Applications of Commuting Difference Operators to Orthogonal Polynomials in Several Variables

Elementary properties of the Koornwinder--Macdonald multivariable Askey--Wilson polynomials are discussed. Studied are the orthogonality, the difference equations, the recurrence relations, and the orthonormalization constants for these polynomials. Essential in our approach are certain commuting difference operators simultaneously diagonalized by the polynomials.     

Making Almost Commuting Matrices Commute

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.     

Bispectral Algebras of Commuting Ordinary Differential Operators

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     

Almost Commuting Unitary Matrices Related to Time Reversal

The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either ${{\bf M}_n(\mathbb{A})}$ M n ( A ) for ${\mathbb{A} = \mathbb{R}}$ A = R , ${\mathbb{A} = \mathbb{C}}$ A = C or ${\mathbb{A} = \mathbb{H}}$ A = H , the algebra of quaternions. There are potential obstructions keeping k-tuples of almost commuting operators from being close to a commuting k-tuple.We consider two-dimensional geometries and so this obstruction lives in ${KO_{-2}(\mathbb{A})}$ K O - 2 ( A ) . This obstruction corresponds to either the Chern number or spin Chern number in physics. We show that if this obstruction is the trivial element in K-theory then the approximation by commuting matrices is possible.     

From Random Matrices to Stochastic Operators

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.     

Commuting Operators and Separation of Variables for Laplacians of Projectively Equivalent Metrics

Let two Riemannian metrics g and g on one manifold M ⁿ have the same geodesics (considered as unparameterized curves). Then we can construct invariantly n commuting differential operators of second order. The Laplacian _g of the metric g is one of these operators. For any x M ⁿ , consider the linear transformation G of T _x M ⁿ given by the tensor g ^Ig_j . If all eigenvalues of G are different at one point of the manifold then they are different at almost every point; the operators are linearly independent and their symbols are functionally independent. If all eigenvalues of G are different at each point of a closed manifold then it can be covered by the n-torus and we can globally separate the variables in the equation _g f = f on this torus.     

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.     

Difference Operators of Sklyanin and van Diejen Type

The Sklyanin algebra ${\mathcal{S}_{\eta}}$ has a well-known family of infinite-dimensional representations ${\mathcal{D}(\mu), {\mu}\,{\in}\,\mathbb{C}^{\ast}}$ , in terms of difference operators with shift η acting on even meromorphic functions. We show that for generic η the coefficients of these operators have solely simple poles, with linear residue relations depending on their locations. More generally, we obtain explicit necessary and sufficient conditions on a difference operator for it to belong to ${\mathcal{D}(\mu)}$ . By definition, the even part of ${\mathcal{D}(\mu)}$ is generated by twofold products of the Sklyanin generators. We prove that any sum of the latter products yields a difference operator of van Diejen type. We also obtain kernel identities for the Sklyanin generators. They give rise to order-reversing involutive automorphisms of ${\mathcal{D}(\mu)}$ , and are shown to entail previously known kernel identities for the van Diejen operators. Moreover, for special μ they yield novel finite-dimensional representations of ${\mathcal{S}_{\eta}}$ .     

Transfer Matrices and Lattice Fermions at Finite Density

I discuss the connection between the Hamiltonian and path integral approaches for fermionic fields. I show how the temporal Wilson projection operators appear naturally in a lattice action. I also carefully treat the insertion of a chemical potential term.     

Commuting Pauli Hamiltonians as Maps between Free Modules

We study unfrustrated spin Hamiltonians that consist of commuting tensor products of Pauli matrices. Assuming translation-invariance, a family of Hamiltonians that belong to the same phase of matter is described by a map between modules over the translation-group algebra, so homological methods are applicable. In any dimension every point-like charge appears as a vertex of a fractal operator, and can be isolated with energy barrier at most logarithmic in the separation distance. For a topologically ordered system in three dimensions, there must exist a point-like nontrivial charge. A connection between the ground state degeneracy and the number of points on an algebraic set is discussed. Tools to handle local Clifford unitary transformations are given.     

Higher Order KFVS Algorithms Using Compact Upwind Difference Operators

A family of high order accurate compact upwind difference operators have been used, together with the split fluxes of the KFVS (kinetic flux vector splitting) scheme to obtain high order semidiscretizations of the 2D Euler equations of inviscid gas dynamics in general coordinates. A TVD multistage Runge–Kutta time stepping scheme is used to compute steady states for selected transonic/supersonic flow problems which indicate the higher accuracy and low diffusion realizable in such schemes.     

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

We study the chromatic polynomial P_G(q) for m× n square- and triangular-lattice strips of widths 2≤ m ≤ 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the antiferromagnetic q-state Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin–Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex q-plane in the limit n→∞. We find that the different phases that appear in this model can be characterized by a topological parameter. We also compute the bulk and surface free energies and the central charge.     

The Four-Pole Transfer Matrices of an Elastic Porous Layer

Acoustical Physics - Relations are proposed for the transfer matrices of linear four-poles that relate the pressure and normal velocity components at free and/or impervious boundaries of a planar...     

Sharp Estimates in Ruelle Theorems for Matrix Transfer Operators

A matrix coefficient transfer operator , on the space of -sections of an m-dimensional vector bundle over n-dimensional compact manifold is considered. The spectral radius of is estimated by\para; and the essential spectral radius by

Reflectionless Analytic Difference Operators II. Relations to Soliton Systems

Abstract

This is the second part of a series of papers dealing with an extensive class of analytic difference operators admitting reflectionless eigenfunctions. In the first part, the pertinent difference operators and their reflectionless eigenfunctions are constructed from given “spectral data”, in analogy with the IST for reflectionless Schrödinger and Jacobi operators. In the present paper, we introduce a suitable time dependence in the data, arriving at explicit solutions to a nonlocal evolution equation of Toda type, which may be viewed as an analog of the KdV and Toda lattice equations for the latter operators. As a corollary, we reobtain various known results concerning reflectionless Schrödinger and Jacobi operators. Exploiting a reparametrization in terms of relativistic Calogero–Moser systems, we also present a detailed study of N-soliton solutions to our nonlocal evolution equation.     

Generalized Farey Trees,Transfer Operators and Phase Transitions

We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral properties of the generalized transfer operator. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function.     

Self-adjoint Operators as Functions I

Observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or affiliated with it in the unbounded case, form a conditionally complete lattice when equipped with the spectral order. Using this order-theoretic structure, we develop a new perspective on quantum observables. In this first paper (of two), we show that self-adjoint operators affiliated with a von Neumann algebra ${\mathcal{N}}$ can equivalently be described as certain real-valued functions on the projection lattice ${\mathcal{P}(\mathcal{N}})$ of the algebra, which we call q-observable functions. Bounded self-adjoint operators correspond to q-observable functions with compact image on non-zero projections. These functions, originally defined in a similar form by de Groote (Observables II: quantum observables, 2005), are most naturally seen as adjoints (in the categorical sense) of spectral families. We show how they relate to the daseinisation mapping from the topos approach to quantum theory (Döring and Isham , New Structures for Physics, Springer, Heidelberg, 2011). Moreover, the q-observable functions form a conditionally complete lattice which is shown to be order-isomorphic to the lattice of self-adjoint operators with respect to the spectral order. In a subsequent paper (Döring and Dewitt, 2012, preprint), we will give an interpretation of q-observable functions in terms of quantum probability theory, and using results from the topos approach to quantum theory, we will provide a joint sample space for all quantum observables.     

Quantum Stochastic Integrals as Operators

We construct quantum stochastic integrals for the integrator being a martingale in a von Neumann algebra, and the integrand—a suitable process with values in the same algebra, as densely defined operators affiliated with the algebra. In the case of a finite algebra we allow the integrator to be an L ²-martingale in which case the integrals are L ²-martingales too.     

Power-Law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension

 We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schr?dinger operators which is based on power-law bounds on transfer matrices. It suffices to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian. Received: 5 June 2002 / Accepted: 20 January 2003 Published online: 28 March 2003 RID="⋆" ID="⋆" D.D. was supported in part by NSF Grant No. DMS–0227289 Communicated by M. Aizenman