Commutative Rings of Differential Operators Corresponding to Multidimensional Algebraic Varieties

We construct new examples of multidimensional commuting matrix differential operators and a multidimensional analog of the Kadomtsev-Petviashvili hierarchy.     

Commuting Differential and Difference Operators Associated to Complex Curves I

The purpose of this paper is to define functional realizations of the Khizhnik–Zamolochikov–Bernard (KZB) connection on the bundle of conformal blocks over the moduli space of curves.     

Commuting Differential Operators of Rank 2 with Rational Coefficients

In this paper we find new pairs of self-adjoint commuting differential operators of rank 2 with rational coefficients and prove that any curve of genus 2 written as a hyperelliptic curve is the spectral curve of a pair of commuting differential operators with rational coefficients. We also study the case where curves of genus 3 are the spectral curves of pairs of commuting differential operators of rank 2 with rational coefficients.     

Commuting Differential Operators of Rank 2 with Polynomial Coefficients

Self-adjoint commuting differential operators with polynomial coefficients are considered. These operators form a commutative subalgebra of the first Weyl algebra. New examples of commuting differential operators of rank 2 are found.     

Mean Value Operators, Differential Operators and D'Atri Spaces

We study commutativity relations between differential operators and spherical means on Riemannian manifolds. The results show that all D'Atri spaces are ball-homogeneous. A further consequence characterizes complete nonpositively curved, simply connected D'Atri spaces by commuting mean value operators.     

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.     

两类四阶微分算子积的自伴性研究

利用算子理论及矩阵运算方法,讨论了由两类不同的对称微分算式D~((4))+D~((2))+q_1(t)和D~((4))+q_2(t)(D=d/dt,t∈I=[a,b])生成的微分算子的积算子的自伴性,获得了积算子是自伴算子的充分必要条件.     

Generators of Rings of Differential Operators on a Class of k—algebras

51.IlltroductionSmithL4JproPoseaso1vingprocedure,whichcIcternllnesl1olyno[nialresolutlonofI3:lrt1itldifferentialequationcorrespondingbysomecoordinateringsofirreduciblel7l;lnecurves.-I'l1lsprogrameventual1ydependsontl1edeterminationofgeneratorsofringsofdif…     

On the Essential Spectrum of a Class of Matrix Differential Operators

We study the essential spectrum of a class of nonelliptic matrix partial differential operators related to a linear magnetohydrodynamic model.     

一类二阶与四阶微分算子积的自伴性

利用矩阵运算及算子的基本理论,讨论了由微分算式L_1=D~((2))+q_1(t)和L_2=D~((4))+q_2(t)其中(D=d/dx,t∈I=[a,b])生成的两个微分算子L_i(i=1,2)积L_1L_2的自伴性问题,并在常型情形下,获得了积算子自伴的充分必要条件.     

Trace Formulas for Some Singular Differential Operators and Applications

We characterize the convergence of the series ∑ λ^–1_n, where λⁿ are the non‐zero eigenvalues of some boundary value problems for degenerate second order ordinary differential operators and we prove a formula for the above sum when the coefficient of the zero‐order term vanishes. We study these operators both in weighted Hilbert spaces and in spaces of continuous functions. After investigating the boundary behaviour of the eigenfunctions, we give applications to the regularity of the generated semigroups.     

Cyclic Maximal Ideals of Rings of Differential Operators over Power Series Rings

For an integral domain R, several necessary and sufficient conditions are given for R to be unibranched inside its absolute integral closure; one such condition is that R^pbe Henselian for each prime ideal P of R. Additional conditions are given in case R is a going-down domain. Unlike the situation in the Noetherian context, such going‐down domains R need not be quasilocal or of Krull dimension at most 1. A number of examples are given for the locally pseudo‐valuation domain case     

Semisimple Lie Algebras of Differential Operators

Starting from the commutation relations in a complex semisimple Lie algebra , one may obtain a space of vector fields on Euclidean space such that and are isomorphic when is equipped with the usual Lie bracket between vector fields and the isotropy subalgebra of is a Borel subalgebra . Furthermore, one may adjoin to the vector fields in multiplication operators to obtain an -parameter family of distinct presentations of as spaces of differential operators, where is the dual of a Cartan subalgebra. Some of these presentations will preserve a space of polynomials on Euclidean space, and, in fact, all the finite-dimensional representations of can be presented in this way. All of this is carried out explicitly for arbitrary . In doing so, one discovers there is a Lie group of diffeomorphisms of the unipotent subgroup N complementary to B which acts on these presentations and preserves a certain notion of weight.     

Hecke Operators on Linear Ordinary Differential Equations

We construct Hecke operators acting on the space of certain linear ordinary differential equations, and describe a Hermitian inner product on the space of such differential equations. We also determine the adjoint of the Hecke operator with respect to this inner product, and prove that the space of ordinary differential equations associated to an automorphic form for a certain discrete subgroup of SL(2, R) has a basis consisting of common eigenvectors of a class of Hecke operators.     

On Differential Operators in White Noise Analysis

White noise analysis is formulated on a general probability space which is such that (1) it admits a standard Brownian motion, and (2) its -algebra is generated by this Brownian motion (up to completion). As a special case, the white noise probability space with time parameter being the half-line is worked out in detail. It is shown that the usual differential operators can be defined on the smooth, finitely based functions of at most exponential growth via the chain rule, without supposing the existence of a linear structure (or translations) on the underlying probability space.     

Hilbert空间L2[a,b]上m个微分算子积的自伴性

考虑 [a ,b]( -∞   

Hilbert空间L^2[α，6]上m个微分算子积的自伴性

考虑[a,b](-∞＜a＜b＜∞)上n阶复值系数正则对称微分算式ly＝∑n j＝0 aj(t)y(j).本文首先给出由lmy(m∈N且m≥2)生成的微分算子T(lm)自伴边条件一种新的描述,其次研究了由微分算式ly生成的m个微分算子Tk(l)(k=1,…,m)的积Tm(l)…T2(l)T1(l)的自伴性并获得其自伴的充分必要条件.     

Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold

Let M be a smooth manifold, the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on . This cohomology space is closely related to the Vect(M)-modules, (M), of linear differential operators on the space of tensor densities on M of degree .     

Regularized Resolvent of Sums of Commuting Operators

For B ₁ and B ₂ commuting linear operators on a Banach space such that B ₁ generates a bounded strongly continuous semigroup and –B ₂ generates an exponentially decaying strongly continuous holomorphic semigroup, it is shown that (B ₁ – B ₂)^–1 B ₂ ^–r and (B ₁ – B ₂)^–1(–B ₁)^–r are bounded and everywhere defined, for any r > 0. Density of domains may also be removed. The results are applied to various abstract Cauchy problems.     

Global Solvability of Linear Differential Operators with Multiple Characteristics

In this paper, we first introduce the irreducible unitary representation of nilpotent Lie groups, then by using the irreducible unitary representation we construct a fundamental solution to a class of left invariant differential operators and thus obtain the global solvability of this kind of operators.