Manfred Möller

Zeros of the Hypergeometric Polynomials F(-n, b; -2n; z)

We investigate the location of the zeros of the hypergeometric polynomial F(-n, b; -2n; z) for b real. The Hilbert-Klein formulas are used to specify the number of real zeros in the intervals (-~, 0), (0, 1), or (1, ~). For b>0 we obtain the equation of the Cassini curve which the zeros of w^nF(-n, b; -2n; 1/w) approach as n->~ and thereby prove a special case of a conjecture made by Marti@?nez-Finkelshtein, Marti@?nez-Gonzalez, and Orive. We also present some numerical evidence linking the zeros of F with more general Cassini curves.

Linearization of boundary eigenvalue problems

It is shown that certain eigenvalue problems for ordinary differential operators with boundary conditions depending holomorphically on the eigenvalue parameter γ can be linearized by making use of the theory of operator colligations. As examples, first order systems with boundary conditions depending polynomially on γ and Sturm-Liouville problems with γ-holomorphic boundary conditions are considered.

The Essential Spectrum of a System of Singular Ordinary Differential Operators of Mixed Order. Part II: The Generalization of Kako's Problem

A system of ordinary differential equations of mixed order on an interval (0, r0) is considered, where some coefficients are singular at 0. Special cases have been dealt with by Kako, where the essential spectrum of an operator associated with a linearized MHD model was calculated, and more recently by Hardt, Mennicken and Naboko. In both papers this operator is a selfadjoint extension of an operator on sufficiently smooth functions. The approach in the present paper is different in that a suitable operator associated with the given system of ordinary differential equations is explicitly defined as the closure of an operator defined on sufficiently smooth functions. This closed operator can be written as a sum of a selfadjoint operator and a bounded operator. It is shown that its essential spectrum is a nonempty compact subset of ℂ, and formulas for the calculation of the essential spectrum in terms of the coefficients are given.

Spectral Properties of a Fourth Order Differential Equation

The eigenvalue problem y(4)(λ, x) − (gy′)′(λ, x) = λ2y(λ, x) with boundary conditions y(λ, 0) = 0, y′′(λ, 0) = 0, y(λ, a) = 0, y′′(λ, a) + iαλy′(λ, a) = 0 is considered, where g ∈ C1[0, a] and α > 0. It is shown that the eigenvalues lie in the closed upper half-plane and on the negative imaginary axis. A formula for the asymptotic distribution of the eigenvalues is given and the location of the pure imaginary spectrum is investigated.

Spectral theory of operator pencils, Hermite-Biehler functions, and their applications

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

Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions

We consider the eigenvalue problem y (4)(λ,x) − (gy′)′(λ,x) = λ 2 y(λ,x) with separated boundary conditions B j (λ)y = 0 for j = 1,…,4, where g ∈ C 1[0, a] is a real valued function, B j (λ)y = y [p j ](a j ) or B j (λ)y = y [pj](a j ) + iϵ j αλy [qj ] (aj ), aj = 0 for j = 1, 2 and a j = a for j = 3, 4, α > 0, ϵ j ∈ {−1, 1}. We will associate to the above eigenvalue problem a quadratic operator pencil L(λ) = λ 2 M − iαλK − A in the space , where and are bounded self-adjoint operators and k is the number of boundary conditions which depend on λ. We give necessary and sufficient conditions for the operator A to be self-adjoint.

Quadratic and cubic transformations and zeros of hypergeometric polynomials

In this paper we consider the location of the zeros of the hypergeometric polynomials that lie in either the quadratic or the cubic class, where each of these classes is determined by a necessary and sufficient condition due to Kummer. We show that the zeros of most polynomials in these classes can be specified by simple applications of the results proved in recent papers of Driver and Duren.

On the essential spectrum of the linearized Navier-Stokes operator

We determine the essential spectrum of the linearized Navier-Stokes operator with physical boundary conditions. In contrast to other approaches we do not make use of pseudo-differential operators. We establish a direct proof using only some fundamental results for matrix operators.

On the Essential Spectrum of a Differentially Rotating Star

Natural oscillations of a differentially rotating star are governed by the linearized Euler equations. Separation of variables leads to a family L k,0 (k ∈ Z) of mixed order partial differential operators. It is shown that for k ¬= 0 their closures L k have nonempty essential spectrum. Indeed, it is shown that the essential spectrum of L k coincides with the essential spectrum of a bounded operator. Some parts of the essential spectrum are calculated explicitly. It is still an open problem if there are more points in the essential spectrum.

ADJOINTS AND FORMAL ADJOINTS OF MATRICES OF UNBOUNDED OPERATORS

In this paper we discuss diverse aspects of the mutual relationship between adjoints and formal adjoints of unbounded operators bearing a matrix structure. We emphasize the behaviour of row and column operators as they turn out to be the germs of an arbitrary matrix operator, providing most of the information about the latter as it is the troublemaker.