Generalized rationality and a 'Jacobi identity' for intertwining operator algebras

We prove a generalized rationality property and a new identity that we call the 'Jacobi identity' for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in [H4] and [H6], the results of the present paper solve completely the problem of finding a natural purely algebraic structure on the direct sum of all inequivalent irreducible modules for a suitable vertex operator algebra. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given.     

A deformation of commutative polynomial algebras in even numbers of variables

We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [18] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture [8] is reduced to an open problem on this deformation of polynomial algebras.     

A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

In this paper, we develop a fractional integro‐differential operator calculus for Clifford algebra‐valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy‐Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel‐Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge‐type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann‐Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.     

Constructively Factoring Linear Partial Differential Operators in Two Variables

We study conditions under which a partial differential operator of arbitrary order n in two variables or an ordinary linear differential operator admits a factorization with a first-order factor on the left.The process of factoring consists of recursively solving systems of linear equations subject to certain differential compatibility conditions.In the general case of partial differential operators, it is not necessary to solve a differential equation. In special degenerate cases, such as an ordinary differential operator, the problem eventually reduces to solving some Riccati equation(s). We give the factorization conditions explicitly for the second and third orders and in outline form for higher orders. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 165–180, November, 2005.     

Left and right generalized Drazin invertible operators

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space. We show that the left (resp. the right) generalized Drazin inverse is a sum of a left invertible (resp. a right invertible) operator and a quasi-nilpotent one. In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations. Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible. An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given. Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory.     

Operator kernel estimates for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

     

Polynomial forms for quantum elliptic Calogero–Moser Hamiltonians

We hypothesize the form of a transformation reducing the elliptic A _N Calogero–Moser operator to a differential operator with polynomial coefficients. We verify this hypothesis for N ≤ 3 and, moreover, give the corresponding polynomial operators explicitly.     

Polynomial Conservation Laws in Quantum Systems

We consider systems with a finite number of degrees of freedom and potential energy that is a finite sum of exponentials with purely imaginary or real exponents. Such systems include the generalized Toda chains and systems with a toric configuration space. We consider the problem of describing all the quantum conservation laws, i.e., the differential operators that are polynomial in the derivatives and commute with the Hamiltonian operator. We prove that in the case where the potential energy spectrum is invariant under reflection with respect to the origin, such nontrivial operators exist only if the system under consideration decomposes into a direct sum of decoupled subsystems. In the general case (without the spectrum symmetry assumption), we prove that the existence of a complete set of independent conservation laws implies the complete integrability of the corresponding classical system.     

Estimates for restrictions of monotone operators on the cone of decreasing functions in Orlicz space

We introduce a number of notions related to the Lyapunov transformation of linear differential operators with unbounded operator coefficients generated by a family of evolution operators. We prove statements about similar operators related to the Lyapunov transformation and describe their spectral properties. One of the main results of the paper is a similarity theorem for a perturbed differential operator with constant operator coefficient, an operator which is the generator of a bounded group of operators. For the perturbation, we consider the operator ofmultiplication by a summable operator function. The almost periodicity (at infinity) of the solutions of the corresponding homogeneous differential equation is established.     

A Generalized Fourier Method for the System of First-Order Differential Equations with an Involution and a Group of Operators

A mixed problem is considered for a differential equation with an involution and matrix potential. The method of similar operators is used to transform the differential operator defined by this equation into the orthogonal direct sum of finite-rank operators. A theorem is established, based on which an operator group is constructed to describe mild solutions of the problem at hand.     

Invertibility Conditions for Second-Order Differential Operators in the Space of Continuous Bounded Functions

We prove the invertibility of second-order differential operators with constant operator coefficients acting on the Banach space of bounded continuous functions on the real line under the condition that they are uniformly injective (in particular, left invertible) or surjective (in particular, right invertible). We show that if these operators are considered on the space of periodic functions, then the unilateral invertibility does not imply the invertibility of such operators. We obtain criteria for the injectivity, surjectivity, and invertibility of differential operators on the space of periodic functions.     

Necessary Covariance Conditions for a One-Field Lax Pair

We study the covariance with respect to Darboux transformations of polynomial differential and difference operators with coefficients given by functions of one basic field. In the scalar (Abelian) case, the functional dependence is established by equating the Frechet differential (the first term of the Taylor series on the prolonged space) to the Darboux transform; a Lax pair for the Boussinesq equation is considered. For a pair of generalized Zakharov-Shabat problems (with differential and shift operators) with operator coefficients, we construct a set of integrable nonlinear equations together with explicit dressing formulas. Non-Abelian special functions are fixed as the fields of the covariant pairs. We introduce a difference Lax pair, a combined gauge-Darboux transformation, and solutions of the Nahm equations.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 122–132, July, 2005.     

Block-matrix generalizations of infinite-dimensional schur products and schur multipliers

We consider an extention of the familiar Schur product to a bilinear product on the space of matrices whose entries are either bounded operators on a fixed Hilbert space or bounded "square" operator matrices. We show that this is a "natural" non-commutative extention of the Schur product, which retains many of its properties. The work is done mainly in infinite dimensions, where we concentrate on the maps induced on the space of bounded operator matrices via left or right "Schur block-multiplication" by a fixed "Schur block-multiplier". Our main goal is to study the distinctions between left and right multipliers, as well as the behaviour of ideals of operators under action of maps induced bu such.     

Darboux transformation,factorization, and supersymmetry in one-dimensional quantum mechanics

We introduce an N-order Darboux transformation operator as a particular case of general transformation operators. It is shown that this operator can always be represented as a product of N first-order Darboux transformation operators. The relationship between this transformation and the factorization method is investigated. Supercharge operators are introduced. They are differential operators of order N. It is shown that these operators and super-Hamiltonian form a superalgebra of order N. For N=2, we have a quadratic superalgebra analogous to the Sklyanin quadratic algebras. The relationship between the transformation introduced and the inverse scattering problem in quantum mechanics is established. An elementary N-parametric potential that has exactly N predetermined discrete spectrum levels is constructed. The paper concludes with some examples of new exactly soluble potentials.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 356–367, August, 1995.     

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.     

The performance of preconditioned waveform relaxation techniques for pseudospectral methods

Waveform relaxation techniques for the pseudospectral solution of the heat conduction problem are discussed. The pseudospectral operator occurring in the equation is preconditioned by transformations in both space and time. In the spatial domain, domain stretching is used to more equally distribute the grid points across the domain, and hence improve the conditioning of the differential operator. Preconditioning in time is achieved either by an exponential or a polynomial transformation. Block Jacobi solutions of the systems are obtained and compared. The preconditioning in space by domain stretching determines the effectiveness of waveform relaxation in this case. Preconditioning in time is also effective in reducing the number of iterations required for convergence. The polynomial transformation is preferred, because it removes the requirement of a matrix exponential calculation in the time-stepping schemes and, at the same time, is no less effective than the direct exponential preconditioning. © 1996 John Wiley & Sons, Inc.     

Operator linear-fractional relations: main properties, some applications

We consider operator linear-fractional relations of the form $$ F(K)=\left\{ {Q:A+BK=Q\left( {C+DK} \right)} \right\}, $$ where A, B, C, D, K, and Q are operators between Hilbert spaces. If C + DK is invertible, the relation F becomes a linear-fractional transformation. In the case where F is the automorphism of a unit operator ball, we study the conditions for F to be represented in the form of a composition of an automorphism and an affine relation. The results obtained are applied to the Abel–Schröder equations, Königs embedding problem, and some other questions.     

Metaplectic operators for finite abelian groups and ?

The Segal-Shale-Weil representation associates to a symplectic transformation of the Heisenberg group an intertwining operator, called metaplectic operator. We develop an explicit construction of metaplectic operators for the Heisenberg group H(G) of a finite abelian group G, an important setting in finite time-frequency analysis. Our approach also yields a simple construction for the multivariate Euclidean case G = ?^d.     

Some algebras similar to the 2×2 Jordanian matrix algebra

The impetus for this study is the work of Dumas and Rigal on the Jordanian deformation of the ring of coordinate functions on 2×2 matrices. We are also motivated by current interest in birational equivalence of noncommutative rings. Recognizing the construction of the Jordanian matrix algebra as a skew polynomial ring, we construct a family of algebras relative to differential operator rings over a polynomial ring in one variable which are birationally equivalent to the Weyl algebra over a polynomial ring in two variables.