3,192 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
Quantization of three-wave equations
The subject of this paper is the consecutive procedure of discretization and
quantization of two similar classical integrable systems in three-dimensional
space-time: the standard three-wave equations and less known modified
three-wave equations. The quantized systems in discrete space-time may be
understood as the regularized integrable quantum field theories. Integrability
of the theories, and in particular the quantum tetrahedron equations for vertex
operators, follow from the quantum auxiliary linear problems. Principal object
of the lattice field theories is the Heisenberg discrete time evolution
operator constructed with the help of vertex operators.Comment: Contribution to J. Phys. A. Special issue "Symmetries and
Integrability of Difference Equations (SIDE) VII
Quantization scheme for modular q-difference equations
Modular pairs of some second order q-difference equations are considered.
These equations may be interpreted as a quantum mechanics of a sort of
hyperelliptic pendulum. It is shown the quantization of a spectrum may be
provided by the condition of the analyticity of the wave function. Baxter's t-Q
equations for the quantum relativistic Toda chain in the ``strong coupling
regime'' are related to the system considered, and the quantization condition
for Q-operator is also considered.Comment: 11 pages, LaTeX2
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