Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models

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.     

Affine R-Matrix and the Generalized Elliptic Ruijsenaars Models

Commutative elliptic difference operators associated with the affine root systems are constructed in terms of affine R-matrices. These operators describe the Ruijsenaars models with elliptic potentials and reduce to the Macdonald operators in the trigonometric limit.     

Harmonic Oscillator in Characteristic p

We construct an irreducible representation of the canonical commutation relations by operators on a certain Banach space over a local field of characteristic p. The Carlitz polynomials forming the basis of the space are shown to be the counterparts of the Hermite functions for this situation. The analogues of coherent states are related to the Carlitz exponential.     

Ruijsenaars' Commuting Difference Operators as Commuting Transfer Matrices

For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set of difference operators acting on functions on the type A weight space. According to the fundamental relation RLL=LLR, taking the trace of the L-operator gives a set of commuting difference operators. We show that for the above mentioned L-operator this approach gives Macdonald type operators with elliptic theta function coefficient, actually equivalent to Ruijsenaars' operators. The relationship between the difference L-operator and Krichever's Lax matrix is given, and an explicit formula for elliptic commuting differential operators is derived. We also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space. Received: 27 December 1995 / Accepted: 11 November 1996     

Diagonalization of the elliptic Ruijsenaars model. Correspondence with the Belavin model

Studied is the elliptic Ruijsenaars model, which is a difference analogue of the Calogero-Sutherland-Moser model. Using a novel relationship between the elliptic Ruijsenaars operator and the transfer matrix of the Belavin model, we diagonalize the Ruijsenaars operator by the algebraic Bethe ansatz method. Received: 29 January 1998 / Accepted: 17 April 1998     

Elliptic Quantum Groups and Ruijsenaars Models

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.     

Deformed Macdonald-Ruijsenaars Operators and Super Macdonald Polynomials

It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald- Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced; the combinatorial formulas for them are given.     

Traces of Intertwiners for Quantum Groups and Difference Equations

In this Letter we study twisted traces of products of intertwining operators for quantum affine algebras. They are interesting special functions, depending on two weights ,, three scalar parameters q,,k, and spectral parameters z ₁,...,z _N , which may be regarded as q-analogs of conformal blocks of the Wess–Zumino–Witten model on an elliptic curve. It is expected that in the rank 1 case they essentially coincide with the elliptic hypergeometric functions defined by Felder and Varchenko. Our main result is that after a suitable renormalization the traces satisfy four systems of difference equations – the Macdonald–Ruijsenaars equation, the q-Knizhnik–Zamolodchikov–Bernard equation, and their dual versions. We also show that in the case when the twisting automorphism is trivial, the trace functions are symmetric under the permutation , k . Thus, our results generalize those of Etingof and Schiffmann, dealing with the case q=1, and Etingof, Varchenko, and Schiffmann, dealing with the finite-dimensional case.     

Algebras of Operators on Holomorphic Functions and Applications

We develop the theory of operators defined on a space of holomorphic functions. First, we characterize such operators by a suitable space of holomorphic functions. Next, we show that every operator is a limit of a sequence of convolution and multiplication operators. Finally, we define the exponential of an operator which permits us to solve some quantum stochastic differential equations.     

Normal form expansion in an algebraic way

It is shown that in an algebraic approach, normal form expansion of operators, bounded or not, but densely defined on Fock space, converge weakly.     

Point Interactions in One Dimension and Holonomic Quantum Fields

We introduce and study a family of quantum fields, associated to δ-interactions in one dimension. These fields are analogous to holonomic quantum fields of Sato et al. in Holonomic quantum fields I–V (Publ. RIMS, Kyoto University, 14: 223–267, 1978; 15: 201–278, 1979; 15: 577–629, 1979; 15: 871-972, 1979; 16: 531–584, 1979). Corresponding field operators belong to an infinite-dimensional representation of the group in the Fock space of ordinary harmonic oscillator. We compute form factors of such fields and their correlation functions, which are related to the determinants of Schroedinger operators with a finite number of point interactions. It is also shown that these determinants coincide with tau functions, obtained through the trivialization of the det^*-bundle over a Grassmannian associated to a family of Schroedinger operators.     

Determinants of Airy Operators and Applications to Random Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener–Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum.     

Degenerate Integrability of the Spin Calogero–Moser Systems and the Duality with the Spin Ruijsenaars Systems

It is shown that spin Calogero–Moser systems are completely integrable in a sense of degenerate integrability. Their Liouville tori have dimension less than half of the dimension of the phase space. It is also shown that rational spin Ruijsenaars systems are degenerately integrable and dual to spin Calogero–Moser systems in a sense that action-angle variables of one are angle-action variables of the other.     

Generalized Lamé Operators

We introduce a class of multidimensional Schrödinger operators with elliptic potential which generalize the classical Lamé operator to higher dimensions. One natural example is the Calogero–Moser operator, others are related to the root systems and their deformations. We conjecture that these operators are algebraically integrable, which is a proper generalization of the finite-gap property of the Lamé operator. Using earlier results of Braverman, Etingof and Gaitsgory, we prove this under additional assumption of the usual, Liouville integrability. In particular, this proves the Chalykh–Veselov conjecture for the elliptic Calogero–Moser problem for all root systems. We also establish algebraic integrability in all known two-dimensional cases. A general procedure for calculating the Bloch eigenfunctions is explained. It is worked out in detail for two specific examples: one is related to the B₂ case, another one is a certain deformation of the A₂ case. In these two cases we also obtain similar results for the discrete versions of these problems, related to the difference operators of Macdonald–Ruijsenaars type.On leave of absence from: Advanced Education and Science Centre, Moscow State University, Moscow 119899, Russia     

Transformation Operators for Sturm-Liouville Operators with Singular Potentials

We construct transformation operators for Sturm-Liouville operators with singular potentials from the space W ⁻¹ ₂(0,1) and show that these transformation operators naturally appear during factorisation of Fredholm operators of a special form. Some applications to the spectral analysis of Sturm-Liouville operators with singular potentials under consideration are also given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Separation of Variables for the Ruijsenaars System

We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variables is verified with the use of the r-matrix technique. The explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n=2 and n=3. Taking the nonrelativistic limit we also construct a separation of variables for the elliptic Calogero-Moser system. Received: 10 January 1997 / Accepted: 1 April 1997     

Dirac Operators on Taub-NUT Space: Relationship and Discrete Transformations

It is shown that the N = 4 superalgebra of the Dirac theory in Taub-NUT space has different unitary representations related among themselves through unitary U(2) transformations. In particular the SU(2) transformations are generated by the spin-like operators constructed with the help of the same covariantly constant Killing-Yano tensors which generate Dirac-type operators. A parity operator is defined and some explicit transformations which connect the Dirac-type operators among themselves are given. These transformations form a discrete group which is a realization of the quaternion discrete group. The fifth Dirac operator constructed using the non-covariant constant Killing-Yano tensor of the Taub-NUT space is quite special. This non-standard Dirac operator is connected with the hidden symmetry and is not equivalent to the Dirac-type operators of the standard N = 4 supersymmetry.     

Trigonometric and Elliptic Ruijsenaars–Schneider Systems on the Complex Projective Space

We present a direct construction of compact real forms of the trigonometric and elliptic \({n}\)-particle Ruijsenaars–Schneider systems whose completed center-of-mass phase space is the complex projective space \({{\mathbb{CP}}^{n-1}}\) with the Fubini–Study symplectic structure. These systems are labeled by an integer \({p\in\{1,\ldots,n-1\}}\) relative prime to \({n}\) and a coupling parameter \({y}\) varying in a certain punctured interval around \({p\pi/n}\). Our work extends Ruijsenaars’s pioneering study of compactifications that imposed the restriction \({0 < y < \pi/n}\), and also builds on an earlier derivation of more general compact trigonometric systems by Hamiltonian reduction.     

Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-smooth Coefficients

The aim of this paper is to give the Weyl formula for eigenvalues of self-adjoint elliptic operators, assuming that first-order derivatives of the coefficients are Lipschitz continuous. The approach is based on the asymptotic formula of Hörmander"s type for the spectral function of pseudodifferential operators having Lipschitz continuous Hamiltonian flow and obtained via a regularization procedure of nonsmooth coefficients.     

Crystal base andq-vertex operators

Theq-deformed vertex operators of Frenkel and Reshetikhin are studied in the framework of Kashiwara's crystal base theory. It is shown that the vertex operators preserve the crystal structure, and are naturally labeled by the global crystal base. As an application the one point functions are calculated for the associated elliptic RSOS models, following the scheme of Kang et al. developed for the trigonometric vertex models.Dedicated to Huzihiro Araki