Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H=AS⁺+A^-S^-+B (with S^± the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E₊], E_-<E₊, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by , 0?E_-<E₊.Our approach is based on trace formulas and matrix-valued (exponential) Herglotz representation theorems. As a by-product of our techniques we obtain the extension of Flaschka's Borg-type result for periodic scalar Jacobi operators to the class of reflectionless matrix-valued Jacobi operators.     

On the singular Weyl-Titchmarsh function of perturbed spherical Schrödinger operators

We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical Schrödinger operators (also known as Bessel operators) under the assumption that the perturbation q(x) satisfies xq(x)∈L¹(0,1). We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.     

M-matrix asymptotics for Sturm–Liouville problems on graphs

We consider a system formulation for Sturm–Liouville operators with formally self-adjoint boundary conditions on a graph. An M-matrix associated with the boundary value problem is defined and related to the matrix Prüfer angle associated with the system boundary value problem, and consequently with the boundary value problem on the graph. Asymptotics for the M-matrix are obtained as the eigenparameter tends to negative infinity. We show that the boundary conditions may be recovered, up to a unitary equivalence, from the M-matrix and that the M-matrix is a Herglotz function. This is the first in a series of papers devoted to the reconstruction of the Sturm–Liouville problem on a graph from its M-matrix.     

Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms

This paper studies the Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms. We first state the existence of standing wave with ground state, based on which we prove a sharp criteria for global existence and blow-up of solutions when E(0)<d. We then introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions for 0<E(0)<d and E(0)≤0, respectively. Furthermore, we prove the global existence and asymptotic behavior of solutions for the case of potential well family with 0<E(0)<d. Finally, a blow-up result for solutions with arbitrarily positive initial energy is obtained.     

Estimates for fundamental solutions and spectral bounds for a class of Schrödinger operators

We study a class of Schrödinger operators of the form , where is a nonnegative function singular at 0, that is V(0)=0. Under suitable assumptions on the potential V, we derive sharp lower and upper bounds for the fundamental solution hε. Moreover, we obtain information on the spectrum of the self-adjoint operator defined by Lε in L²(R). In particular, we give a lower bound for the eigenvalues.     

Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems

Hyperspace dynamical system (E²,f²) induced by a given dynamical system (E,f) has been recently investigated regarding topological mixing, weak mixing and transitivity that characterize orbit structure. However, the Vietoris topology on E² employed in these studies is non-metrizable when E is not compact metrizable, e.g., E=Rn. Consequently, metric related dynamical concepts of (E²,f²) such as sensitivity on initial conditions and metric-based entropy, could not even be defined. Moreover, a condition on (E²,f²) equivalent to the transitivity of (E,f) has not been established in the literature. On the other hand, Hausdorff locally compact second countable spaces (HLCSC) appear naturally in dynamics. When E is HLCSC, the hit-or-miss topology on E² is again HLCSC, thus metrizable. In this paper, the concepts of co-compact mixing, co-compact weak mixing and co-compact transitivity are introduced for dynamical systems. For any HLCSC system (E,f), these three conditions on (E,f) are respectively equivalent to mixing, weak mixing and transitivity on (E²,f²) (hit-or-miss topology equipped). Other noticeable properties of co-compact mixing, co-compact weak mixing and co-compact transitivity such as invariants for topological conjugacy, as well as their relations to mixing, weak mixing and transitivity, are also explored.     

Nodal solutions of multi-point boundary value problems

We study the nonlinear boundary value problem consisting of the equation y^″+w(t)f(y)=0 on [a,b] and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes in the existence question for different types of nodal solutions as the problem changes.     

The method of external excitation for solving generalized Sturm-Liouville problems

A new numerical technique for solving the generalized Sturm-Liouville problem , b_l[w(0),λ]=b_r[w(1),λ]=0 is presented. In particular, we consider the problems when the coefficient q(x,λ) or the boundary conditions depend on the spectral parameter λ in an arbitrary nonlinear manner. The method presented is based on mathematically modelling the physical response of a system to excitation over a range of frequencies. The response amplitudes are then used to determine the eigenvalues.The results of the numerical experiments justifying the method are presented.     

Existence of Positive Solutions for Operator Equations and Applications to Semipositone Problems

In this paper we study the existence of positive solutions of the following operator equation in a Banach space E: where G(x, λ) = λKFx+e₀, K: E↦ E is a linear completely continuous operator, F: P↦ E is a nonlinear continuous , bounded operator, e₀∈ E, λ is a parameter and P is a cone of Banach space E. Since F is not assumed to be positive and e₀ may be a negative element, the operator equation is a so-called semipositone problem. We prove that under certain super-linear conditions on the operator F the operator equation has at least one positive solution for λ > 0 sufficiently small, and that under certain sub-linear conditions on the operator F the operator equation has at least one positive solution for λ > 0 sufficiently large. In addition, we briefly outline an application of our results which simplify previous theorems in the literature.     

Quantum effects, sequential independence and majorization

A quantum effect is a positive Hilbert space contraction operator. If {E_i}, 1?i?n, are n quantum effects (defined on some Hilbert space H), then their sequential product is the operator . It is proved that the quantum effects {E_i}, 1?i?n, are sequentially independent if and only if for every permutation r₁r₂…r_n of the set S_n={1,2,…,n}. The sequential independence of the effects E_i, 1?i?n, implies E_noE_n-1o…oE_j+1oE_jo…oE₁=(E_noE_n-1o…E_j+1)oE_jo…oE₁ for every 1?j?n. It is proved that if there exists an effect E_j, 1?j?n, such that E_j?(E_noE_n-1o…E_j+1)oE_jo…oE₁, then the effects {E_i} are sequentially independent and satisfy .     

Characterization of domains of self-adjoint ordinary differential operators

The GKN (Glazman, Krein, Naimark) Theorem characterizes all self-adjoint realizations of linear symmetric (formally self-adjoint) ordinary differential equations in terms of maximal domain functions. These functions depend on the coefficients and this dependence is implicit and complicated. In the regular case an explicit characterization in terms of two-point boundary conditions can be given. In the singular case when the deficiency index d is maximal the GKN characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter λ. In the much more difficult intermediate cases, not all solutions contribute to the singular self-adjoint conditions. In 1986 Sun found a representation of the self-adjoint singular conditions in terms of certain solutions for nonreal values of λ. In this paper we give a representation in terms of certain solutions for real λ. This leads to a classification of solutions as limit-point (LP) or limit-circle (LC) in analogy with the celebrated Weyl classification in the second-order case. The LC solutions contribute to the singular boundary conditions, the LP solutions do not. The advantage of using real λ is not only because it is, in general, easier to find explicit solutions but, more importantly, it yields information about the spectrum.     

Approximate invariant manifolds up to exponentially small terms

This paper is devoted to analytic vector fields near an equilibrium for which the linearized system is split in two invariant subspaces E₀ (), E₁ (). Under light Diophantine conditions on the linear part, we prove that there is a polynomial change of coordinate in E₁ allowing to eliminate, in the E₁ component of the vector field, all terms depending only on the coordinate u₀∈E₀, up to an exponentially small remainder. This main result enables to prove the existence of analytic center manifolds up to exponentially small terms and extends to infinite-dimensional vector fields. In the elliptic case, our results also proves, with very light assumptions on the linear part in E₁, that for initial data very close to a certain analytic manifold, the solution stays very close to this manifold for a very long time, which means that the modes in E₁ stay very small.     

Left-definite theory with applications to orthogonal polynomials

In the past several years, there has been considerable progress made on a general left-definite theory associated with a self-adjoint operator A that is bounded below in a Hilbert space H; the term ‘left-definite’ has its origins in differential equations but Littlejohn and Wellman [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280-339] generalized the main ideas to a general abstract setting. In particular, it is known that such an operator A generates a continuum {H_r}_r>0 of Hilbert spaces and a continuum of {A_r}_r>0 of self-adjoint operators. In this paper, we review the main theoretical results in [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280-339]; moreover, we apply these results to several specific examples, including the classical orthogonal polynomials of Laguerre, Hermite, and Jacobi.     

Fixed point theorems for mixed monotone operators with PPF dependence

In this paper, fixed point theorems for mixed monotone operators are investigated by weakening the requirements in the contractive assumption and strengthening the metric space utilized with a partial order, taking the domain space C[[a,b],E]$C[[a,b],E]$ different from the range E$E$ and using the method of reflection operators.     

Algebraic reflexivity of some subsets of the isometry group

Let X be a compact first countable space. In this paper we show that the set of isometries of C(X) that are involutions is algebraically reflexive. As a consequence of a recent work of Botelho and Jamison this leads to the conclusion that the set of generalized bi-circular projections on C(X) is also algebraically reflexive. We also consider these questions for the space C(X,E) where E is a uniformly convex Banach space.     

Interpretation of De Finetti coherence criterion in ?ukasiewicz Logic

De Finetti gave a natural definition of “coherent probability assessment” β:E→[0,1] of a set E={X₁,…,X_m} of “events” occurring in an arbitrary set W⊆[0,1]^E of “possible worlds”. In the particular case of yes-no events, (where W⊆{0,1}^E), Kolmogorov axioms can be derived from his criterion. While De Finetti’s approach to probability was logic-free, we construct a theory Θ in infinite-valued ?ukasiewicz propositional logic, and show: (i) a possible world of W is a valuation satisfying Θ, (ii) β is coherent iff it is a convex combination of valuations satisfying Θ, (iii) iff β agrees on E with a state of the Lindenbaum MV-algebra of Θ, (iv) iff for some Borel probability measure μ on W. Thus ?ukasiewicz semantics, MV-algebraic (finitely additive) states, and (countably additive) Borel probability measures provide a universal representation of coherent assessments of events occurring in any conceivable set of possible worlds.     

Geometric topological completions with universal final lifts

In this paper necessary and sufficient conditions are given on a concrete category over a category B so that it can be densely embedded (over B) into a geometric topological category E that admits certain universal final lifts. These conditions, as well as the class of universal final lifts, depend upon an a priori given full subcategory Δ of B. For example, E may have, depending upon Δ and B, universal coproducts or quotients or colimits. For appropriate Δ's, if B is cartesian closed then so is E.