Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

An ordinary differential operator of arbitrary order is considered. We find necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of a system of root functions of this operator in L _p .     

Biorthogonal expansions in root functions of differential operators

We consider the problem on the convergence of biorthogonal expansions in a system of eigenfunctions and associated functions for a wide class of operators, whose special cases include nonself-adjoint differential operators. We introduce the notion of almost basis property of systems of root functions of a linear operator. We demonstrate the necessity to use a new method, earlier introduced by the authors, for defining associated functions.     

On an analog of the Riesz theorem and the basis property of the system of root functions of a differential operator in L p : II

We consider an ordinary differential operator of arbitrary order, obtain necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of the system of root functions of the given operator in L _p .     

Riesz inequality for systems of root vector functions of the Dirac operator

We establish a criterion for the Riesz property of systems of root vector functions of the one-dimensional Dirac operator.     

Some properties of root functions of ordinary second-order differential operators in the absence of the carleman condition

On a finite interval G of the real line, we consider the root functions of an ordinary second-order differential operator without any boundary conditions for the case in which the imaginary part of the spectral parameter is unbounded.We refine the estimates for the C-and L _p -norms of a root function and its first derivative on a compact set contained in the interior of G for the case in which the Carleman condition fails.A sufficient condition is obtained for the root functions of an ordinary second-order differential operator to have the Bessel property, assuming that the Carleman condition fails. We show that, under certain conditions, this problem can be reduced to analyzing the Bessel property of systems of exponentials.     

On a problem with a boundary condition of the second kind,a complex-valued coefficient,and a spectral parameter in the other boundary condition

We study a classical problem that arises in the analysis of natural vibrations of a loaded string with a free endpoint. We assume that the coefficient occurring in the boundary condition of the third kind with a spectral parameter instead of a physical parameter can take complex values. We discuss the traditional aspects of the completeness, minimality, and basis property of the system of root functions. Special attention is paid to the structure of root subspaces.     

Basis Property of the System of Root Functions of a Classical Spectral Problem with a Multiple Eigenvalue

The problem of describing the biorthogonal system of a classical system of root functions is considered for a loaded string problem with a multiple eigenvalue. The basis property of selected complete and minimal subsystems of the system of root functions in the presence of one or two associated functions is studied.     

On the influence of the potential on the convergence rate of expansions in root functions of the Shrödinger operator

We consider the one-dimensional Schrödinger operator with integrable potential. We analyze the rate of the uniform equiconvergence of the biorthogonal expansion of an absolutely continuous function in the root functions of this operator with its Fourier trigonometric series on a compact set. For this convergence rate, we obtain an estimate depending on the modulus of continuity of the potential. We extract subclasses of absolutely continuous functions on which these estimates can be improved.     

Recurrence and Transience Criteria for Symmetric Hunt Processes

In this paper, we will discuss recurrence, transience and other potential theoretic aspects based on symmetric regular Dirichlet space. We will first deal with Dirichlet space with the strong local property and give a recurrence criterion in terms of exhaustion function. This criterion shows that recurrence automatically provides us with an exhaustion function which is usable to verify a Liouville property on subharmonic functions. Secondly, a recurrence criterion and a transience criterion for a Nonlocal Dirichlet space will be presented. Those criteria can be applied to Albeverio–Karwowski"s random walks on p-adic number field. Lastly, we will prove the assertions which cover other potential theoretic aspect of p-adic number field such as Liouville property on harmonic functions.     

On a classical problem with a complex-valued coefficient and the spectral parameter in a boundary condition

We consider a classical problem that arises when studying natural vibrations of a loaded string. We assume that the coefficient playing the role of a physical parameter can take complex values. We discuss the completeness, minimality, and basis property of the system of root functions.     

Uniqueness of Solutions of the Derrida-Lebowitz-Speer-Spohn Equation and Quantum Drift-Diffusion Models

We prove uniqueness of solutions of the DLSS equation in a class of sufficiently regular functions. The global weak solutions of the DLSS equation constructed by Jüngel and Matthes belong to this class of uniqueness. We also show uniqueness of solutions for the quantum drift-diffusion equation, which contains additional drift and second-order diffusion terms. The results hold in case of periodic or Dirichlet-Neumann boundary conditions. Our proof is based on a monotonicity property of the DLSS operator and sophisticated approximation arguments; we derive a PDE satisfied by the pointwise square root of the solution, which enables us to exploit the monotonicity property of the operator.     

第n根性质与根环

令E为 (MR)的自同态环 ,本文首先证明了MR 有第n根性质的充要条件是EE 有第n根性质 ,由此引进了根环的概念 ,并研究了根环的性质。特别地我们证明了如nA nB且A拟内射 ,则A B .     

On the solvability of a nonlocal boundary value problem with the equality of fluxes at a part of the boundary and of the adjoint problem

In the present paper, we consider a nonlocal boundary value problem for the Laplace operator in a circular sector with the equality of fluxes on the radii and with zero value of the solution on one of the radii. We also consider the adjoint problem. We prove the uniqueness of the solution of these problems and obtain an explicit form for the solution by the spectral method. When proving the solvability of the problems, we study the completeness and the basis property of systems of root functions for problems of the type of the Samarskii-Ionkin problem in L _p , which can be of interest in itself.     

On the basis property of root functions of a periodic problem with an integral perturbation of the boundary condition

We consider the spectral problem for the Schrödinger operator with an integral perturbation in the periodic boundary conditions. The unperturbed problem is assumed to have multiple eigenvalues and a system of eigenfunctions forming a Riesz basis in L ₂(0, 1). We show that the basis property of systems of root functions of the problem can change under arbitrarily small changes in the kernel of the integral perturbation.     

On the geometry of the Calogero-Moser system

We discuss a special eigenstate of the quantized periodic Calogero—Moser system associated to a root system. This state has the property that its eigenfunctions, when regarded as multivalued functions on the space of regular conjugacy classes in the corresponding semisimple complex Lie group, transform under monodromy according to the complex reflection representation of the affine Hecke algebra. We show that this endows the space of conjugacy classes in question with a projective structure. For a certain parameter range this projective structure underlies a complex hyperbolic structure. If in addition a Schwarz type of integrality condition is satisfied, then it even has the structure of a ball quotient minus a Heegner divisor. For example, the case of the root system E₈ with the triflection monodromy representation describes a special eigenstate for the system of 12 unordered points on the projective line under a particular constraint.     

A class of completely integrable quantum systems associated with classical root systems

We classify the completely integrable systems associated with classical root systems whose potential functions are meromorphic at an infinite point.     

On generalizations of root systems

We define a generalization of a root system as a set of vectors in a vector space with some symmetry property. The main difference with the usual root systems is the existence of isotropic roots. We classify irreducible generalized root systems. As follows from our classification all such root systems are root systems of contragredient Lie superalgebras which were classified by V.Kac in 1977.     

The mean value property for harmonic functions on graphs and trees

Following the Euclidean example, we introduce the strong and weak mean value property for finite variation measures on graphs. We completely characterize finite variation measures with bounded support on radial trees which have the strong mean value property. We show that for counting measures on bounded subsets of a tree with root o, the strong mean value property is equivalent to the invariance of the subset under the action of the stabilizer of o in the automorphism group. We finally characterize, using the discrete Laplacian, the finite variation measures on a generic graph which have the weak mean value property and we give a non-trivial example. Received: July 21, 2000; in final form: March 13, 2001?Published online: March 19, 2002     

Estimates of the local convergence rate of spectral expansions for even-order differential operators

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a wide class of even-order ordinary differential operators defined on a finite interval. These expansions are compared with the trigonometric Fourier series expansions of the same functions in the integral or uniform metric on an arbitrary interior compact set of the main interval as well as on the entire interval. We show the dependence of the equiconvergence rate of these expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the existence of infinitely many associated functions in the system of root functions.     

Dependence of estimates of the local convergence rate of spectral expansions on the distance from an interior compact set to the boundary

We consider the problem on the convergence rate of biorthogonal expansions of functions in systems of root functions of a wide class of ordinary second-order differential operators defined on a finite interval. These expansions are compared with expansions of the same functions in Fourier trigonometric series in an integral or uniform metric on any interior compact subset of the main interval. We find the dependence of the equiconvergence rate of resulting expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the presence of infinitely many associated functions in the system of root functions.