Symmetric Differential Operators Research Articles

Eigenvalue estimates for Dirac operators in geometries with torsion

On a Riemannian spin manifold (Mn, g), equipped with a non-integrable geometric structure and characteristic connection ▽c with parallel torsion ▽cT c = 0, we can introduce the Dirac operator D1/3, which is constructed by lifting the affine metric connection with torsion 1/3 T c to the spin structure. D1/3 is a symmetric elliptic differential operator, acting on sections of the spinor bundle and can be identified in special cases with Kostant’s cubic Dirac operator or the Dolbeault operator. For compact (Mn, g), we investigate the first eigenvalue of the operator \({\left(D^{1/3} \right)^{2}}\) . As a main tool, we use Weitzenbock formulas, which express the square of the perturbed operator D1/3 + S by the Laplacian of a suitable spinor connection. Here, S runs through a certain class of perturbations. We apply our method to spaces of dimension 6 and 7, in particular, to nearly Kahler and nearly parallel G2-spaces.

Analysis of differential operators associated with boundary value problems

We consider boundary value problems in a two-component domain of the Euclidean space R n obtained by eliminating from R n the boundary G. Traces on both sides of G are defined without limit passages. In a Hilbert trace space, we introduce orthogonal projections, analogs of the Calderon projections, which are used for constructing operators whose continuous invertibility implies the solvability of the corresponding boundary value problems. For the resolvents we obtain representations similar to the Krein formula. For a symmetric differential operator we show that the constructed resolvents of boundary value problems correspond to closed (not necessarily self-adjoint) extensions of this operator in the sense of von Neumann. Bibliography: 9 titles.

A permutation model for free random variables and its classical analogue

In this paper, we generalize a permutation model for free random variables which was first proposed by Biane in \cite{biane}. We also construct its classical probability analogue, by replacing the group of permutations with the group of subsets of a finite set endowed with the symmetric difference operation. These constructions provide new discrete approximations of the respective free and classical Wiener chaos. As a consequence, we obtain explicit examples of non random matrices which are asymptotically free or independent. The moments and the free (resp. classical) cumulants of the limiting distributions are expressed in terms of a special subset of (noncrossing) pairings. At the end of the paper we present some combinatorial applications of our results.

Asymptotic Normality in Density Support Estimation

Let $X_1,\ldots,X_n$ be $n$ independent observations drawn from a multivariate probability density $f$ with compact support $S_f$. This paper is devoted to the study of the estimator $\hat{S}_n$ of $S_f$ defined as the union of balls centered at the $X_i$ and with common radius $r_n$. Using tools from Riemannian geometry, and under mild assumptions on $f$ and the sequence $(r_n)$, we prove a central limit theorem for $\lambda (S_n \Delta S_f)$, where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}^d$ and $\Delta$ the symmetric difference operation.

A Convergent Adaptive Method for Elliptic Eigenvalue Problems

We prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second-order symmetric elliptic partial differential operators. The weak form is assumed to yield a bilinear form which is bounded and coercive in $H^1$. Each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation. The error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems, and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem. Because of this nonlinearity, the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine.

A Method to Find Weight Matrices Having Symmetric Second-Order Differential Operators with Matrix Leading Coefficient

We develop a method to construct examples of weight matrices of size N×N having symmetric second-order differential operators of the form $$D^{2}A_{2}(t)+D^{1}A_{1}(t)+D^{0}A_{0},$$ where A2, A1 and A0 are matrix polynomials of degrees not larger than 2, 1 and 0, respectively. The main feature of this method is that in some cases it finds weight matrices having such differential operators even though none of these operators has diagonal leading coefficient.

Krein's resolvent formula for self-adjoint extensions of symmetric second-order elliptic differential operators

Given a symmetric, semi-bounded, second-order elliptic differential operator A on a bounded domain with C1,1 boundary, we provide a Kreĭn-type formula for the resolvent difference between its Friedrichs extension and an arbitrary self-adjoint one.

STRUCTURAL CLASSIFICATION OF XML DOCUMENTS USING MULTISETS

In this paper, we investigate the problem of clustering XML documents based on their structure. We represent the paths in an XML document as a multiset and use the symmetric difference operation on multisets to define certain metrics. These metrics are then used to obtain a measure of similarity between any two documents in a collection. Our technique was successfully applied to real and synthesized XML documents yielding high-quality clusterings.

On the Cauchy problem for the Schrödinger equation degenerating outside a segment: Properties of solutions and spectral aspects of the regularization

The evolutionary Schrödinger equation with a second-order generator is considered on the line. For the Schrödinger equation with a degenerate operator, the characteristic form of which vanishes outside a segment I = [−l, l] ⊂ ℝ, a well-posed setting of the Cauchy problem is investigated. We find conditions for the initial-value data of the problem that are necessary and sufficient for its unique solvability in a given interval of time. A sequence of regularized Cauchy problems with uniformly elliptic operators is considered as well; we study the convergence of the sequence of solutions for nondegenerate problems to the solution of the degenerate problem as well as the convergence of regularized semigroups of transformations in the strong operator topology. We prove that any arbitrary sequence of solutions of regularized problems with initial data that does not satisfy the existence condition for the solution diverges. However, one cannot exclude that there exists a subsequence of the regularization parameters such that the corresponding sequence of regularized semigroups converges (in the strong operator topology) uniformly on each segment. We provide a description of the set of all possible partial limits for the sequence of regularized semigroups; this description is given in terms of a collection of self-adjoint extensions for the degenerate operator. It is still an open question if all those partial limits are accessible. The Cauchy problem for the Schrödinger equation, the generator of which is a symmetric linear differential operator in the Hilbert space H = L 2(ℝ), is considered as well. We investigate if the behavior of the sequence of regularized semigroups depends on the choice of the regularization for the generator. We define a linear self-adjoint regularization of the Cauchy problem with a degenerate operator as a directed set of well-posed problems approximating the original one. We define a correct regularization as a linear self-adjoint regularization of the degenerate operator such that its index of error determines the well-posedness and the convergence and weak convergence of the sequence of regularized solutions. We find necessary and sufficient conditions for the convergence (in the strong and weak operator topologies) of the sequence of correctly regularized semigroups.

Fast way to compute functional determinants of radially symmetric partial differential operators in general dimensions

Recently the partial-wave cutoff method was developed as a new calculational scheme for a functional determinant of quantum field theory in radial backgrounds. For the contribution given by an infinite sum of large partial waves, we derive explicitly radial-WKB series in the angular momentum cutoff for $d=2$, 3, 4, and 5 ($d$ is the space-time dimension), which has uniform validity irrespectively of any specific values assumed for other parameters. Utilizing this series, precision evaluation of the renormalized functional determinant is possible with a relatively small number of low partial-wave contributions determined separately. We illustrate the power of this scheme in a numerically exact evaluation of the prefactor (expressed as a functional determinant) in the case of the false vacuum decay of 4D scalar field theory.

SPECTRAL FLOW AND WINDING NUMBER IN VON NEUMANN ALGEBRAS

We introduce a new topology, weaker than the gap topology, on the space of self-adjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of self-adjoint Breuer–Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips's analytical definition and derive integral formulae for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey and Phillips. Furthermore, we prove the homotopy invariance of the real-valued index. As an example we consider invariant symmetric elliptic differential operators on Galois coverings.

Second-Order Differential Operators Having Several Families of Orthogonal Matrix Polynomials as Eigenfunctions

The aim of this paper is to bring into the picture a new phenomenon in the theory of orthogonal matrix polynomials satisfying second order differential equations. The last few years have witnessed some examples of a (fixed) family of orthogonal matrix polynomials whose elements are common eigenfunctions of several linearly independent second order differential operators. We show that the dual situation is also possible: there are examples of different families of matrix polynomials, each family orthogonal with respect to a different weight matrix, whose elements are eigenfunctions of a common second order differential operator. These examples are constructed by adding a discrete mass at certain point to a weight matrix: $\widetilde{W}=W+\delta_{t_0}M(t_0)$. Our method consists in showing how to choose the discrete mass $M(t_0)$, the point $t_0$ where the mass lives and the weight matrix $W$ so that the new weight matrix $\widetilde{W}$ inherits some of the symmetric second order differential operators associated with $W$. It is well known that this situation is not possible for the classical scalar families of Hermite, Laguerre and Jacobi.

Fuzzy implication operators for difference operations for fuzzy sets and cardinality-based measures of comparison

In this paper, we determine by means of fuzzy implication operators, two classes of difference operations for fuzzy sets and two classes of symmetric difference operations for fuzzy sets which preserve properties of the classical difference operation for crisp sets and the classical symmetric difference operation for crisp sets respectively. The obtained operations allow us to construct as in [B. De Baets, H. De Meyer, Transitivity-preserving fuzzification schemes for cardinality-based similarity measures, European Journal of Operational Research 160 (2005) 726–740], cardinality-based similarity measures which are reflexive, symmetric and transitive fuzzy relations and, to propose two classes of distances (metrics) which are fuzzy versions of the well-known distance of cardinality of the symmetric difference of crisp sets.

Robust a-posteriori estimator for advection-diffusion-reaction problems

We propose an almost-robust residual-based a-posteriori estimator for the advection-diffusion-reaction model problem. The theory is developed in the one-dimensional setting. The numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators. In particular, the mentioned norm possesses features that allow us to obtain a meaningful a-posteriori estimator, robust up to a √log(Pe) factor, where Pe is the global Peclet number of the problem. Various numerical tests are performed in one dimension, to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature. We also consider a possible two-dimensional extension of our result and only present a few basic numerical tests, indicating that the estimator seems to preserve the good features of the one-dimensional setting.

Kreĭn–Saakyan and Trace Formulas for a Pair of Canonical Differential Operators

An analog of the Kreĭn–Saakyan formula is derived for any pair of relatively prime self-adjoint extensions of a minimal symmetric canonical differential operator. This allows us to deduce a trace formula in the matrix case.

Resonances of a two-state semiclassical Schrödinger Hamiltonians

The goal of this article is to generalize some travels made in (Helffer, B. and Sjöstrand, J., 1986, Résonances en limite semi-classique. Mémoires de la Société Mathématique de France Sér. 2; Hunziker, W., 1986, Distorsion analyticity, and molecular resonance curves. Ann. I. H. P. (section Physique Théorique), 45, 339–358.; Messirdi, B., 1994, Asymptotique de Born-Oppenheimer pour la prédissociation moléculaire (cas de potentiels réguliers). Annales de l'IHP, Section Physique Théorique, 61(3), 255–292.; Messirdi, B., 1993, Asymptotique de Born-Oppenheimer pour la prédissociation moléculaire. Thèse de Doctorat de l'Université de Paris 13) to multidimensional two-state perturbed and semiclassical systems. More precisely, we study the Schrödinger Hamiltonians on where V is flat at infinity, R is a bounded symmetric differential operator of first order and h tends to 0+, in the case where resonances appear. Using a microlocal estimates on resolvents of P(h) and the so-called Grushin problem the spectral study of the distorded operator of P(h) is reduced to the resolution of an analytic algebraic equation. Under the assumptions that V(x) admits a relatively compact well U and the Dirichlet realization of P(h) near U has a simple eigenvalue, it is then showed that P(h) has a unique resonance. We obtain also that the width of this resonance is exponentially small as h tends to zero.

A self adjoint expansion of a symmetric differential operator with operator coefficient

In this work, we prove that the closure of a symmetric operator L0 which is formed by differential expression (L0y)(x )= −(p(x)y � (x)) � − Q(x)y(x) and with the boundary condition cos α.y(0) + sin α.y � (0) = 0 is self adjoint where α ∈ (−∞, ∞) in the space L2(0, ∞; H ). Furthermore, we investigate some properties of this operator.

Deficiency indices and spectrum of self-adjoint extensions of some classes of differential operators

Problems relating to the asymptotic behaviour in the neighbourhood of the point and in the neighbourhood of the origin of a solution of an equation of arbitrary (even or odd) order with complex-valued coefficients are studied. It is assumed here that the coefficients of the quasidifferential expression have the following property: if one reduces the equation to a system of first-order differential equations, then one can transform that system to a system of differential equations with regular singular point at or . The results obtained allow one to determine the deficiency indices of the corresponding minimal symmetric differential operators and the structure of the spectrum of self-adjoint extensions of these operators. In addition, on the basis of refined asymptotic formulae for solutions to the equation the deficiency numbers of a certain differential operator generated by a differential expression with leading coefficient vanishing in the interior of the interval in question are found.

Matrix Valued Orthogonal Polynomials Arising from the Complex Projective Space

The main purpose of this paper is to display new families of matrix valued orthogonal polynomials satisfying second-order differential equations, obtained from the representation theory of U(n). Given an arbitrary positive definite weight matrix W(t) one can consider the corresponding matrix valued orthogonal polynomials. These polynomials will be eigenfunctions of some symmetric second-order differential operator D only for very special choices of W(t). Starting from the theory of spherical functions associated to the pair (SU(n+1), U(n)) we obtain new families of such pairs {W,D}. These depend on enough integer parameters to obtain an immediate extension beyond these cases.

P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation

The solution of several instances of the Schrodinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585–600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Grunbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461–84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein.