Non-real eigenvalues of indefinite Sturm–Liouville problems

The present paper deals with non-real eigenvalues of regular indefinite Sturm–Liouville problems. A priori bounds and sufficient conditions of the existence for non-real eigenvalues are obtained under mild integrable conditions of coefficients.     

Non-real Zeros of Derivatives of Real Entire Functions and the Pólya-Wiman Conjectures

A function f is in the class $ V_2p $ iff $ f(z) = e^{-az^{2p+2}}g(z) $ where a S 0 and g is a constant multiple of a real entire function of genus h 2 p + 1 with only real zeros. The class $ U_2p $ is defined as follows: $ U_0 = V_0 $ , $ U_{2p} = V_{2p}-V_{2p-2} $ . Functions in the class $ U_{2p}^{*} $ are represented as $ g(z) = c(z)f(z) $ where $ f\in U_{2p} $ and c is a real polynomial with no real zeros. Every real entire function g , of finite order with at most finitely many non-real zeros satisfies $ g\in U_{2p}^{*} $ for a unique p . We show the exact number of non-real zeros of f" , for $ f\in U_{2p} $ , in terms of the number of non-real zeros of f' and a geometrical condition on the components of Im Q ( z ) > 0, where $ \displaystyle Q(z) = z-({f(z)}/{f'(z)}) $ . Further, for a subclass of $ f\in U_{2p} $ , we show necessary and sufficient conditions for f" to have exactly 2 p non-real zeros. For a subclass of $ U_{2p}^{*} $ we show that if f' has only real zeros, then f" has exactly 2 p non-real zeros. For $ f\in U_{2p}^{*} $ we show that 2 p is a lower bound for the number of non-real zeros of $ f^{(k)} $ for k S 2.     

Bounds on Non-Real Eigenvalues of Indefinite Sturm-Liouville Problems

It is known since the early 20th century that regular indefinite Sturm-Liouville problems may possess non-real eigenvalues. However, finding bounds for this set in terms of the coefficients of the differential expression has remained an open problem until recently. In this note we prove a variant of a recent result in [1] on the bounds for the non-real eigenvalues of an indefinite Sturm-Liouville problem with Dirichlet boundary conditions. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

ON THE TITCHMARSH-WEYL THEORY OF ORDINARY SYMMETRIC ODD-ORDER DIFFERENTIAL EXPRESSIONS AND A DIRECT CONVERGENCE THEOREM

Abstract

In this paper the odd-order differential equation M[y] λ wy on the interval (O,∞), associated with the symmetric differential expression M of (2k-1)st order (k ≥ 2) with w a positive weight function and λ a complex number, is shown to possess k-Titchmarsh-Weyl solutions for every non-real λ in the underlying Hilbert space L² _w(O, ∞) having identical representation for every non-real λ. In terms of these solutions the Green's function associated with the singular boundary value problem is shown to possess identical representation for all non-real λ which has been further made use of in the third-order case to establish a direct convergence eigenfunction expansion theorem. The symmetric spectral matrix appearing in the expansion theorem has been characterized in terms of the Titchmarsh-Weyl m-coefficients.     

Non-real zeros of linear differential polynomials

Let f be a real entire function with finitely many non-real zeros, not of the form f = Ph with P a polynomial and h in the Laguerre-Pólya class. Lower bounds are given for the number of non-real zeros of f″ + ω f, where ω is a positive real constant.     

Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the nth derivative tends to infinity, as . We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.     

On Normal Subgroups of Coxeter Groups Generated by Standard Parabolic Subgroups

We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots. Partially supported by a KBN grant 2 P03A 017 25     

The octonionic eigenvalue problem

We discuss the eigenvalue problem for 2×2 and 3×3 octonionic Hermitian matrices. In both cases, we give the general solution for real eigenvalues, and we show there are also solutions with non-real eigenvalues.     

Bounds on the Non-real Spectrum of a Singular Indefinite Sturm-Liouville Operator on ℝ

A simple explicit bound on the absolute values of the non-real eigenvalues of a singular indefinite Sturm-Liouville operator on the real line with the weight function sgn(·) and an integrable, continuous potential q is obtained. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Propagation of singularities for non-real pseudo-differential operators

The purpose of this work is to prove a theorem of propagation of singularities for a class of non-real pseudo-differential operator with multiple characteristics. The main tools are L² estimates on the time-dependent Schrödinger equation related toP. We extend here the results of [6]; we improve the results announced by the second author in [7]. The second part of this work consists in an extension of the result of [5] to complex-valued symbols.     

Spectral properties of primal-based penalty preconditioners for saddle point problems

For large and sparse saddle point linear systems, this paper gives further spectral properties of the primal-based penalty preconditioners introduced in [C.R. Dohrmann, R.B. Lehoucq, A primal-based penalty preconditioner for elliptic saddle point systems, SIAM J. Numer. Anal. 44 (2006) 270-282]. The regions containing the real and non-real eigenvalues of the preconditioned matrix are obtained. The model of the Stokes problem is supplemented to illustrate the theoretical results and to test the quality of the primal-based penalty preconditioner.     

Non-real zeros of higher derivatives of real entire functions of infinite order

Letf be a real meromorphic function of infinite order in the plane such thatf has finitely many poles. Then for eachk≥3, at least one off andf ^(k) has infinitely many non-real zeros. Together with a result of Edwards and Hellerstein, this establishes the analogue for higher derivatives of a conjecture going back to Wiman around 1911.     

Second Order Linear Differential Polynomials and Real Meromorphic Functions

The main result determines all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non-real poles, while f′′ + a ₁ f′ + a ₀ f has finitely many non-real zeros, where a ₁ and a ₀ are real rational functions which satisfy a ₁(∞) = 0 and a ₀(x) ≥ 0 for all real x with |x| sufficiently large. This is accomplished by refining some earlier results on the zeros in a neighbourhood of infinity of meromorphic functions and second order linear differential polynomials. Examples are provided illustrating the results.     

Bounds on the non-real spectrum of differential operators with indefinite weights

Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and $\infty $ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm–Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.     

The damped wave equation with unbounded damping

We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.     

Geometrical solutions of some quadratic equations with non-real roots

This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax ² + bx + c = 0, a p 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,b,c ] R, the set of real numbers, are presented. Case II deals with the geometrical solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when b ] R, the set of real numbers; and a,c ] C, the set of complex numbers. Finally, the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,c ] R, the set of real numbers, and b ] C, the set of complex numbers, are presented in case III.     

Liouville-type theorems for some complex hypoelliptic operators

It is shown that the natural analogue of Liouville's theorem holds for the well-known hypoelliptic operators _α (for ¦α¦ < n) introduced and studied by Folland and Stein on the Heisenberg group H_n. Since these operators are non-real for α ≠ 0, the usual methods of potential theory fail and are replaced by an explicit use of the fundamental solution.     

Witt equivalence of algebraic function fields over real closed fields

We examine the conditions for two algebraic function fields over real closed fields to be Witt equivalent. We show that there are only two Witt classes of algebraic function fields with a fixed real closed field of constants: real and non-real ones. The first of them splits further into subclasses corresponding to the tame equivalence. This condition has a natural interpretation in terms of both: orderings (the associated Harrison isomorphism maps 1-pt fans onto 1-pt fans), and geometry and topology of associated real curves (the bijection of points is a homeomorphism and these two curves have the same number of semi-algebraically connected components). Finally, we derive some immediate consequences of those theorems. In particular we describe all the Witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field. Received: 16 February 2000; in final form: 7 December 2000 / Published online: 18 January 2002