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.     

Commuting Rank 2 Differential Operators Corresponding to a Curve of Genus 2

We construct commuting rank 2 formally self-adjoint ordinary differential operators corresponding to a curve of genus 2.     

Commuting Nilpotent Operators and Maximal Rank

Let X, [(X)\tilde]{\widetilde X} be commuting nilpotent matrices over k with nilpotency p ^t , where k is an algebraically closed field of positive characteristic p. We show that if X- [(X)\tilde]{X- \widetilde X} is a certain linear combination of products of pairwise commuting nilpotent matrices, then X is of maximal rank if and only if [(X)\tilde]{\widetilde X} is of maximal rank.     

Real Commuting Differential Operators Connected with Two-Dimensional Abelian Varieties

We indicate smooth real commuting differential operators whose eigenvalues and eigenfunctions are parametrized by principally polarized abelian varieties.     

Real Commuting Differential Operators Connected with Two-Dimensional Abelian Varieties

We indicate smooth real commuting differential operators whose eigenvalues and eigenfunctions are parametrized by principally polarized abelian varieties.     

复系数2n阶微分算子的谱

本文研究了复系数２ｎ阶微分算式（２．１）生成的Ｊ－自伴微分算子谱,对两类微分算子的本质谱,离散谱作了定性研究,得到了所生成微分算子本质谱的存在范围,以及所生成微分算子的谱是离散的充分条件．     

The Dressing Chain and One-Point Commuting Difference Operators of Rank 1

We construct solutions to the difference-differential equation that are associated with onepoint commuting difference operators of rank 1 in the case of spectral curves of genus 1.     

Some Results on Ordinary Differential Operators with Periodic Coefficients

For a general ordinary differential operator \(\mathcal {L}\) with periodic coefficients we prove that the characteristic polynomial of the Floquet matrix is irreducible over the field of meromorphic functions. We also consider a multipoint eigenvalue problem and show that its eigenspaces are spanned by pure or generalized Floquet solutions. Finally, at the end of the paper we mention some relevant conjectures and open questions.     

Gauge-Related Deformations of Ordinary Linear Differential Operators with Constant Coefficients

Relations between the coefficients of gauge-related deformations and the solutions of a certain system of ordinary nonlinear differential equations are studied. These coefficients are found in an explicit form. Bibliography: 14 titles. Dedicated to P. V. Krauklis on the occasion of his seventieth birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 235–251.     

Analytic Functions with Negative Coefficients and Differential and Integral Operators

In this paper integral properties of some analytic functions with negative coefficients from certain classes of functions defined using Ruscheweyh differential operator are studied. The obtained results are sharp and they are improvement of some known results.     

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.     

The Structure of the Spectrum of Fourth-Order Differential Operators with Random Coefficients

In this paper it is proved that under certain conditions on the coefficients the random operator H = $H = \frac{1}{{r\left( {x_t } \right)}}\left[ {\frac{{d^2 }}{{dt^2 }}\left( {\frac{1}{{p\left( {x_t } \right)}}\frac{{d^2 }}{{dt^2 }}} \right) + q\left( {x_t } \right)} \right], \in R^1$ being a stationary, ergodic Markov process with compact State space K, has almost surely pure point spectrum and exponentially decreasing eigenfunctions. The method used here can be extended to operators corresponding to certain matrix Sturm-Liouville problems.     

具有内部奇异点的实系数微分算子的自共轭域

研究了一类带有内部奇异点的实系数微分算子自共轭域的描述问题.通过构造相应的直和空间,应用直和空间的相关理论及对相应最大算子域进行分解,在直和空间上生成的相应最小算子具有实正则型域的情形下,利用微分方程的实参数解给出此类算子的自共轭域的完全解析描述,并且确定其边界条件的矩阵仅由微分方程的解在正则点的初始值决定.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

具有负指数系数的微分算子的谱

本文研究了形如∑_n~k=o~((α_k)(e~((α_k)x))D~k(a_k≤0)及∑_k~n=o((-1)~k)α_(2k)D~ke~(α_(2k)x)D~k+i/2∑_k~n=o(α_(2k+1))(D~ke~((α_(2k+1))x)D~(k+1)+D~(k+1)e~((α_(2k+1)x)D~k)(α_k≤0)的算式的谱问题,分别得到了它们的本质谱或本质谱所在的范围.     

Elliptic Operators with Singular Coefficients

Mathematical Notes -     

C0半群高阶微分算子的谱

本文研究了Ｃ０半群Ｔ（ｔ）的高阶微分算子Ｔ（ｎ）（ｔ）的谱给出了Ｔ＾（ｎ）（ｔ）的谱集的一种构造方法,讨论了Ｔ（ｎ）（ｔ）的谱点与Ｔ（ｔ）的无穷小生成元Ａ的谱点之间的关系。     

一类具指数系数的对称微分算子谱的离散性

研究了具指数函数系数的2n阶实系数微分算式生成的对称微分算子,利用算子的直和分解法及不等式估计得到此类微分算子谱是离散的充分条件.     

A Family of Fundamental Solutions of Elliptic Partial Differential Operators with Real Constant Coefficients

We present a construction of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. The aim is to write detailed expressions for such functions. Such expressions are then exploited to prove regularity properties in the frame of Schauder spaces and jump properties of the corresponding single layer potentials.