Fourth-order Differential Operator Research Articles

Variational characterizations of the total scalar curvature and eigenvalues of the Laplacian

For the dual operator $s_g'^*$ of the linearization $s_g'$ of the scalar curvature function, it is well-known that if $\ker s_g'^*\neq 0$, then $s_g$ is a non-negative constant. In particular, if the Ricci curvature is not flat, then $ {s_g}/(n-1)$ is an eigenvalue of the Laplacian of the metric $g$. In this work, some variational characterizations were performed for the space $\ker s_g'^*$. To accomplish this task, we introduce a fourth-order elliptic differential operator $\mathcal A$ and a related geometric invariant $\nu$. We prove that $\nu$ vanishes if and only if $\ker s_g'^* \ne 0$, and if the first eigenvalue of the Laplace operator is large compared to its scalar curvature, then $\nu$ is positive and $\ker s_g'^*= 0$. Furthermore, we calculated the lower bound on $\nu$ in the case of $\ker s_g'^* = 0$. We also show that if there exists a function which is $\mathcal A$-superharmonic and the Ricci curvature has a lower bound, then the first non-zero eigenvalue of the Laplace operator has an upper bound.

Spectral properties of some regular boundary value problems for fourth order differential operators

In this paper we consider the problem $\begin{gathered} y^{iv} + p_2 (x)y'' + p_1 (x)y' + p_0 (x)y = \lambda y,0 < x < 1, \hfill \\ y^{(s)} (1) - ( - 1)^\sigma y^{(s)} (0) + \sum\limits_{l = 0}^{s - 1} {\alpha _{s,l} y^{(l)} (0) = 0,} s = 1,2,3, \hfill \\ y(1) - ( - 1)^\sigma y(0) = 0, \hfill \\ \end{gathered} $ where λ is a spectral parameter; p j (x) ∈ L 1(0, 1), j = 0, 1, 2, are complex-valued functions; α s;l , s = 1, 2, 3, \(l = \overline {0,s - 1} \), are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α 3,2 + α 1,0 ≠ α 2,1. It is proved that the system of root functions of this spectral problem forms a basis in the space L p (0, 1), 1 < p < ∞, when α 3,2+α 1,0 ≠ α 2,1, p j (x) ∈ W j 1 (0, 1), j = 1, 2, and p 0(x) ∈ L 1(0, 1); moreover, this basis is unconditional for p = 2.

Conical instabilities on paper

The stability of the fundamental defects of an unstretchable flat sheet is examined. This involves expanding the bending energy to second order in deformations about the defect. The modes of deformation occur as eigenstates of a fourth-order linear differential operator. Unstretchability places a global linear constraint on these modes. Conical defects with a surplus angle exhibit an infinite number of states. If this angle is below a critical value, these states possess an n-fold symmetry labeled by an integer, n ⩾ 2. A nonlinear stability analysis shows that the twofold ground state is stable, whereas excited states possess 2(n − 2) unstable modes which come in even and odd pairs.

Orthogonal polynomials associated to a certain fourth order differential equation

We introduce orthogonal polynomials \(M_{j}^{\mu,\ell}(x)\) as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters μ∈ℂ and ℓ∈ℕ0.These polynomials arise as K-finite vectors in the L 2-model of the minimal unitary representations of indefinite orthogonal groups, and reduce to the classical Laguerre polynomials \(L_{j}^{\mu}(x)\) for ℓ=0.We establish various recurrence relations and integral representations for our polynomials, as well as a closed formula for the L 2-norm. Further we show that they are uniquely determined as polynomial eigenfunctions.

Special functions associated with a certain fourth-order differential equation

We develop a theory of "special functions" associated to a certain fourth order differential operator $\mathcal{D}_{\mu,\nu}$ on $\mathbb{R}$ depending on two parameters $\mu,\nu$. For integers $\mu,\nu\geq-1$ with $\mu+\nu\in2\mathbb{N}_0$ this operator extends to a self-adjoint operator on $L^2(\mathbb{R}_+,x^{\mu+\nu+1}dx)$ with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, $L^2$-norms, integral representations and various recurrence relations. This fourth order differential operator $\mathcal{D}_{\mu,\nu}$ arises as the radial part of the Casimir action in the Schr\"odinger model of the minimal representation of the group $O(p,q)$, and our "special functions" give $K$-finite vectors.

Homogenization of a thin plate reinforced with periodic families of rigid rods

The asymptotics of the solution to the elastic bending problem for a thin plate reinforced with several periodic families of closely spaced but disjoint rods are constructed and justified, the result of homogenization being substantially different from the case when the rods are welded together into a single periodic mesh. The material in the rods is assumed to be appreciably more rigid than that in the plate. An averaged fourth-order differential operator is obtained from summing the nonelliptic operators generated by each of the families of the rods. This operator is shown to be elliptic if and only if the rods from at least two families are nonparallel. As a simplified example, the paper examines a similar stationary heat conduction problem.Bibliography: 24 titles.

Spectral Asymptotics of Self-Adjoint Fourth Order Differential Operators with Eigenvalue Parameter Dependent Boundary Conditions

A fourth-order regular ordinary differential operator with eigenvalue dependent boundary conditions is considered. This problem is realized by a quadratic operator pencil with self-adjoint operators. The location of the eigenvalues is discussed and the first four terms of the eigenvalue asymptotics are evaluated explicitly.

An integral formula for L 2-eigenfunctions of a fourth-order Bessel-type differential operator

We find an explicit integral formula for the eigenfunctions of a fourth-order differential operator against the kernel involving two Bessel functions. Our formula establishes the relation between K-types in two different realizations of the minimal representation of the indefinite orthogonal group, namely the L 2-model and the conformal model.

Scalar contribution to the graviton self-energy during inflation

We use dimensional regularization to evaluate the one loop contribution to the graviton self-energy from a massless, minimally coupled scalar on a locally de Sitter background. For noncoincident points our result agrees with the stress tensor correlators obtained recently by Perez-Nadal, Roura, and Verdaguer. We absorb the ultraviolet divergences using the ${R}^{2}$ and ${C}^{2}$ counterterms first derived by 't Hooft and Veltman, and we take the $D=4$ limit of the finite remainder. The renormalized result is expressed as the sum of two transverse, fourth order differential operators acting on nonlocal, de Sitter invariant structure functions. In this form it can be used to quantum correct the linearized Einstein equations so that one can study how the inflationary production of infrared scalars affects the propagation of dynamical gravitons and the force of gravity.

The Ising model: from elliptic curves to modular forms and Calabi–Yau equations

We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contributions 's of the susceptibility of the Ising model for n ⩽ 6 are linear differential operators associated with elliptic curves. Beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral E, the second and third order differential operators Z2, F2, F3, can actually be interpreted as modular forms of the elliptic curve of the Ising model. A last order-4 globally nilpotent linear differential operator is not reducible to this elliptic curve, modular form scheme. This operator is shown to actually correspond to a natural generalization of this elliptic curve, modular form scheme, with the emergence of a Calabi–Yau equation, corresponding to a selected 4F3 hypergeometric function. This hypergeometric function can also be seen as a Hadamard product of the complete elliptic integral K, with a remarkably simple algebraic pull-back (square root extension), the corresponding Calabi–Yau fourth order differential operator having a symplectic differential Galois group . The mirror maps and higher order Schwarzian ODEs, associated with this Calabi–Yau ODE, present all the nice physical and mathematical ingredients we had with elliptic curves and modular forms, in particular an exact (isogenies) representation of the generators of the renormalization group, extending the modular group to a symmetry group.

On Computing the Instability Index of a Non-Self-Adjoint Differential Operator Associated with Coating and Rimming Flows

We study the problem of finding the instability index of certain non-selfadjoint fourth order differential operators that appear as linearizations of coating and rimming flows, where a thin layer of fluid coats a horizontal rotating cylinder. The main result reduces the computation of the instability index to a finite-dimensional space of trigonometric polynomials. The proof uses Lyapunov's method to associate the differential operator with a quadratic form, whose maximal positive subspace has dimension equal to the instability index. The quadratic form is given by a solution of Lyapunov's equation, which here takes the form of a fourth order linear PDE in two variables. Elliptic estimates for the solution of this PDE play a key role. We include some numerical examples.

Spectral analysis of fourth order differential operators III

AbstractThis paper extends the results of the two previous papers in several directions. For one we allow slower decay of the coefficients, but higher order differentiability. For this an expansion for the diagonalizing transformations is derived. Secondly unbounded coefficients are permitted. This requires further transformations in order to achieve Levinson's form, but also a modification with the usual M‐matrix approach. While the standard results carry over to weakly singular coefficients, very singular coefficients will generally lead to discrete spectra (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Spectral properties of a fourth-order differential operator with integrable coefficients

The aim of this paper is to study spectral properties of differential operators with integrable coefficients and a constant weight function. We analyze the asymptotic behavior of solutions to a differential equation with integrable coefficients for large values of the spectral parameter. To find the asymptotic behavior of solutions, we reduce the differential equation to a Volterra integral equation. We also obtain asymptotic formulas for the eigenvalues of some boundary value problems related to the differential operator under consideration.

The volume of the past light-cone and the Paneitz operator

We study a conjecture involving the invariant volume of the past light-cone from an arbitrary observation point back to a fixed initial value surface. The conjecture is that a fourth order differential operator which occurs in the theory of conformal anomalies gives 8π when acted upon the invariant volume of the past light-cone. We show that an extended version of the conjecture is valid for an arbitrary homogeneous and isotropic geometry. First order perturbation theory about flat spacetime reveals a violation of the conjecture which, however, vanishes for any vacuum solution of the Einstein equation. These results may be significant for constructing quantum gravitational observables, for quantifying the back-reaction on spacetime expansion and for alternate gravity models which feature a timelike vector field.

Asymptotics of the spectrum of a singular nonsemibounded fourth-order vector differential operator in a vector function space

Asymptotics of the spectrum of a singular nonsemibounded fourth-order vector differential operator in a vector function space

Basis properties of a fourth order differential operator with spectral parameter in the boundary condition

Abstract We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L p(0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.

Accurate Asymptotic Formulas for Eigenvalues and Eigenfunctions of a Boundary-Value Problem of Fourth Order

In the present paper, we consider a nonself-adjoint fourth-order differential operator with the periodic boundary conditions. We compute new accurate asymptotic expression of the fundamental solutions of the given equation. Then, we obtain new accurate asymptotic formulas for eigenvalues and eigenfunctions.

The asymptotics of the eigenvalues of a fourth order differential operator with summable coefficients

A fourth order differential operator with summable coefficients and some boundary conditions is considered. Asymptotics of solutions to a fourth order differential equation is studied. The equation for eigenvalues is also studied and an asymptotics of the eigenvalues of the considered boundary value problem is obtained.

Time evolution of the scattering data for a fourth-order linear differential operator

The time evolution of the scattering and spectral data is obtained for the differential operator , where u(x, t) and v(x, t) are real-valued potentials decaying exponentially as x → ±∞ at each fixed t. The result is relevant in a crucial step of the inverse scattering transform method that is used in solving the initial-value problem for a pair of coupled nonlinear partial differential equations satisfied by u(x, t) and v(x, t).

The Paneitz equation in hyperbolic space

The Paneitz operator is a fourth order differential operator which arises in conformal geometry and satisfies a certain covariance property. Associated to it is a fourth order curvature – the Q-curvature.We prove the existence of a continuum of conformal radially symmetric complete metrics in hyperbolic space Hn, n>4, all having the same constant Q-curvature.Moreover, similar results can be shown also for suitable non-constant prescribed Q-curvature functions.