3,196 research outputs found
Ansatz of Hans Bethe for a two-dimensional Bose gas
The method of q-oscillator lattices, proposed recently in [hep-th/0509181],
provides the tool for a construction of various integrable models of quantum
mechanics in 2+1 dimensional space-time. In contrast to any one dimensional
quantum chain, its two dimensional generalizations -- quantum lattices -- admit
different geometrical structures. In this paper we consider the q-oscillator
model on a special lattice. The model may be interpreted as a two-dimensional
Bose gas. The most remarkable feature of the model is that it allows the
coordinate Bethe Ansatz: the p-particles' wave function is the sum of plane
waves. Consistency conditions is the set of 2p equations for p one-particle
wave vectors. These "Bethe Ansatz" equations are the main result of this paper.Comment: LaTex2e, 12 page
Explicit Free Parameterization of the Modified Tetrahedron Equation
The Modified Tetrahedron Equation (MTE) with affine Weyl quantum variables at
N-th root of unity is solved by a rational mapping operator which is obtained
from the solution of a linear problem. We show that the solutions can be
parameterized in terms of eight free parameters and sixteen discrete phase
choices, thus providing a broad starting point for the construction of
3-dimensional integrable lattice models. The Fermat curve points parameterizing
the representation of the mapping operator in terms of cyclic functions are
expressed in terms of the independent parameters. An explicit formula for the
density factor of the MTE is derived. For the example N=2 we write the MTE in
full detail. We also discuss a solution of the MTE in terms of bosonic
continuum functions.Comment: 28 pages, 3 figure
Geometry of quadrilateral nets: second Hamiltonian form
Discrete Darboux-Manakov-Zakharov systems possess two distinct Hamiltonian
forms. In the framework of discrete-differential geometry one Hamiltonian form
appears in a geometry of circular net. In this paper a geometry of second form
is identified.Comment: 6 page
Tetrahedron equations and nilpotent subalgebras of U_q(sl_n)
A relation between q-oscillator R-matrix of the tetrahedron equation and
decompositions of Poinkare-Birkhoff-Witt type bases for nilpotent subalgebras
of U_q(sl_n) is observed.Comment: 4 page
Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations
In this paper we consider three-dimensional quantum q-oscillator field theory
without spectral parameters. We construct an essentially big set of eigenstates
of evolution with unity eigenvalue of discrete time evolution operator. All
these eigenstates belong to a subspace of total Hilbert space where an action
of evolution operator can be identified with quantized discrete BKP equations
(synonym Miwa equations). The key ingredients of our construction are specific
eigenstates of a single three-dimensional R-matrix. These eigenstates are
boundary states for hidden three-dimensional structures of U_q(B_n^1) and
U_q(D_n^1)$.Comment: 13 page
Spectral equations for the modular oscillator
Motivated by applications for non-perturbative topological strings in toric
Calabi--Yau manifolds, we discuss the spectral problem for a pair of commuting
modular conjugate (in the sense of Faddeev) Harper type operators,
corresponding to a special case of the quantized mirror curve of local
and complex values of Planck's constant. We
illustrate our analytical results by numerical calculations.Comment: 23 pages, 9 figures, references added and interpretation of the
numerical results of Section 6 correcte
Quantum 2+1 evolution model
A quantum evolution model in 2+1 discrete space - time, connected with 3D
fundamental map R, is investigated. Map R is derived as a map providing a zero
curvature of a two dimensional lattice system called "the current system". In a
special case of the local Weyl algebra for dynamical variables the map appears
to be canonical one and it corresponds to known operator-valued R-matrix. The
current system is a kind of the linear problem for 2+1 evolution model. A
generating function for the integrals of motion for the evolution is derived
with a help of the current system. The subject of the paper is rather new, and
so the perspectives of further investigations are widely discussed.Comment: LaTeX, 37page
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