N = 2 Supercomplexification of the Korteweg-de Vries,Sawada-Kotera and Kaup-Kupershmidt Equations

The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd Hamiltonian structures and superfermionic conservation laws. The odd bi-Hamiltonian structure, Lax representation and superfermionic conserved currents for new N = 2 supersymmetric Korteweg-de Vries equation and for Sawada-Kotera one, are given.     

The Perturbation of the Quantum¶Calogero–Moser–Sutherland System¶and Related Results

The Hamiltonian of the trigonometric Calogero–Sutherland model coincides with a certain limit of the Hamiltonian of the elliptic Calogero–Moser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one. In this article we show the essential self-adjointness of the Hamiltonian of the elliptic Calogero–Moser model and the regularity (convergence) of the perturbation for the arbitrary root system. We also show the holomorphy of the joint eigenfunctions of the commuting Hamiltonians w.r.t the variables (x ₁, …,x _N ) for the A _{N -1}-case. As a result, the algebraic calculation of the perturbation is justified. Received: 30 May 2001 / Accepted: 27 November 2001     

Shell model calculations with modified empirical Hamiltonian in the <Superscript>132</Superscript>Sn region

Using recent experimental information for the ¹³²Sn region, an empirical Hamiltonian is obtained by some modifications of a Hamiltonian (CW5082) originally derived from the ²⁰⁸Pb region. Shell model calculations with the new Hamiltonian show a remarkable improvement in the predictive power when compared with the available experimental results. It overcomes many limitations of the CW5082 Hamiltonian in this region, specially for isotones. The calculated level spectra and B(E2) values with the new Hamiltonian, also compare well with the available results calculated with the CD-Bonn and SKX Hamiltonians, reflecting consistency in the wave function structure at least in the low-lying regions. An interesting behaviour of effective charges is revealed in this region. It is shown that a drastic reduction of proton effective charge is necessary for obtaining B(E2) values for the N = 84 isotones. Neutron effective charge is found to be in the range (0.62-0.72)e. We predict the spectroscopic properties of ^135,136Sn not yet known experimentally. Further improvement of the modified Hamiltonian is also initiated.Received: 25 September 2003, Revised: 20 January 2004, Published online: 13 July 2004PACS: 21.60.Cs Shell model - 21.10.-k Properties of nuclei, nuclear energy levels - 23.20.-g Electromagnetic transitions - 27.60. + j      

supersymmetric 3-particles Calogero model

We constructed the most general N=4 superconformal 3-particles systems with translation invariance. In the basis with decoupled center of mass the supercharges and Hamiltonian possess one arbitrary function which defines all potential terms. We have shown that with the proper choice of this function one may describe the standard, A₂ Calogero model as well as G₂ and BC₂ Calogero models, which, by construction, possess N=4 superconformal symmetry. The main property of all these systems is that even with the coupling constant equal to zero they still contain nontrivial interactions in the fermionic sector. In other words, there are infinitely many non-equivalent N=4 supersymmetric extensions of the free action depending on one arbitrary function. We also considered quantization and explicitly showed how the supercharges and Hamiltonian are modified. In the quantum case the constructed systems exhibit only invariance with respect to N=4 Poincaré supersymmetry.     

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet

In its ground state representation, the infinite, spin 1/2 Heisenberg chain provides a model for spin wave scattering, which entails many features of the quantum mechanicalN-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, forall numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.Research supported in part by NSF Grant No. MCS-76-05857Research supported in part by NSF Grant No. MCS-74-07313-A02     

Lyapunov Bounds for Lattice Gauge Dynamics

We study the classical Hamiltonian dynamics of the Kogut–Susskind model for lattice gauge theories on a finite box in a d-dimensional integer lattice. The coupling constant for the plaquette interaction is denoted λ². When the gauge group is a real or a complex subgroup of a unitary matrix group U(N), N≥ 1, we show that the maximal Lyapunov exponent is bounded by , uniformly in the size of the lattice, the energy of the system as well as the order, N, of the gauge group. Received: 20 December 1997 / Accepted: 21 July 1998     

On the Inozemtsev Model

The BC _N Inozemtsev model is investigated. Finite-dimensional spaces which are invariant under the action of the Hamiltonian of the BC _N Inozemtsev model are introduced and it is shown that commuting operators of conserved quantities also preserve the finite-dimensional spaces. The BC ₂ Inozemtsev model is studied in more detail.     

The N = 2 supersymmetric Toda lattice hierarchy

We propose a new integrable N = 2 supersymmetric Toda lattice hierarchy which may be relevant for constructing a supersymmetric one-matrix model. We define its first two Hamiltonian structures, the recursion operator and Lax-pair representation. We provide partial evidence for the existence of an infinite-dimensional N = 2 superalgebra of its flows. We study its bosonic limit and introduce new Lax-pair representations for the bosonic Toda lattice hierarchy. Finally we discuss the relevance this approach for constructing N = 2 supersymmetric generalized Toda lattice hierarchies.     

On the Sutherland Spin Model of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">N</Emphasis></Subscript> Type and Its Associated Spin Chain

 The B _N hyperbolic Sutherland spin model is expressed in terms of a suitable set of commuting Dunkl operators. This fact is exploited to derive a complete family of commuting integrals of motion of the model, thus establishing its integrability. The Dunkl operators are shown to possess a common flag of invariant finite-dimensional linear spaces of smooth scalar functions. This implies that the Hamiltonian of the model preserves a corresponding flag of smooth spin functions. The discrete spectrum of the restriction of the Hamiltonian to this spin flag is explicitly computed by triangularization. The integrability of the hyperbolic Sutherland spin chain of B _N type associated with the dynamical model is proved using Polychronakos's ``freezing trick'. Received: 14 February 2002 / Accepted: 19 June 2002 Published online: 10 December 2002 RID="*" ID="*" Corresponding author. E-mail: artemio@fis.ucm.es RID="**" ID="**" On leave of absence from Institute of Mathematics, 3 Tereschenkivska St., 01601 Kyiv-4 Ukraine Communicated by L. Takhtajan     

Spin Coherent State Representation of the Crow-Kimura and Eigen Models of Quasispecies Theory

We present a spin coherent state representation of the Crow-Kimura and Eigen models of biological evolution. We deal with quasispecies models where the fitness is a function of Hamming distances from one or more reference sequences. In the limit of large sequence length N, we find exact expressions for the mean fitness and magnetization of the asymptotic quasispecies distribution in symmetric fitness landscapes. The results are obtained by constructing a path integral for the propagator on the coset SU(2)/U(1) and taking the classical limit. The classical limit gives a Hamiltonian function on a circle for one reference sequence, and on the product of 2 ^m −1 circles for m reference sequences. We apply our representation to study the Schuster-Swetina phenomena, where a wide lower peak is selected over a narrow higher peak. The quadratic landscape with two reference sequences is also analyzed specifically and we present the phase diagram on the mutation-fitness parameter phase space. Furthermore, we use our method to investigate more biologically relevant system, a model of escape from adaptive conflict through gene duplication, and find three different phases for the asymptotic population distribution.     

Energy Correlations in O(N) Models¶and the Wolff Representation

We prove that energy functions are positively correlated in isotropic, ferromagnetic O(N) models on an arbitrary graph. In our inductive proof, this is used to prove the strong FKG property of the Wolff representation for isotropic, ferromagnetic O(N+ 1) models. This strong FKG property is then used to prove energy correlations for the O(N+ 1) model. Furthermore, percolation in the Wolff representation is proved to be a necessary and sufficient condition for positivity of the spontaneous magnetization (previously known only for N= 3). Received: 7 March 2000 / Accepted: 31 October 2000     

Hamiltonian limit of the 3D Zamolodchikov model

A two-dimensional quantum Hamiltonian _N,M commuting with the layer-to-layer transfer matrix of the three-dimensional Zamolodchikov model is derived. This Hamiltonian is defined on a lattice ofN×M sites. The special casesN×2, 2×M, and 3×M are studied.This paper is dedicated to Cyril Domb.     

Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

Solvability of the rational quantum integrable systems related to exceptional root spaces G₂,F₄ is re-examined and for E_6,7,8 is established in the framework of a unified approach. It is shown that Hamiltonians take algebraic form being written in certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for arbitrary values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.Alexander V. Turbiner: On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia.     

Flow equations and extended Bogoliubov transformation for the Heisenberg antiferromagnet near the classical limit

The Heisenberg spin-S quantum antiferromagnet is studied near the large-spin limit, applying a new continuous unitary transformation which extends the usual Bogoliubov transformation to higher order in the 1/S-expansion of the Hamiltonian. This allows to diagonalize the bosonic Hamiltonian resulting from the Holstein-Primakoff representation beyond the conventional spin-wave approximation. The zero-temperature flow equations derived from the extension of the Bogoliubov transformation to order for the ground-state energy, the spin-wave velocity, and the staggered magnetization are solved exactly and yield results which are in agreement with those obtained by a perturbative treatment of the magnon interactions. Received: 19 March 1998 / Revised: 2 June 1998 / Accepted: 8 June 1998     

Unitary Positive-Energy Representations of Scalar Bilocal Quantum Fields

The superselection sectors of two classes of scalar bilocal quantum fields in D ≥ 4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification in terms of the dual of the respective gauge groups U(N) and O(N) confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory, but the approach using standard Lie algebra methods rather than abstract duality theory is complementary. The result indicates that one does not lose interesting models if one postulates the absence of scalar fields of dimension D−2 in models with global conformal invariance. Another remarkable outcome is the observation that, with an appropriate choice of the Hamiltonian, a Lie algebra embedded into the associative algebra of observables completely fixes the representation theory.     

The plancherel formula for the infinite XXZ Heisenberg spin chain

In this Letter we discuss the explicit Plancherel formula for the Bethe Ansatz eigenstates for the Hamiltonian of the infinite XXZ Heisenberg-Ising spin chain in the N-magnon sectors: N=2, 3,... In particular, we shall point out that a striking spectral phenomenon occurs when the coupling constant c is such that 0<|c|<1.Partially supported by the NSF     

Nuclear structure of <Superscript>128–140</Superscript>Nd in IBM

We determined the most appropriate Hamiltonian that is needed for calculations of nuclei in the A ≅ 130 region from the viewpoint of the interacting boson model (IBM). Using the best-fitted values of parameters in the Hamiltonian, we have calculated energy levels and B(E2) values for a number of transitions in some doubly even Nd nuclei of 128 ⩽ N ⩽ 140. The results were compared with the previous experimental and theoretical data, and it is observed that they are in good agreement. The calculations have been extended to Nd isotopes for which some B(E2) values are still not known. The text was submitted by the author in English.     

Mixed spin transverse Ising model with longitudinal random crystal field interactions

The effect of a longitudinal random crystal field interaction on the phase diagrams of the mixed spin transverse Ising model consisting of spin-1/2 and spin-1 is investigated within the finite cluster approximation based on a single-site cluster theory. In order to expand a cluster identity of spin-1, we transform the spin-1 to spin-1/2 representation containing Pauli operators. We derive the state equations applicable to structures with arbitrary coordination number N. The phase diagrams obtained in the case of a honeycomb lattice (N=3) and a simple-cubic lattice (N=6), are qualitatively different and examined in detail. We find that both systems exhibit a variety of interesting features resulting from the fluctuation of the crystal field interactions. Received: 13 February 1998 / Accepted: 17 March 1998     

Representations of the algebra sl(2, R) and integration of Hamiltonian systems

A new representation of the sl(2, R) is given, which is related to the integrable N-particle system with inversely quadratic potential. The Hamiltonian plays the role of a raising operator and integrals correspond to highest weight vectors.     

Unstructured Adiabatic Quantum Search

In the adiabatic quantum computation model, a computational procedure is described by the continuous time evolution of a time dependent Hamiltonian. We apply this method to the Grover's problem, i.e., searching a marked item in an unstructured database. Classically, the problem can be solved only in a running time of order O(N) (where N is the number of items in the database), whereas in the quantum model a speed up of order has been obtained. We show that in the adiabatic quantum model, by a suitable choice of the time-dependent Hamiltonian, it is possible to do the calculation in constant time, independent of the the number of items in the database. However, in this case the initial time-complexity of is replaced by the complexity of implementing the driving Hamiltonian.