Aspects of Generalized Calogero Model

A multispecies model of Calogero type in D 1 dimensions is constructed. The model includes harmonic, two-body and three-body interactions. Using the underlying conformal SU(1,1) algebra, we find the exact eigenenergies corresponding to a class of the exact global collective states. Analyzing corresponding Fock space, we detect the universal critical point at which the model exhibits singular behavior.     

Density waves in the Calogero model - revisited

The Calogero model bears, in the continuum limit, collective excitations in the form of density waves and solitary modulations of the density of particles. This sector of the spectrum of the model was investigated, mostly within the framework of collective-field theory, by several authors, over the past 15 years or so. In this work we shall concentrate on periodic solutions of the collective BPS-equation (also known as “finite amplitude density waves”), as well as on periodic solutions of the full static variational equations which vanish periodically (also known as “large amplitude density waves”). While these solutions are not new, we feel that our analysis and presentation add to the existing literature, as we explain in the text. In addition, we show that these solutions also occur in a certain two-family generalization of the Calogero model, at special points in parameter space. A compendium of useful identities associated with Hilbert transforms, including our own proofs of these identities, appears in Appendix A. In Appendix B we also elucidate in the present paper some fine points having to do with manipulating Hilbert-transforms, which appear ubiquitously in the collective field formalism. Finally, in order to make this paper self-contained, we briefly summarize in Appendix C basic facts about the collective field formulation of the Calogero model.     

Near-horizon states of black holes and Calogero models

We find self-adjoint extensions of the rational Calogero model in the presence of the harmonic interaction. The corresponding eigenfunctions may describe the near-horizon quantum states of certain types of black holes.     

Supertrace and Superquadratic Lie Structure on the Weyl Algebra,and Applications to Formal Inverse Weyl Transform

Using the Moyal *-product and orthosymplectic supersymmetry, we construct a natural nontrivial supertrace and an associated nondegenerate invariant supersymmetric bilinear form for the Lie superalgebra structure of the Weyl algebra W. We decompose adjoint and twisted adjoint actions. We define a renormalized supertrace and a formal inverse Weyl transform in a deformation quantization framework and develop some examples Mathematics Subject Classification: 53D55, 17B05, 17B10, 17B20, 17B60, 17B65     

Symmetric and Non-Symmetric Bases of Quantum Integrable Particle Systems with Long-Range Interactions

We study in an algebraic manner the symmetric basis of the Calogero model and the non-symmetric basis of the corresponding Calogero model with distinguishable particles. The Rodrigues formulas are presented for the polynomial parts of both bases. The square norm of the non-symmetric basis is evaluated. Symmetrization of the non-symmetric basis reproduces the symmetric basis and enables us to calculate its square norm.     

W∞ Algebra and High-Order Virasoro Algebra

We give the relation between W_∞ algebra and high-order Virasoro algebra (HOVA), i.e., W_∞ algebra is the limit of HOVA. Then we give the super high-order Virasoro algebra from super W_∞ algebra.     

Fun and frustration with Calogero model

We review some algebraical (oscillator) aspects of N-body single-species and multispecies Calogero models in one dimension. We treat them as a particular cases of deformed harmonic oscillators and discuss the corresponding Fock spaces. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

The effect of singular potentials on the harmonic oscillator

We address the problem of a quantum particle moving under interactions presenting singularities. The self-adjoint extension approach is used to guarantee that the Hamiltonian is self-adjoint and to fix the choice of boundary conditions. We specifically look at the harmonic oscillator added of either a δ-function potential or a Coulomb potential (which is singular at the origin). The results are applied to Landau levels in the presence of a topological defect, the Calogero model and to the quantum motion on the noncommutative plane.     

Differential calculi over quantum groups and twisted cyclic cocycles

We study some aspects of the theory of non-commutative differential calculi over complex algebras, especially over the Hopf algebras associated to compact quantum groups in the sense of S.L. Woronowicz. Our principal emphasis is on the theory of twisted graded traces and their associated twisted cyclic cocycles. One of our principal results is a new method of constructing differential calculi, using twisted graded traces.     

The extended trace identity and its application

The trace identity is extended to the general loop algebra. The Hamiltonian structures of the integrable systems concerning vector spectral problems and the multi-component integrable hierarchy can be worked out by using the extended trace identity. As its application, we have obtained the Hamiltonian structures of the Yang hierarchy, the Korteweg-de--Vries (KdV) hierarchy, the multi-component Ablowitz--Kaup--Newell--Segur (M-AKNS) hierarchy, the multi-component Ablowitz--Kaup--Newell--Segur Kaup--Newell (M-AKNS--KN) hierarchy and a new multi-component integrable hierarchy separately.     

Bound and Scattering States of Extended Calogero Model with an Additional PT Invariant Interaction

Here we discuss two many-particle quantum systems, which are obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the Calogero model with and without confining potential. It is shown that the energy eigenvalues are real for both of these quantum systems. For the case of extended Calogero model with confining potential, we obtain discrete bound states satisfying generalised exclusion statistics. On the other hand, the extended Calogero model without confining term gives rise to scattering states with continuous spectrum. The scattering phase shift for this case is determined through the exchange statistics parameter. We find that, unlike the case of usual Calogero model, the exclusion and exchange statistics parameters differ from each other in the presence of PT invariant interaction.     

Nonlinear Deformed Algebra Realized in a Physical System

We show that nonlinear deformed algebra can exist in a physical system with Poschl-Teller potential. Due to this algebra, the eigenvalue problem of the system can be exactly solved by operator method. The raising and lowering operators satisfying this algebra are constructed. And the physical meaning of two deforming functions involving in this algebra is given. In addition, the SU(1,1) symmetry is exhibited in such a system by the operator method.     

Mathematical structure of the three-dimensional（3D） Ising model

An overview of the mathematical structure of the three-dimensional(3D) Ising model is given from the points of view of topology,algebra,and geometry.By analyzing the relationships among transfer matrices of the 3D Ising model,Reidemeister moves in the knot theory,Yang-Baxter and tetrahedron equations,the following facts are illustrated for the 3D Ising model.1) The complex quaternion basis constructed for the 3D Ising model naturally represents the rotation in a(3+1)-dimensional space-time as a relativistic quantum statistical mechanics model,which is consistent with the 4-fold integrand of the partition function obtained by taking the time average.2) A unitary transformation with a matrix that is a spin representation in 2 n·l·o-space corresponds to a rotation in 2n·l·o-space,which serves to smooth all the crossings in the transfer matrices and contributes the non-trivial topological part of the partition function of the 3D Ising model.3) A tetrahedron relationship would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model,and its existence is guaranteed by the Jordan algebra and the Jordan-von Neumann-Wigner procedures.4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φx,φy,and φz.The relationship with quantum field and gauge theories and the physical significance of the weight factors are discussed in detail.The conjectured exact solution is compared with numerical results,and the singularities at/near infinite temperature are inspected.The analyticity in β=1/(kBT) of both the hard-core and the Ising models has been proved only for β0,not for β=0.Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.     

A Maple Package on Symbolic Computation of Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations with logarithmic transformations is presented. In the algorithm, the general assumption of Hirota bilinear form is successfully reduced based on the property of uniformity in rank. Furthermore, we discard the integral operation in the traditional algorithm. The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety soliton equations.     

High—Dimensional Integrable Models with Infinitely Dimensional Virasoro—Type Symmetry Algebra

Using every realization of the Virasoro-type symmetry algebra[σ(f1),σ(f2)]=σ(f1f2-f2f1),we can obtain various high-dimensional integrable models under the meaning that they possess infinitely many symmetries,By means of a concrete realization ,many(3 1)-dimensional equations which possess Kac-Moody-Virasoro-type infinite dimensional symmetry algebras are obtained.