3, 2).Presented at the 4th Colloquium Quantum groups and integrable systems, Prague, 22–24 June 1995.On leave from Bulgarian Acad. Sci., Institute of Nuclear Research and Nuclear Energy, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria.On leave from Pennsylvania State University (Fulbright scholar).     

Study of anomalous lasing behavior on the twoJ=0−1 transitions in Ne-like V

We present an experimental report on the recent observation of lasing at 26.1 and 30.4 nm on the 3p ¹ S ₀–3s ³ P ₁ [termedG((0–1)] and 3p ¹ S ₀–3s ¹ P ₁ [termedE(0–1)] transitions in Ne-like V, in which the normally weakG(0–1) transition was observed to lase more strongly than theE(0–1) transition. The experiment was performed on the Asterix IV iodine laser with a prepulse 5.23 ns before the main pulse. At a target length of about 2.5 cm, it was found that, while theG(0–1) andE(0–1) lines have comparable intensities in V, theE(0–1) line dominates spectra from Mn, Cr, Ti and Sc, which have adjacent nuclear charges. It was also found that the two lasers in V also have different temporal histories and spatial distributions. This is in contrast to the LASNEX + XRASER simulation, which predicts virtually similar temporal and spatial behavior for the two transitions. On leave from: Shanghai Institute of Optics and Fine Mechanics, P.O. Box 800211, Shanghai, People's Republic of China     

Saturation of Coulomb sum rules in the <Superscript>6</Superscript>Li case

The Coulomb sums S _L(q) of the ⁶Li nucleus have been obtained from electron scattering measurements at 3-momentum transfers q = 1.125–1.625 fm⁻¹. It is found that at q > 1.35 fm⁻¹ the Coulomb sum of the nucleus becomes saturated: S _L(q) = 1 .     

Representations of the nonstandard (twisted) deformationU′ q(so n ) forq a root of unity

Irreducible representations of the algebrasU′ _q(so _n ) forq a root of unityq ^p=1 are given. The main class of these representations act onp ^N-dimensional linear space (whereN is a number of positive roots of the Lie algebra so _n ) and are given byr = dim so _n complex parameters. Some classes of degenerate irreducible representations are also described. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. The research described in this publication was made possible in part by Grant UP1-2115 of the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF).     

A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group

We construct a quantum version of the SU(2) Hopf bundle S⁷→S⁴. The quantum sphere S⁷_q arises from the symplectic group Sp_q(2) and a quantum 4-sphere S⁴_q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S⁴_q) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S⁴. We compute the fundamental K-homology class of S⁴_q and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SU_q(2) on S⁷_q such that the algebra A(S⁷_q) is a non-trivial quantum principal bundle over A(S⁴_q) with structure quantum group A(SU_q(2)).     

Electromagnetic form factors of nucleons in a relativistic square-root potential model of independent quarks

The nucleon electromagnetic form factorsG _E ^P (q²),G _M ^P (q²) and the axial-vector form factor G_A(q²) are studied in a relativistic model of independent quarks confined by an equally mixed scalar-vector square root potentialV _q(r)=1/2(1+γ ⁰)(ar ^1/2+ν ₀) taking into account the appropriate centre-of-mass corrections. The respective root-mean-square radii associated withG _E ^P (q²) and G _A (q²) come out as [〈r ²〉 _E ^P ]^1/2=0.86 fm and 〈r _A ² 〉^1/2=0.88 fm. Restoration of chiral symmetry in this model is discussed to derive the pion-nucleon form factorG _πNN(q²) and consequently the pion-nucleon coupling constant is obtained asg _πNN(q²)=12.81 as compared tog _πNN(q²)_exp⋍13.     

Schur duality in the toroidal setting

The classical Frobenius-Schur duality gives a correspondence between finite dimensional representations of the symmetric and the linear groups. The goal of the present paper is to extend this construction to the quantum toroidal setup with only elementary (algebraic) methods. This work can be seen as a continuation of [J, D1 and C2] (see also [C-P and G-R-V]) where the cases of the quantum groups U _q (sl(n)), Y(sl(n)) (the Yangian) and U _q (sl(n)) are given. In the toroidal setting the two algebras involved are deformations of Cherednik's double affine Hecke algebra introduced in [C1] and of the quantum toroidal group as given in [G-K-V]. Indeed, one should keep in mind the geometrical construction in [G-R-V] and [G-K-V] in terms of equivariant K-theory of some flag manifolds. A similar K-theoretic construction of Cherednik's algebra has motivated the present work. At last, we would like to lay emphasis on the fact that, contrary to [J, D1 and C2], the representations involved in our duality are infinite dimensional. Of course, in the classical case, i.e.,q=1, a similar duality holds between the toroidal Lie algebra and the toroidal version of the symmetric group. The authors would like to thank V. Ginzburg for a useful remark on a preceding version of this paper. Communicated by M. Jimbo     

Potts Model on Infinite Graphs and the Limit of Chromatic Polynomials

 Given an infinite graph 𝔾 quasi-transitive and amenable with maximum degree Δ, we show that reduced ground state degeneracy per site W _r (𝔾, q) of the q-state antiferromagnetic Potts model at zero temperature on 𝔾 is analytic in the variable 1/q, whenever |2Δe ³ /q|<1. This result proves, in an even stronger formulation, a conjecture originally sketched in [12] and explicitly formulated in [16 and 19], based on which a sufficient condition for W _r (𝔾, q) to be analytic at 1/q=0 is that 𝔾 is a regular lattice. Received: 16 January 2002 / Accepted: 17 October 2002 Published online: 18 February 2003 RID="*" ID="*" Partially supported by CNPq (Brazil) RID="**" ID="**" Partially supported by CNR, G.N.F.M. (Italy) Communicated by H. Spohn     

Metastable Behavior for Bootstrap Percolation on Regular Trees

We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least θ occupied neighbors, occupied sites remain occupied forever. It is known that, when b>θ≥2, the limiting density q=q(p) of occupied sites exhibits a jump at some p _T=p _T(b,θ)∈(0,1) from q _T:=q(p _T)<1 to q(p)=1 when p>p _T. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p _T+h with h>0 and show that, as h ↓0, the system lingers around the “critical” state for time order h ^−1/2 and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q∈(q _T,1) converges, as h ↓0, to a well-defined measure.     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue

Let H=−Δ+V(x) be a three dimensional Schrödinger operator. We study the time decay in L^p spaces of scattering solutions e^−itHP_cu, where P_c is the orthogonal projection onto the continuous spectral subspace of L²(R³) for H. Under suitable decay assumptions on V(x) it is shown that they satisfy the so-called L^p-L^q estimates ||e^−itHP_cu||_p≤(4π|t|)^{−3(1/2−1/p)}||u||_q for all 1≤q≤2≤p≤∞ with 1/p+1/q=1 if H has no threshold resonance and eigenvalue; and for all 3/2<q≤2≤p<3 if otherwise.     

Finite Dimensional Unitary Representations of Quantum Anti–de Sitter Groups at Roots of Unity

We study irreducible unitary representations of U _q (SO(2,1)) and U _q (SO(2,?3)) for q a root of unity, which are finite dimensional. Among others, unitary representations corresponding to all classical one-particle representations with integral weights are found for , with M being large enough. In the “massless” case with spin bigger than or equal to 1 in 4 dimensions, they are unitarizable only after factoring out a subspace of “pure gauges” as classically. A truncated associative tensor product describing unitary many-particle representations is defined for . Received: 27 November 1996 / Accepted: 28 July 1997     

Clusters of cycles

A cluster of cycles (or (r,q)-polycycle) is a simple planar 2-connected finite or countable graph G of girth r and maximal vertex-degree q, which admits an (r,q)-polycyclic realization P(G) on the plane. An (r,q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q; (ii) all interior faces (denote their number by p_r) are combinatorial r-gons; (iii) all vertices, edges and interior faces form a cell-complex.An example of (r,q)-polycycle is the skeleton of (r^q), i.e. of the q-valent partition of the sphere, Euclidean plane or hyperbolic plane by regular r-gons. Call spheric pairs (r,q)=(3,3),(4,3),(3,4),(5,3),(3,5). Only for those five pairs, P((r^q)) is (r^q) without exterior face; otherwise, P((r^q))=(r^q).Here we give a compact survey of results on (r,q)-polycycles. We start with the following general results for any (r,q)-polycycle G: (i) P(G) is unique, except of (easy) case when G is the skeleton of one of the five Platonic polyhedra; (ii) P(G) admits a cell-homomorphism f into (r^q); (iii) a polynomial criterion to decide if given finite graph is a polycycle, is presented.Call a polycycle proper if it is a partial subgraph of (r^q) and a helicene, otherwise. In [ARS Comb. A 29 (1990) 5], all proper spheric polycycles are given. An (r,q)-helicene exists if and only if p_r>(q−2)(r−1) and (r,q)≠(3,3). We list the (4,3)-, (3,4)-helicenes and the number of (5,3)-, (3,5)-helicenes for first interesting p_r. Any outerplanar (r,q)-polycycle G is a proper (r,2q−2)-polycycle and its projection f(P(G)) into (r^2q−2) is convex. Any outerplanar (3,q)-polycycle G is a proper (3,q+2)-polycycle.The symmetry group Aut(G) (equal to Aut(P(G)), except of Platonic case) of an (r,q)-polycycle G is a subgroup of Aut((r^q)) if it is proper and an extension of Aut(f(P(G))), otherwise. Aut(G) consists only of rotations and mirrors if G is finite, so its order divides one of the numbers 2r, 4 or 2q. Almost all polycycles G have trivial AutG.Call a polycycle G isotoxal (or isogonal, or isohedral) if AutG is transitive on edges (or vertices, or interior faces); use notation IT (or IG, or IH), for short. Only r-gons and non-spheric (r^q) are isotoxal. Let T^*(l,m,n) denote Coxeter’s triangle group of a triangle on S², E² or H² with angles π/l, π/m, π/n and let T(l,m,n) denote its subgroup of index 2, excluding motions of 2nd kind. We list all IG- or IH-polycycles for spheric (r,q) and construct many examples of IH-polycycles for general case (with AutG being above two groups for some parameters, including strip and modular groups). Any IG-, but not IT-polycycle is infinite, outerplanar and with same vertex-degree, we present two IG-, but not IH-polycycles with (r,q)=(3,5),(4,4) and AutG=T(2,3,∞)PSL(2,Z), T^*(2,4,∞). Any IH-polycycle has the same number of boundary edges for each its r-gon. For any r≥5, there exists a continuum of quasi-IH-polycycles, i.e. not isohedral, but all r-gons have the same 1-corona.On two notions of extremal polycycles:

1. We found for the spheric (r,q) the maximal number n_int of interior points for an (r,q)-polycycle with given p_r; in general case, (p_r/q)≤n_int<(rp_r/q) if any r-gon contains an interior point.

2. All non-extendible (r,q)-polycycles (i.e. not a proper subgraph of another (r,q)-polycycle) are (r^q), four special ones, (possibly, but we conjecture their non-existence) some other finite (3,5)-polycycles, and, for any (r,q)≠(3,3),(3,4),(4,3), a continuum of infinite ones.

On isometric embedding of polycycles into hypercubes Q_m, half-hypercubes and, if infinite, into cubic lattices Z_m, : for (r,q)≠(5,3),(3,5), there are exactly three non-embeddable polycycles (including (4³)−e, (3⁴)−e); all non-embeddable (5,3)-polycycles are characterized by two forbidden sub-polycycles with p₅=6.     

Nonintegrability of some Hamiltonian systems,scattering and analytic continuation

We consider canonical two degrees of freedom analytic Hamiltonian systems with Hamiltonian functionH=1/2[p ₁ ² +p ₂ ² ]+U(q ₁,q ₂), where U(q₁, q₂) = 1/2[– v²q ₁ ² + ²q ₂ ² ] +O(q ₁ ² + q ₂ ² )^3/2) and _q2 U(q₁, 0) = 0. Under some additional, not so restrictive hypothesis, we present explicit conditions for the exisstence of transversal homoclinic orbits to some periodic orbits of these systems. We use a theorem of Lerman (1991) and an analogy between one of its conditions with the usual one dimensional quantum scattering problem. The study of the scattering equation leads us to an analytic continuation problem for the solutions of a linear second order differential equation. We apply our results to some classical problems.Partially supported by FAPESP (Fundação de Amparo a Pesquisa do Estado de São Paulo), Projeto Temático Transição de Fase em Sistemas Evolutivos, Grant 90/3918-5.Visiting scholar (acadmic year 92/93) financially supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), Brasil.     

Generalized Position and Momentum Tsallis Entropies

We use the generalized Tsallis entropy S _v(q = (1 –( ^w _i=1 p ^q _i )/(q – 1) to studythe information measurement in position and momentum space for simple quantummechanical systems. We consider here the hydrogen atom in three dimensions andthe D-dimensional harmonic oscillator to calculate the position and momentumentropies analytically for ground and excited states which involve classicalorthognal polynomials. In both the cases we verify the generalized entropicuncertainty relation and pseudoadditivity relation. We also study the effect ofscreening on the entropies. We compare the present results with the correspondingresults of the Shannon formalism.     

The hyperfine splittings in heavy-light mesons and quarkonia

Hyperfine splittings (HFS) are calculated within the Field Correlator Method, taking into account relativistic corrections. The HFS in bottomonium and the B _q (q = n, s) mesons are shown to be in full agreement with experiment if a universal coupling α _HF = 0.310 is taken in perturbative spinspin potential. It gives M(B*) −M(B) = 45.7(3) MeV, M(B _s ^* ) − M(B _s ) = 46.7(3) MeV (n _f = 4), while in bottomonium Δ_HF(b $ \bar b $ \bar b ) = M(Υ(9460)) − M(η _b (1S)) = 63.4 MeV for n _f = 4 and 71.1 MeV for n _f = 5 are obtained; just the latter agrees with recent BaBar data. For unobserved excited states we predict M(Υ(2S))−M(η _b (2S)) = 36(2)MeV,M(Υ(3S))−M(η _b (3S)) = 28(2)MeV, and also M(B _c ^*) = 6334(4) MeV, M(B _c (2S)) = 6868(4) MeV, M(B _c ^* (2S)) = 6905(4) MeV. The mass splittings between D(2³ S ₁) − D(2¹ S ₀), D _s (2³ S ₁) − D _s (2¹ S ₀) are predicted to be ∼75 MeV, which are significantly smaller than in several other studies but agree with the mass splitting between recently observed D(2533) and D*(2610).