On the similarity of a <Emphasis Type="Italic">J</Emphasis>-nonnegative Sturm-Liouville operator to a self-adjoint operator

In terms of Weyl-Titchmarsh m-functions, we obtain a new necessary condition for an indefinite Sturm-Liouville operator to be similar to a self-adjoint operator. This condition is used to construct examples of J-nonnegative Sturm-Liouville operators with singular critical point zero.     

A dissipative singular Sturm-Liouville problem with a spectral parameter in the boundary condition

In the Hilbert space , we consider nonselfadjoint singular Sturm-Liouville boundary value problem (with two singular end points a and b) in limit-circle cases at a and b, and with a spectral parameter in the boundary condition. The approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Sturm-Liouville equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and Sturm-Liouville boundary value problem.     

On the asymptotics of eigenfunctions of Sturm-Liouville operators with distribution potentials

We consider a Sturm-Liouville operator in the space L ₂[0, π] and derive asymptotic formulas for the eigenvalues and eigenfunctions of this operator for the case of Dirichlet-Neumann boundary conditions. The leading and second terms of the asymptotics are found in closed form.     

Sélection automatique du paramètre de lissage pour l'estimation non paramétrique de la régression pour des données fonctionnelles

We study regression estimation when the explanatory variable is functional. Nonparametric estimates of the regression operator have been recently introduced. They depend on a smoothing factor which controls its behaviour, and the aim of our Note is to construct some data-driven criterion for choosing this smoothing parameter. The criterion can be formulated in terms of a functional version of cross-validation ideas. Under mild assumptions on the unknown regression operator, it is seen that this rule is asymptotically optimal. As by-products of this result, we state asymptotic equivalences for several measures of accuracy for nonparametric estimate of the regression operator. To cite this article: M. Rachdi, P. Vieu, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On the continuous analog of Rakhmanov's Theorem for orthogonal polynomials

We obtain the continuous analogs of Rakhmanov's Theorem for polynomials orthogonal on the unit circle. Sturm-Liouville operators and Krein systems are considered. For a Sturm-Liouville operator with bounded potential q, we prove the following statement. If the essential spectrum and absolutely continuous component of the spectral measure fill the whole positive half-line, then q decays at infinity in the certain integral sense.     

The similarity problem for J-nonnegative Sturm-Liouville operators

Sufficient conditions for the similarity of the operator with an indefinite weight r(x)=(sgnx)|r(x)| are obtained. These conditions are formulated in terms of Titchmarsh-Weyl m-coefficients. Sufficient conditions for the regularity of the critical points 0 and ∞ of J-nonnegative Sturm-Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm-Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case r(x)=sgnx and , we prove that A is similar to a self-adjoint operator if and only if A is J-nonnegative. The latter condition on q is sharp, i.e., we construct such that A is J-nonnegative with the singular critical point 0. Hence A is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that J-positivity is sufficient for the similarity of A to a self-adjoint operator. In the case q≡0, we prove the regularity of the critical point 0 for a wide class of weights r. This yields new results for “forward-backward” diffusion equations.     

Recovering the Sturm-Liouville Operator with Singular Potential Using Nodal Data

The paper deals with the Sturm-Liouville operator with singular potential. We assume that the potential is a sum of an a priori known distribution from a certain class and an unknown sufficiently smooth function. The inverse problem is to recover the operator using zeros of eigenfunctions (nodes) as an input data. For this inverse problem we obtain a procedure for constructing the solution.     

On non-self-adjoint Sturm-Liouville operators in the space of vector functions

In this article we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operator generated in L ₂ ^m [0, 1] by the Sturm-Liouville equation with m × m matrix potential and the boundary conditions which, in the scalar case (m = 1), are strongly regular. Using these asymptotic formulas, we find a condition on the potential for which the root functions of this operator form a Riesz basis.     

Inverse spectral problem for integro-differential operators

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ? which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L ₂[0, π] and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential q, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for ? and by two pairs of boundary conditions coinciding at one of the finite points.     

Half-inverse problem for diffusion operators on the finite interval

The potential function q(x) in the regular and singular Sturm-Liouville problem can be uniquely determined from two spectra. Inverse problem for diffusion operator given at the finite interval eigenvalues, normal numbers also on two spectra are solved. Half-inverse spectral problem for a Sturm-Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. In this study, by using the Hochstadt and Lieberman's method we show that if q(x) is prescribed on , then only one spectrum is sufficient to determine q(x) on the interval for diffusion operator.     

Formulation of a Ritz-Galerkin type procedure for the approximate solution of the neutron transport equation

The one-group neutron transport equation is commonly given as an integrodifferential equation for the neutron density ψ(x, ω) over a domain G × S in the five-dimensional phase space $\text{E}{}_3\text{× S(¦ ω ¦ = 1)}$. In this paper we show how, by decomposing the domain of the transport operator into a complementary pair of manifolds by means of a projection operator, any transport problem can be formulated, on either manifold, in terms of a symmetric positive definite operator. We use Friedrichs' method to extend the operator to a selfadjoint operator and look for a generalized solution by minimizing a certain functional over the appropriate Hilbert space. A Ritz-Galerkin type approximation procedure is formulated, and an estimate for the difference between the exact and approximate solution is given. The procedure is illustrated for a special choice of finite dimensional subspace.     

AMBARZUMYAN’S THEOREM WITH EIGENPARAMETER IN THE BOUNDARY CONDITIONS

In this paper, the classical Ambarzumyan’s theorem for the regular SturmLiouville problem is extended to the case in which the boundary conditions are eigenparameter dependent. Specifically, we show that if the spectrum of the operator D 2 +q with eigenparameter dependent boundary conditions is the same as the spectrum belonging to the zero potential, then the potential function q is actually zero.     

Classification of Non-symmetric Sturm-Liouville Systems and Operator Realizations

The main objective of this paper is to give a classification of Sturm-Liouville differential equations with non-symmetric matrices coefficients in terms of the number of square integrable solutions of the system and its conjugate system. The conjugate system is innovatively introduced to get the classification. Furthermore, the asymptotic behavior of elements in the maximal domain is studied. As applications, the J-self-adjoint operator realizations are given for a special case in the classification.     

Characterization of the spectrum of regular boundary value problems for the Sturm-Liouville operator

On the interval (0, τ), we consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L ₂(0, τ) and with regular (but not strengthened-regular) boundary conditions. Under certain additional assumptions, we establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.     

On the best approximations and rate of convergence of decompositions in the root vectors of an operator

We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator A that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of K-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of f from B in the root vectors of the operator A to f     

Inversion of Trace Formulas for a Sturm-Liouville Operator

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.     

On the divergence of expansions in systems of root functions of the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville operator with an arbitrary complex-valued potential q(x) ?? L ₁(0, ??) and with degenerate boundary conditions. We show that, under some additional condition, the system of root functions of that operator is not a basis in the space L ₂(0, ??).     

<Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Mappings. Classification of Invariant Subspaces

We continue the study of an operator algebra associated with a self-mapping ? on a countable setX which can be represented as a directed graph. This C*-algebra belongs to a class of operator algebras, generated by a family of partial isometries satisfying some relations on their source and range projections. Earlier we have formulated the irreducibility criterion of such algebras, which give us a possibility to examine the structure of the corresponding Hilbert space. We will show that for reducible algebras the underlying Hilbert space can be represented either as an infinite sum of invariant subspaces or as a tensor product of a finite-dimensional Hilbert space with l²(Z). In the first case we present a conditions under which the studied algebra has an irreducible representation into a C*-algebra generated by a weighted shift operator. In the second case, the algebra has the irreducible finite-dimensional representations indexed by the unit circle.     

Asymptotic analysis of non-self-adjoint Hill operators

We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L _t (q) with a potential q ∈ L ₁[0,1] and t-periodic boundary conditions, t ∈ (?π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L ₂(?∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.     

Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution

We consider the spectral problem for a model second-order differential operator with an involution. The operator is given by the differential expression Lu = ?u??(?x) and boundary conditions of general form. We obtain a criterion for the basis property of the systems of eigenfunctions of this operator in terms of the coefficients in the boundary conditions.