The Bethe Equation at q = 0, The Möbius Inversion Formula,and Weight Multiplicities: III. The X

It is shown that the numbers of off-diagonal solutions to the U _q (X ^(r) _N ) Bethe equation at q = 0 coincide with the coefficients in the recently introduced canonical power series solution of the Q-system. Conjecturally, the canonical solutions are characters of KR (Kirillov–Reshetikhin) modules. This implies that the numbers of off-diagonal solutions agree with the weight multiplicities, which is interpreted as a formal completeness of the U _q (X ^(r) _N ) Bethe ansatz at q = 0.     

Nested bethe ansatz for <Emphasis Type="Italic">y</Emphasis>(<Emphasis Type="Italic">gl</Emphasis>(n)) open spin chains with diagonal boundary conditions

In this proceeding we present the nested Bethe ansatz for open spin chains of XXX-type, with arbitrary representations (i.e. “spins”) on each site of the chain and diagonal boundary matrices (K ⁺(u), K ⁻(u)). The nested Bethe ansatz applies for a general K ⁻(u), but a particular form of the K ⁺(u) matrix. We give the eigenvalues, Bethe equations and the form of the Bethe vectors for the corresponding models. The Bethe vectors are expressed using a trace formula     

Asymptotic Bethe ansatz for the Sutherland–Römer model with open boundary conditions

By constructing the reflection spin-Dunkl operators, the integrable Sutherland–Römer model (SRM) with open boundary condition is established, which describes a one-dimensional, two-component, quantum many-particle system in which like particles interact with a pair potential g(g+1)/sinh²(r), while unlike particles interact with a pair potential −g(g+1)/cosh²(r). By solving the Schrödinger equation and using the properties of the hypergeometric functions and gamma functions, the two-particle scattering matrix and the reflection matrix are obtained in the framework of the asymptotic Bethe ansatz method. The Bethe ansatz equations of the system are obtained. The Hamiltonians of SRM with some other open boundary conditions are expressed explicitly. Our method can be generalized, as a example, to the boundary Calogero–Sutherland model which is also constructed by the reflection spin-Dunkl operators.     

Elliptic Quantum Groups and Ruijsenaars Models

We construct symmetric and exterior powers of the vector representation of the elliptic quantum groupsE _Τ,η(slN). The corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle by the nested Bethe ansatz method. In special cases we recover the Ruijsenaars systems of commuting difference operators.     

The Bethe ansatz equations for reflecting magnons

We derive the Bethe ansatz equations on the half line for particles interacting through factorized S-matrices invariant relative to the centrally extended su(2|2) Lie superalgebra and su(1|2) open boundaries. These equations may be of relevance for the study of the spectrum of open strings on AdS₅×S⁵ background attached to Y=0 giant graviton branes. A one-dimensional spin chain Hamiltonian associated to this system is also derived.     

Algebraic properties of an integrable t-J model with impurities

We investigate the algebraic structure of a recently proposed integrable t-J model with impurities. Three forms of the Bethe ansatz equations are presented corresponding to the three choices for the grading. We prove that the Bethe ansatz states are highest weight vectors of the underlying gl(2′1) supersymmetry algebra. By acting with the gl(2′1) generators we construct a complete set of states for the model.     

Coordinate space form of interacting reference response function ofd-dimensional jellium

Summary The interacting reference response functionX _I ^[3](k) of three-dimensional jellium ink space was defined by Niklasson in terms of the momentum distribution of the interacting electron assembly. Here the Fourier transformF _I ^[d](r) ofX _I ^[d] (k) is studied for the jellium model withe ²/r interactions in dimensionalityd=1,2 and 3, in an extension of recent work by Holas, March and Tosi for the cased=3. The small-r and large-r forms ofF _I ^[d] (r) are explicitly evaluated from the analytic behaviour of the momentum distributionn _d(p). In the appendix, a model ofn _d (p) is constructed which interpolates between these limits.     

A Note on the Bethe Ansatz Solution of the Supersymmetric t-J Model

The three different sets of Bethe ansatz equations describing the Bethe ansatz solution of the supersymmetric t-J model are known to be equivalent. Here we give a new, simplified proof of this fact which relies on the properties of certain polynomials. We also show that the corresponding transfer matrix eigenvalues agree.     

基于火焰发射光谱的转动温度和振动温度的测量

利用光学多通道分析仪(OMA)对酒精和煤油在大气中燃烧时的火焰发射光谱进行了分析.实验发现上述火焰发射光谱在275-600 nm波段范围内存在多支辐射强度很大的带状分子光谱,其中以OH自由基A²Σ⁺→X²Π_r(300-330 nm),CH自由基A²Δ→X²Π(410-440 nm)和C₂自由基A³Π_g→X³Π_u(500-520 nm)电子带系的发射光谱最为强烈.实验中采用高分辨率光栅对OH自由基A²Σ⁺→X²Π_r和CH自由基A²Δ→X²Π电子带系发射光谱的精细结构进行了分析.与此同时,本文基于分子光谱理论计算了不同转动温度和振动温度条件下OH自由基A²Σ⁺→X²Π_r和CH自由基A²Δ→X²Π电子带系发射光谱的强度分布,同时通过理论计算光谱和实验光谱进行比较确定了酒精燃烧火焰的转动温度和振动温度. 关键词： 火焰发射光谱 谱线强度 转动温度 振动温度     

Intertwining relations,asymmetric face model,and algebraic Bethe ansatz forSU p,q (2) invariant spin chain

Intertwining relations for the quantumR-matrix of theSU _p,q (2) invariant spin chain are obtained and the corresponding face model is deduced. An important difference is seen to arise due to the asymmetry generated by the parametersp andq, which leads to a asymmetric face model. An algebraic Bethe ansatz is set up and solved with the help of these intertwining vectors.     

An Algebraic Derivation of the Eigenspaces Associated with an Ising-Like Spectrum of the Superintegrable Chiral Potts Model

In terms of the loop algebra and the algebraic Bethe-ansatz method, we derive the invariant subspace associated with a given Ising-like spectrum consisting of 2 ^r eigenvalues of the diagonal-to-diagonal transfer matrix of the superintegrable chiral Potts (SCP) model with arbitrary inhomogeneous parameters. We show that every regular Bethe eigenstate of the τ ₂-model leads to an Ising-like spectrum and is an eigenvector of the SCP transfer matrix which is given by the product of two diagonal-to-diagonal transfer matrices with a constraint on the spectral parameters. We also show in a sector that the τ ₂-model commutes with the loop algebra, , and every regular Bethe state of the τ ₂-model is of highest weight. Thus, from physical assumptions such as the completeness of the Bethe ansatz, it follows in the sector that every regular Bethe state of the τ ₂-model generates an -degenerate eigenspace and it gives the invariant subspace, i.e. the direct sum of the eigenspaces associated with the Ising-like spectrum.     

Quantum group invariant integrable n-state models with periodic boundary conditions

Using the methods of topological quantum field theory we construct aU _q [sl(n)] invariant integrable transfer matrix for the case ofq being a root of unity. It corresponds to a 2-dimensional vertex model on a torus with topological interaction w.r.t. its interior. By means of the nested Bethe ansatz method we analyse conformai properties and discuss the representational content of the Bethe ansatz solutions.     

Quantum Integrable Models and Discrete Classical Hirota Equations

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A _k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived. Received: 15 May 1996 / Accepted: 25 November 1996     

Exact diagonalization of the quantum supersymmetric SUq(n|m) model

We use the algebraic nested Bethe ansatz to solve the eigenvalue and eigenvector problem of the supersymmetric SU_q(n|m) model with open boundary conditions. Under an additional condition this model is related to a multicomponent supersymmetric t-J model. We also prove that the transfer matrix with open boundary conditions is SU_q(n|m) invariant.     

Conformal blocks and generalized theta functions

LetSU _X r be the moduli space of rankr vector bundles with trivial determinant on a Riemann surfaceX. This space carries a natural line bundle, the determinant line bundleL. We describe a canonical isomorphism of the space of global sections ofL ^k with the space of conformal blocks defined in terms of representations of the Lie algebrasl _r (C((z))). It follows in particular that the dimension ofH ⁰(SU _X r,L ^k ) is given by the Verlinde formula.Both authors were partially supported by the European Science Project Geometry of Algebraic Varieties, Contract no. SCI-0398-C(A)     

The spectrum of the transfer matrices connected with Kac-Moody algebras

The spectrum of the transfer matrices corresponding to trigonometrical Bazhanov-Jimbo R matrices is found. The Bethe equations characterizing the eigenvalues of the transfer matrices are written down in terms of root systems. Using the generalization of the Bethe equations for Kac-Moody algebras D _inf4 ^sup(3) , G _inf2 ^sup(1) , E _inf6 ^sup(1) and E _inf6 ^sup(2) , we give conjectures for the eigenvalues of the corresponding transfer matrices.     

On Nested Bethe Ansatz for RTT Algebra of o(2n) Type

We study the highest weight representations of the RTT algebras for the R matrix of o(2n) type by the nested algebraic Bethe ansatz. We show how auxiliary RTT algebra à can be used to find Bethe vectors and Bethe conditions. For special representations, in which representation of RTT algebra à is trivial, the problem was solved by Reshetikhin.

     

Bethe ansatz equations for the brokenZ N -symmetric model

We obtain the Bethe ansatz equations for the brokenZ _N -symmetric model by constructing a functional relation of the transfer matrix ofL-operators. This model is an elliptic off-critical extension of the Fateev-Zamolodchikov model. We calculate the free energy of this model on the basis of the string hypothesis.     

Between Poisson and GUE Statistics: Role of the Breit–Wigner Width

We consider the spectral statistics of the superposition of a random diagonal matrix and a GUE matrix. By means of two alternative superanalytic approaches, the coset method and the graded eigenvalue method, we derive the two-level correlation functionX₂(r) and the number varianceΣ²(r). The graded eigenvalue approach leads to an expression forX₂(r) which is valid for all values of the parameterλgoverning the strength of the GUE admixture on the unfolded scale. A new twofold integration representation is found which can be easily evaluated numerically. Forλ?1 the Breit–Wigner widthΓ₁measured in units of the mean level spacingDis much larger than unity. In this limit, closed analytical expressions forX₂(r) andΣ²(r) can be derived by (i) evaluating the double integral perturbatively or (ii) anab initioperturbative calculation employing the coset method. The instructive comparison between both approaches reveals that random fluctuations ofΓ₁manifest themselves in modifications of the spectral statistics. The energy scale which determines the deviation of the statistical properties from GUE behavior is given by. This is rigorously shown and discussed in great detail. The Breit–WignerΓ₁width itself governs the approach to the Poisson limit forr→∞. Our analytical findings are confirmed by numerical simulations of an ensemble of 500×500 matrices, which demonstrate the universal validity of our results after proper unfolding.     

Off-shell Bethe ansatz equation for osp(21) Gaudin magnets

The semi-classical limit of the algebraic Bethe ansatz method is used to solve the theory of Gaudin models. Via off-shell Bethe ansatz method we find the spectra and eigenvectors of the N−1 independents Gaudin Hamiltonians with symmetry osp(21). We also show how the off-shell Gaudin equation solves the trigonometric Knizhnik–Zamolodchikov equation.