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1.
Boris Feigin Michio Jimbo Tetsuji Miwa Alexandr Odesskii Yaroslav Pugai 《Communications in Mathematical Physics》1998,191(3):501-541
We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed
W
n
algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a
set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the
deformed W
3 algebra.
Received: 3 March 1997 / Accepted: 20 May 1997 相似文献
2.
S. N. M. Ruijsenaars 《Communications in Mathematical Physics》2009,286(2):629-657
In this series of papers we study Hilbert-Schmidt integral operators acting on the Hilbert spaces associated with elliptic
Calogero-Moser type Hamiltonians. As shown in this first part, the integral kernels are joint eigenfunctions of differences
of the latter Hamiltonians. On the relativistic (difference operator) level the kernel is built from the elliptic gamma function,
whereas the building block in the nonrelativistic (differential operator) limit is basically the Weierstrass sigma-function.
For the A
N−1 case we consider all of the commuting Hamiltonians at once, the eigenfunction properties reducing to a sequence of elliptic
identities. For the BC
N
case we only treat the defining Hamiltonians. The functional identities encoding the eigenfunction properties have a remarkable
corollary in the relativistic BC
1 case: They imply that the sum over eight-fold products of the four Jacobi theta functions is invariant under the Weyl group
of E
8. 相似文献
3.
S. N. M. Ruijsenaars 《Communications in Mathematical Physics》2009,286(2):659-680
This is the second part of a series of papers concerning Hilbert-Schmidt integral operators acting on the Hilbert spaces associated
with elliptic Calogero-Moser type Hamiltonians. We present an explicit diagonalization of special Hilbert-Schmidt operators
arising in the free A
N−1 case, and a spectral structure analysis of the commuting family of Hilbert-Schmidt operators associated with the general
A
N−1 case. 相似文献
4.
We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank
N. It combines and unifies the ideas of Duistermaat–Grünbaum and Wilson. Our construction is completely algorithmic and enables
us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad
families of bispectral algebras which may help to penetrate deeper into the problem.
Received: 8 April 1996 / Accepted: 9 April 1997 相似文献
5.
We construct symmetric and exterior powers of the vector representation of the elliptic quantum groupsE
Τ,η(slN). The corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle
by the nested Bethe ansatz method. In special cases we recover the Ruijsenaars systems of commuting difference operators. 相似文献
6.
M. B. Hastings 《Communications in Mathematical Physics》2009,291(2):321-345
Suppose two Hermitian matrices A, B almost commute (\({\Vert [A,B] \Vert \leq \delta}\)). Are they close to a commuting pair of Hermitian matrices, A′, B′, with \({\Vert A-A' \Vert,\Vert B-B'\Vert \leq \epsilon}\) ? A theorem of H. Lin [3] shows that this is uniformly true, in that for every \({\epsilon > 0}\) there exists a δ > 0, independent of the size N of the matrices, for which almost commuting implies being close to a commuting pair. However, this theorem does not specify how δ depends on \({\epsilon}\) . We give uniform bounds relating δ and \({\epsilon}\) . The proof is constructive, giving an explicit algorithm to construct A′ and B′. We provide tighter bounds in the case of block tridiagonal and tridiagonal matrices. Within the context of quantum measurement, this implies an algorithm to construct a basis in which we can make a projective measurement that approximately measures two approximately commuting operators simultaneously. Finally, we comment briefly on the case of approximately measuring three or more approximately commuting operators using POVMs (positive operator-valued measures) instead of projective measurements. 相似文献
7.
F. J. van de Bult E. M. Rains J. V. Stokman 《Communications in Mathematical Physics》2007,275(1):37-95
Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework
for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each
case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E
7 (elliptic, hyperbolic) and of type E
6 (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations
using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric
hypergeometric functions are identified with Ruijsenaars’ relativistic hypergeometric function and the Askey-Wilson function,
respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric
special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an
expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions. 相似文献
8.
We consider the relativistic generalization of the quantum A N-1 Calogero–Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh–Feigin–Veselov–Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results. 相似文献
9.
The Lamé polynomials naturally arise when separating variables in Laplace's equation in elliptic coordinates. The products
of these polynomials form a class of spherical harmonics, which are joint eigenfunctions of a quantum completely integrable
(QCI) system of commuting, second-order differential operators P
0=Δ, P
1,…,P
N
−1 acting on C
∞(?
N
). These operators naturally depend on parameters and thus constitute an ensemble. In this paper, we compute the limiting
level-spacings distributions for the zeroes of the Lamé polynomials in various thermodynamic, asymptotic regimes. We give
results both in the mean and pointwise, for an asymptotically full set of values of the parameters.
Received: 17 January 2001 / Accepted: 14 May 2001 相似文献
10.
F. Finkel D. Gómez-Ullate A. González-López M.A. Rodríguez R. Zhdanov 《Communications in Mathematical Physics》2003,233(2):191-209
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 相似文献
11.
We construct the spaces that the elliptic Ruijsenaars operators act on. It is shown that they are extensible to nonnegative selfadjoint operators on a space of square integrable functions, or preserve a finite dimensional vector space of entire functions. These facts are shown in terms of the R-operators which satisfy the Yang–lBaxter equation. The elliptic Ruijsenaars operators are considered as the elliptic analogues of the Macdonald operators or the difference analogues of the Lamé operators. 相似文献
12.
Brian Jefferies Gerald W. Johnson Lance Nielsen 《Mathematical Physics, Analysis and Geometry》2007,10(1):65-80
The spectral theorem for commuting self-adjoint operators along with the associated functional (or operational) calculus is
among the most useful and beautiful results of analysis. It is well known that forming a functional calculus for noncommuting
self-adjoint operators is far more problematic. The central result of this paper establishes a rich functional calculus for
any finite number of noncommuting (i.e. not necessarily commuting) bounded, self-adjoint operators A
1,..., A
n
and associated continuous Borel probability measures μ
1, ⋯, μ
n
on [0,1]. Fix A
1,..., A
n
. Then each choice of an n-tuple of measures determines one of Feynman’s operational calculi acting on a certain Banach algebra of analytic functions even
when A
1, ..., A
n
are just bounded linear operators on a Banach space. The Hilbert space setting along with self-adjointness allows us to extend
the operational calculi well beyond the analytic functions. Using results and ideas drawn largely from the proof of our main
theorem, we also establish a family of Trotter product type formulas suitable for Feynman’s operational calculi.
相似文献
13.
14.
Arthur Jaffe 《Communications in Mathematical Physics》2001,219(1):89-124
We study the elliptic genus (a partition function) in certain interacting, twist quantum field theories. Without twists,
these theories have N=2 supersymmetry. The twists provide a regularization, and also partially break the supersymmetry. In spite of the regularization,
one can establish a homotopy of the elliptic genus in a coupling parameter. Our construction relies on a priori estimates and other methods from constructive quantum field theory; this mathematical underpinning allows us to justify evaluating
the elliptic genus at one endpoint of the homotopy. We obtain a version of Witten's proposed formula for the elliptic genus
in terms of classical theta functions. As a consequence, the elliptic genus has a hidden SL(2, ℤ) symmetry characteristic of conformal theory, even though the underlying theory is not conformal.
Received: 7 January 2000 / Accepted: 10 April 2000 相似文献
15.
S. Pulmannová 《International Journal of Theoretical Physics》1978,17(9):665-675
A joint distribution of a set of observables on a quantum logic in a statem is defined and its properties are derived. It is shown that if the joint distribution exists, then the observables can be represented in the statem by a set of commuting operators on a Hilbert space. 相似文献
16.
A new formula connecting the elliptic 6j-symbols and the fusion of the vertex-face intertwining vectors is given. This is based on the identification of the k fusion intertwining vectors with the change of base matrix elements from Sklyanin’s standard base to Rosengren’s natural base in the space of even theta functions of order 2k. The new formula allows us to derive various properties of the elliptic 6j-symbols, such as the addition formula, the biorthogonality property, the fusion formula and the Yang-Baxter relation. We also discuss a connection with the Sklyanin algebra based on the factorised formula for the L-operator.Mathematics Subject Classification (2000). 33D15, 81R50, 82B23 相似文献
17.
Cherednik attached to an affine Hecke algebra module a compatible system of difference equations, called quantum affine Knizhnik–Zamolodchikov (KZ) equations. In the case of a principal series module, we construct a basis of power series solutions of the quantum affine KZ equations. Relating the bases for different asymptotic sectors gives rise to a Weyl group cocycle, which we compute explicitly in terms of theta functions.For the spin representation of the affine Hecke algebra of type C, the quantum affine KZ equations become the boundary qKZ equations associated to the Heisenberg spin-\({\frac{1}{2}}\) XXZ chain. We show that in this special case the results lead to an explicit 4-parameter family of elliptic solutions of the dynamical reflection equation associated to Baxter’s 8-vertex face dynamical R-matrix. We use these solutions to define an explicit 9-parameter elliptic family of boundary quantum Knizhnik–Zamolodchikov–Bernard (KZB) equations. 相似文献
18.
The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space
by q-deformation. Simultaneously, angular momentum is deformed to , it acts on the q-Euclidean space that becomes a -module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on
functions on . On a factorspace of a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined
on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert
space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a q-lattice.
Received: 27 June 2000 / Published online: 9 August 2000 相似文献
19.
V. Janiš 《Czechoslovak Journal of Physics》1990,40(8):836-856
We present an operator quantization scheme on a continuous direct product of Hilbert spaces over a time interval as an extension of the quantization using Feynman path integrals. We define the continuous direct product as a Hilbert space with two principal bases: the Fock and the Feynman ones. The Fock basis, defined by a complete set of commuting operators at different times, serves for a definition of the operator calculus. The Feynman basis, simultaneously diagonalizing the complete set of commuting operators, leads to path integrals constructed without time slicing as a spectral representation of certain operator functions. The construction of quantum theory and the corresponding path integrals for the harmonic oscillator is demonstrated both in the configuration and phase spaces. The extension of the theory to coherent states and anticommuting variables is performed. 相似文献
20.
Atsushi Nakayashiki 《Communications in Mathematical Physics》2000,212(1):29-61
The trace of intertwining operators over the level one irreducible highest weight modules of the quantum affine algebra of type ANу(1) is studied. It is proved that the trace function gives a basis of the solution space of the qKZ equation at a generic level. The highest-highest matrix elements of the composition of intertwining operators are explicitly determined as rational functions up to an overall scalar function. The integral formula for the trace is presented. 相似文献