2,650 research outputs found
Lattice electrons in constant magnetic field: Bethe like ansatz
We use the functional representation of Heisenberg-Weyl group and obtain
equation for the spectrum of the model, which is more complicated than Bethes
ones, but can be written explicitly through theta functions.Comment: 8 pages, LATE
On a Generalized Oscillator: Invariance Algebra and Interbasis Expansions
This article deals with a quantum-mechanical system which generalizes the
ordinary isotropic harmonic oscillator system. We give the coefficients
connecting the polar and Cartesian bases for D=2 and the coefficients
connecting the Cartesian and cylindrical bases as well as the cylindrical and
spherical bases for D=3. These interbasis expansion coefficients are found to
be analytic continuations to real values of their arguments of the
Clebsch-Gordan coefficients for the group SU(2). For D=2, the superintegrable
character for the generalized oscillator system is investigated from the points
of view of a quadratic invariance algebra.Comment: 13 pages, Latex file. Submitted for publication to Yadernaya Fizik
Antiferromagnetic ordering of energy levels for spin ladder with four-spin cyclic exchange: Generalization of the Lieb-Mattis theorem
The Lieb-Mattis theorem is generalized to an antiferromagnetic spin-ladder
model with four-spin cyclic exchange interaction. We prove that for J>2K, the
antiferromagnetic ordering of energy levels takes place separately in two
sectors, which remain symmetric and antisymmetric under the reflection with
respect to the longitudinal axis of the ladder. We prove also that at the
self-dual point J=2K, the Lieb-Mattis rule holds in the sectors with fixed
number of rung singlets. In both cases, it agrees with the similar rule for
Haldane chain with appropriate spin number.Comment: 4 pages, some references updated and added, typos corrected, to
appear in Phys. Rev.
Ferromagnetic Ordering of Energy Levels for Symmetric Spin Chains
We consider the class of quantum spin chains with arbitrary
-invariant nearest neighbor interactions, sometimes
called for the quantum deformation of , for
. We derive sufficient conditions for the Hamiltonian to satisfy the
property we call {\em Ferromagnetic Ordering of Energy Levels}. This is the
property that the ground state energy restricted to a fixed total spin subspace
is a decreasing function of the total spin. Using the Perron-Frobenius theorem,
we show sufficient conditions are positivity of all interactions in the dual
canonical basis of Lusztig. We characterize the cone of positive interactions,
showing that it is a simplicial cone consisting of all non-positive linear
combinations of "cascade operators," a special new basis of
intertwiners we define. We also state applications to
interacting particle processes.Comment: 23 page
Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries
We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions mFn and is QES if the Schrödinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze's tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set
Conformal mechanics inspired by extremal black holes in d=4
A canonical transformation which relates the model of a massive relativistic
particle moving near the horizon of an extremal black hole in four dimensions
and the conventional conformal mechanics is constructed in two different ways.
The first approach makes use of the action-angle variables in the angular
sector. The second scheme relies upon integrability of the system in the sense
of Liouville.Comment: V2: presentation improved, new material and references added; the
version to appear in JHE
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