79,303 research outputs found
Summation formulas for GJMS-operators and Q-curvatures on the M\"obius sphere
For the M\"obius spheres , we give alternative elementary proofs of
the recursive formulas for GJMS-operators and -curvatures due to the first
author [Geom. Funct. Anal. 23, (2013), 1278-1370; arXiv:1108.0273]. These
proofs make essential use of the theory of hypergeometric series.Comment: 21 pages; final version; updated presentation in view of more recent
work of the first author and of Fefferman and Graha
The Quantum Dynamics of the Compactified Trigonometric Ruijsenaars-Schneider Model
We quantize a compactified version of the trigonometric
Ruijse\-naars-Schneider particle model with a phase space that is
symplectomorphic to the complex projective space CP^N. The quantum Hamiltonian
is realized as a discrete difference operator acting in a finite-dimensional
Hilbert space of complex functions with support in a finite uniform lattice
over a convex polytope (viz., a restricted Weyl alcove with walls having a
thickness proportional to the coupling parameter). We solve the corresponding
finite-dimensional (bispectral) eigenvalue problem in terms of discretized
Macdonald polynomials with q (and t) on the unit circle. The normalization of
the wave functions is determined using a terminating version of a recent
summation formula due to Aomoto, Ito and Macdonald. The resulting eigenfunction
transform determines a discrete Fourier-type involution in the Hilbert space of
lattice functions. This is in correspondence with Ruijsenaars' observation
that---at the classical level---the action-angle transformation defines an
(anti)symplectic involution of CP^N. From the perspective of algebraic
combinatorics, our results give rise to a novel system of bilinear summation
identities for the Macdonald symmetric functions
A New Method of the High Temperature Series Expansion
We formulate a new method of performing high-temperature series expansions
for the spin-half Heisenberg model or, more generally, for SU() Heisenberg
model with arbitrary . The new method is a novel extension of the
well-established finite cluster method. Our method emphasizes hidden
combinatorial aspects of the high-temperature series expansion, and solves the
long-standing problem of how to efficiently calculate correlation functions of
operators acting at widely separated sites. Series coefficients are expressed
in terms of cumulants, which are shown to have the property that all deviations
from the lowest-order nonzero cumulant can be expressed in terms of a
particular kind of moment expansion. These ``quasi-moments'' can be written in
terms of corresponding ``quasi-cumulants'', which enable us to calculate
higher-order terms in the high-temperature series expansion. We also present a
new technique for obtaining the low-order contributions to specific heat from
finite clusters.Comment: 20 pages, 30 figures, to appear in J. Stat. Phy
Matrix representation of the time operator
In quantum mechanics the time operator satisfies the commutation
relation , and thus it may be thought of as being canonically
conjugate to the Hamiltonian . The time operator associated with a given
Hamiltonian is not unique because one can replace by , where satisfies the homogeneous condition
. To study this nonuniqueness the matrix elements of
for the harmonic-oscillator Hamiltonian are calculated in the
eigenstate basis. This calculation requires the summation of divergent series,
and the summation is accomplished by using zeta-summation techniques. It is
shown that by including appropriate homogeneous contributions, the matrix
elements of simplify dramatically. However, it is still not clear
whether there is an optimally simple representation of the time operator.Comment: 13 pages, 3 figure
- …