Semi-classical mass asymptotics on stationary spacetimes

Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface and a curvature function F, which is either the mean curvature, the root of the second symmetric polynomial or a curvature function of class (K*). We consider curvature flows with curvature function F and a volume preserving term and prove long time existence of the flow and exponential convergence of the graphs in the C-infinity topology to a hypersurface of constant F-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces.

We study focal points and Maslov index of a horizontal geodesic γ : I → M in the total space of a semi-Riemannian submersion π : M → B by determining an explicit relation with the corresponding objects along the projected geodesic \({\pi\circ\gamma:I\to B}\) in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.

It is well known that the conformable and the symmetric differential operators have formulas in terms of the first derivative. In this document, we combine the two definitions to get the symmetric conformable derivative operator (SCDO). The purpose of this effort is to provide a study of SCDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to impose two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SCDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function and convolution structures. Moreover, by using the SCDO, we introduce a generalized class of Briot–Bouquet differential equations to introduce, what is called the symmetric conformable Briot–Bouquet differential equations. We shall show that the upper bound of this class is symmetric in the open unit disk.

Orthogonal beamforming is the name given to certain high-resolution methods for estimating the spectra of a wave field received by an array of sensors. These methods use the eigenvalues and eigenvectors of the spectral matrix of the sensor outputs. The problem is to predict the behaviour of such methods when only an estimate of the matrix is known. The sensor outputs may consist of sensor noise, ambient noise and noise from a finite set of discrete sources. The properties of the eigensystem of the spectral matrix in the case of weak ambient noise motivate the methods of orthogonal beamforming, for example Pisarenko's nonlinear peak estimates and the projection estimates of Owsley and Liggett. If the spectral matrix is estimated by one of the classical methods, some asymptotic distributional properties of the matrix estimate and its eigensystem are well known. They can be used to determine asymptotic expressions, for example for the first and second moments of the peak estimators, and to approximate the distributions. The parameters, however, cannot be calculated in applications since the eigensystem of the exact spectral matrix is required. Therefore we recently developed bounds for the deviation of the peak estimates which only use limited knowledge about the matrix. We applied some results on perturbations of Hermitian operators. The asymptotic behaviour of the bounds for the projection estimator is investigated, and possibilities for their estimation are indicated. Finally, we report on extensive simulations with random matrices to evaluate the new bounds. As a result, we found that the projection estimator behaves stably and that there are tight bounds if the eigenvalues of interest are sufficiently separated from the rest.

The theoretical part of this monograph examines the distribution of the spectrum of operator polynomials, focusing on quadratic operator polynomials with discrete spectra. The second part is devoted to applications. Standard spectral problems in Hilbert spaces are of the form A- I for an operator A, and self-adjoint operators are of particular interest and importance, both theoretically and in terms of applications. A characteristic feature of self-adjoint operators is that their spectra are real, and many spectral problems in theoretical physics and engineering can be described by using them. However, a large class of problems, in particular vibration problems with boundary conditions depending on the spectral parameter, are represented by operator polynomials that are quadratic in the eigenvalue parameter and whose coefficients are self-adjoint operators. The spectra of such operator polynomials are in general no more real, but still exhibit certain patterns. The distribution of these spectra is the main focus of the present volume. For some classes of quadratic operator polynomials, inverse problems are also considered. The connection between the spectra of such quadratic operator polynomials and generalized Hermite-Biehler functions is discussed in detail. Many applications are thoroughly investigated, such as the Regge problem and damped vibrations of smooth strings, Stieltjes strings, beams, star graphs of strings and quantum graphs. Some chapters summarize advanced background material, which is supplemented with detailed proofs. With regard to the reader s background knowledge, only the basic properties of operators in Hilbert spaces and well-known results from complex analysis are assumed

On eigenfunction expansions of differential equations with degenerating weight

The function , where is Lerch's transcendent, satisfies the following two-dimensional formally self- adjoint second-order hyperbolic differential equation: where . The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space , where . We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of . We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.

then f(x) = ce-x. In ?2 a new proof is given using Liouville's theorem. So far as I am aware, the method of proof does not apply to any essentially more general problems, depending as it does on unique properties of the analytic function ez. However, I have obtained a few similar results by other methods, and to establish a sufficiently broad framework to contain all cases, it is convenient to pose a somewhat indefinite general problem: PROBLEM A. Let C1 be a linear space of functions; let T be an operator on Ci to C1; and let .| be a norm defined on C1. When does fEC1 and II TkfII _1, k=O, 1, 2, ,imply that fC C2 where C2 is a finite-dimensional subspace of C1? A narrower framework will also be considered: PROBLEM B. Let T be a differential operator and let g be a function such that Tg -g (or +g). When does I Tkf| <gi, k=O, 1, 2, imply f = cg? A very simple result of type A is obtained in ?3 by taking T to be a self-adjoint operator on a Hilbert space. Differential operators are natural choices for T, and some special cases are spelled out using Fourier and Hermite expansions. A corollary of type B is given. In ?4, I consider the possibility of generalizing results for particular operators T to results for conjugate operators of the form STS-1. For Problem B a natural choice for S is an operator which acts in effect to change the variables of the differential equation Tf = -f. By employing this device, results are obtained for example for T=pl(x)D +po(x) and for T=AD2+BD+C using respectively Tagamlitzki's theorem and a Fourier expansion.

We discuss a problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators outlined in [1]. We describe one of the possible ways of constructing in terms of the closure of an initial symmetric operator associated with a given differential expression and deficient spaces. Particular attention is focused on the features peculiar to differential operators, among them on the notion of natural domain and the representation of asymmetry forms generated by adjoint operators in terms of boundary forms. Main assertions are illustrated in detail by simple examples of quantum-mechanical operators like the momentum or Hamiltonian.

Let A be a self-adjoint operator, let (α,β) be a gap in the spectrum of A, and let B=A+V, where, in general, the perturbation operator V is unbounded. We establish some abstract conditions under which the spectrum of B in (α,β) is discrete; does not accumulate to β; is finite. An estimate of the number of eigenvalues of B in (α,β) is obtained. Bibliography: 3 titles.

Under certain conditions on the coefficients of symmetric singular differential operators of order 2n, selfadjoint extensions are shown to have a discrete spectrum. The results are proven specifically for the Friedrichs extension.

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H1 whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula A≥E, B(x)≥0, and the operator B(x) is bounded for all x. An operator L0 is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) eH1 according to the formula: L0y=−y″ + Q(x)y (−∞<x<∞). It is proved that the closure of the operator L0 is a self-adjoint operator in H1 under the given assumptions.

A continuum of unusual self-adjoint linear partial differential operators

Relative form boundedness and compactness for a second-order differential operator

Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces