Limiting Absorption Principle for Stratified Operators with Thresholds: A General Method

Out problem is about propagation of waves in stratified strips. The operators are quite general, a typical example being a coupled elasto-acoustic operator H defined in ?² × I where I is a bounded interval of ? with coefficients depending only on z ∈ I. The “conjugate operator method” will be applied to an operator obtained by a spectral decomposition of the partial Fourier transform ? of H. Around each value of the spectrum (except the eigenvalues) including the thresholds, a conjugate operator is constructed which permits to get the ”good properties“ of regularity for H. A limiting absorption principle is then obtained for a large class of operators at every point of the spectrum (except eigenvalues).     

Perturbed boundary eigenvalue problem for the Schrödinger operator on an interval

A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with Dirichlet conditions. The perturbation is described by the potential μ⁻¹ V((x − x ₀)ɛ⁻¹), where 0 < ɛ ≪ 1 and μ is an arbitrary parameter such that there exists δ > 0 for which ɛ/μ = o(ɛ^δ). It is shown that the eigenvalues of this operator converge, as ɛ → 0, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.     

Geometry of Local Adaptive Galerkin Bases

The local adaptive Galerkin bases for large-dimensional dynamical systems, whose long-time behavior is confined to a finite-dimensional manifold, are optimal bases chosen by a local version of a singular decomposition analysis. These bases are picked out by choosing directions of maximum bending of the manifold restricted to a ball of radius ɛ . We show their geometrical meaning by analyzing the eigenvalues of a certain self-adjoint operator. The eigenvalues scale according to the information they carry, the ones that scale as ɛ ² have a common factor that depends only on the dimension of the manifold, the ones that scale as ɛ ⁴ give the different curvatures of the manifold, the ones that scale as ɛ ⁶ give the third invariants, as the torsion for curves, and so on. In this way we obtain a decomposition of phase space into orthogonal spaces E _m , where E _m is spanned by the eigenvectors whose corresponding eigenvalues scale as ɛ ^m . This decomposition is analogous to the Frenet frames for curves. We also discover a practical way to compute the dimension and local structure of the invariant manifold. Accepted 14 October 1998     

Generalized slant Toeplitz operators on H2

For an integer k ≥ 2, k^th‐order slant Toeplitz operator U_φ [1] with symbol φ in L^∞(??), where ?? is the unit circle in the complex plane, is an operator whose representing matrixM = (α_ij ) is given by α_ij = 〈φ, z^ki–j〉, where 〈. , .〉 is the usual inner product in L²(??). The operator V_φ denotes the compression of U_φ to H²(??) (Hardy space). Algebraic and spectral properties of the operator V_φ are discussed. It is proved that spectral radius of V_φ equals the spectral radius of U_φ, if φ is analytic or co‐analytic, and if T_φ is invertible then the spectrum of V_φ contains a closed disc and the interior of the disc consists of eigenvalues of infinite multiplicities. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Perturbation Theory for the Two-Particle Schrodinger Operator on a One-Dimensional Lattice

We consider the two-particle Schrodinger operator H(k) on the one-dimensional lattice ℤ. The operator H(π) has infinitely many eigenvalues z_m(π) = v(m), m ∈ ℤ₊. If the potential v increases on ℤ₊, then only the eigenvalue z₀(π) is simple, and all the other eigenvalues are of multiplicity two. We prove that for each of the doubly degenerate eigenvalues z_m(π), m ∈ ℕ, the operator H(π) splits into two nondegenerate eigenvalues z _m ⁻ (k) and z _m ⁺ (k) under small variations of k ∈ (π − δ, π). We show that z _m ⁻ (k) < z _m ⁺ (k) and obtain an estimate for z _m ⁺ (k) − z _m ⁻ (k) for k ∈ (π − δ, π). The eigenvalues z₀(k) and z ₁ ⁻ (k) increase on [π − δ, π]. If (Δv)(m) > 0, then z _m ^± (k) for m ≥ 2 also has this property. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 212–220, November, 2005.     

Spectre de l'opérateur de transfert en dimension 1

We study spectral properties of a transfer operator ℳΦ(x)=∑_ω g _ω(x)Φ(ψ_ω x) acting on functions of bounded variation. Using a symmetrical integral, we first obtain bounds on its spectral and essential spectral radii. We then consider the dynamical determinant Det^#(Id +zℳ). Our main theorem generalizes to discontinuous weights the result of Baladi and Ruelle (for continuous weights) on the link between zeroes of the sharp determinant and eigenvalues of the transfer operator. The proof is based on regularizing the weights and uses a (new) spectral result giving the surjectivity of some applications between eigenspaces of operators. Received: 8 May 2001     

Asymptotic methods for spherically symmetric MHD α2-dynamos

We consider two models of spherically-symmetric MHD α²–dynamos; one with idealized boundary conditions (BCs); and one with physically realistic BCs. As it has been shown in our previous work, the eigenvalues λ of a model with idealized BCs and constant α–profile α₀ are linear functions of α₀ and form a mesh in the (α₀, λ)–plane. The nodes of the spectral mesh correspond to double-degenerate eigenvalues of algebraic and geometric multiplicity 2 (diabolical points). It was found that perturbations of the constant α –profile lead to a resonant unfolding of the diabolical points with selection rules of the resonant unfolding defined by the Fourier coefficients of the perturbations. In the present contribution we present new exact results on the spectrum of the model with physically realistic BCs and constant α. For non-degenerate (simple) eigenvalues perturbation gradients are found at any particular α₀. We briefly discuss the spectral behavior of the α²–dynamo operator over a family of homotopic deformations of the BCs between idealized ones and physically realistic ones. Furthermore, we demonstrate that although the spectral singularities are lifted, a memory about their locations remains deeply imprinted in the homotopic family of spectral deformations due to a hidden underlying invariance. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algebraic Order Bounded Disjointness Preserving Operators and Strongly Diagonal Operators

Let T be an order bounded disjointness preserving operator on an Archimedean vector lattice. The main result in this paper shows that T is algebraic if and only if there exist natural numbers m and n such that n ≥ m, and T^n!, when restricted to the vector sublattice generated by the range of T^m, is an algebraic orthomorphism. Moreover, n (respectively, m) can be chosen as the degree (respectively, the multiplicity of 0 as a root) of the minimal polynomial of T. In the process of proving this result, we define strongly diagonal operators and study algebraic order bounded disjointness preserving operators and locally algebraic orthomorphisms. In addition, we introduce a type of completeness on Archimedean vector lattices that is necessary and sufficient for locally algebraic orthomorphisms to coincide with algebraic orthomorphisms.     

Eigenvalues and Pole Functions of Hamiltonian Systems with Eigenvalue Depending Boundary Conditions

This paper consists of two chapters. The first chapter concerns matrix functions belonging to the generalized Nevanlinna class N_k^{m × m}. We present results about the operator representation of such functions. These representations are then used to obtain information about the (generalized) poles of generalized Nevanlinna functions. The second chapter may be viewed as a continuation of our paper [DLS3] and treats Hamiltonian systems of differential equations with boundary conditions depending on the eigenvalue parameter. In particular we study the eigenvalues, both isolated and embedded eigenvalues.     

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.     

Some Applications of the Regularity and Irreducibility on Transport Theory

This paper is concerned with the spectral analysis of transport operator with general boundary conditions in L ₁-setting. This problem will be investigated under results from the theory of positive linear operators, irreducibility and regularity of the collision operator. The basic problems treated here are notions of essential spectra, spectral bound and leading eigenvalues.     

Building and solving matrix spectral problems

When an infinite dimensional operator T: X → X is approximated with (a slight perturbation of) an operator T_n : X → X of finite rank less than or equal to n, the spectral elements of an auxiliary matrix Z ∈ ℂ^{n ×n} , lead to those of Tn, if they are computed exactly. This contribution covers a general theoretical framework for matrix problems issued from finite rank discretizations and perturbed variants, the stop criterion of the QR method for eigenvalues, the possibility of using the Newton method to compute a Schur form, and the use of Newton method to refine coarse approximate bases of spectral subspaces. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trace formulae forS 1 invariant green’s operators onS 2

Summary We study the spectrum of the Laplacian acting on 1-forms for a surface of revolution diffeomorphic toS ² and obtain, for theS ¹ invariant spectrum, a trace formula for the eigenvalues of its Green’s operator. The trace formula is used to “hear” negative curvature for some metrics and to prove the existence of metrics onS ² with rather unusual spectral characteristics. This article was processed by the author using the L^AT_EX style file from Springer-Verlag.     

Eigenvalue distribution theorems for certain homogeneous spaces

A classical theorem of Szegö describes the limiting behavior of the eigenvalues of P_nAP_n, where A is a multiplication operator on L₂(S¹) and P_n is the projection on the subspace spanned by e^ikθ (0 ? k ? n). A generalization of this is proved, whereby S¹ is replaced by an arbitrary rank one homogeneous space (e.g. S^m), A by a pseudodifferential operator of order zero, and P_n by a sequence of spectral projections for the Laplace-Beltrami operator. A by-product is an alternate proof of a theorem of A. Weinstein on the eigenvalue clusters of the Laplacian plus a potential.     

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript> by the use of finite submatrices

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l ²(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J _n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].     

Asymptotics Near ±m of the Spectral Shift Function for Dirac Operators with Non-Constant Magnetic Fields

We consider a 3-dimensional Dirac operator H ₀ with non-constant magnetic field of constant direction, perturbed by a sign-definite matrix-valued potential V decaying fast enough at infinity. Then we determine asymptotics, as the energy goes to +m and ?m, of the spectral shift function for the pair (H ₀ + V, H ₀). We obtain, as a by-product, a generalized version of Levinson's Theorem relating the eigenvalues asymptotics of H ₀ + V near +m and ?m to the scattering phase shift for the pair (H ₀ + V, H ₀).     

The negative discrete spectrum of the operator (-τ) l -αV inL 2(R d ) ford even and 2l≥dinL 2(R d ) ford even and 2l≥d

We study the asymptotic behaviour ofN(α)—the number of negative eigenvalues of the operator (-τ) ^l -αV inL ₂(R ^d ) for an evend and2l≥d. This is the only case where the previously known results were far from being complete. In order to describe our results we introduce an auxiliary ordinary differential operator (system) on the semiaxis. Depending on the spectral properties of this operator we can distinguish between three cases whereN(α) is of the Weyl-type,N(α) is of the Weyl-order but not the Weyl-type coefficient and finally whereN(α)=O(α^q) withq>d/2l.     

Algebraic Topological Closure Operators

For any closure operator c there is a T_o-closure operator whose lattice of closed subsets are isomorphic to that of c. A correspondence between algebraic topological (T_o) closure operators on a nonempty set X and pre-orderes (partial orders) on X is established. Equivalent conditions are obtained for a T_o-lattice to be a complete atomic Boolean algebra and for the lattice of closed subsets of an algebraic topological closure operator to be a complete atomic Boolean algebra. Further it is proved that a complete lattice is an algebraic T_o-lattice if and only if it is isomorphic to the lattice of closed subsets of some algebraic topological closure operator on a suitable set.AMS Subject Classification (1991): 06A23, 54D65.     

Integral Equations and Operator Theory

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C₀ contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of Aⁿ, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A^* has trivial kernel.