Denk, Robert
0000-0002-5406-4486 Berufsbeschreibung
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On a queer binomial sum
In this paper the binomial sum Sn(r) = ∑m=0n ((n) || (m)) [((-1)m)/(mr+1)] (r > 0, n e N) is investigated. It turns out that the behaviour of this sum if n tends to infinity depends on the parameter r and changes dramatically at the values r=1 and r=2. In particular, for r greater or equal to 2 we obtain an oscillatory behaviour while for r
Hill's equation systems and infinite determinants
In this paper a method is developed to calculate the Floquet exponents of the matrix-valued version of Hill's equation using infinite determinants. It is shown that the Floquet exponents are precisely the zeros of an infinite determinant corresponding to the differential equation. The proof of this result uses the continuity and holomorphy of the infinite determinant.
The Newton polygon and elliptic problems with parameter
In the study of the resolvent of a scalar elliptic operator, say, on a manifold without boundary there is a well-known Agmon-Agranovich-Vishik condition of ellipticity with parameter which guarantees the existence of a ray of minimal growth of the resolvent. The paper is devoted to the investigation of the same problem in the case of systems which are elliptic in the sense of Douglis-Nirenberg. We look for algebraic conditions on the symbol providing the existence of the resolvent set containing a ray on the complex plane. We approach the problem using the Newton polyhedron method. The idea of the method is to study simultaneously all the quasihomogeneous parts of the system obtained by assigning to the spectral parameter various weights, defined by the corresponding Newton polygon. On this way several equivalent necessary and sufficient conditions on the symbol of the system guaranteeing the existence and sharp estimates for the resolvent are found. One of the equivalent conditions can be formulated in the following form: all the upper left minors of the symbol satisfy ellipticity conditions. This subclass of systems elliptic in the sense of Douglis-Nirenberg was introduced by A. Kozhevnikov.
The determinantal method for Hill systems
The Floquet exponents of Hill systems can be described as the zeros of the determinant of an infinite block matrix. Using acceleration of convergence, this approach can be applied numerically. On the other hand, the determinantal method provides the starting values for the calculation of the Floquet exponents as selected eigenvalues of an infinite block matrix.
Weakly smooth nonselfadjoint spectral elliptic boundary problems
The paper is devoted to general elliptic boundary problems (A, B1 , ..., Bm) with a differential operator A of order 2m and general boundary conditions, acting in a bounded domain G of the n-dimensional space. No self-adjointness is assumed. The main goal is to minimize, to some extent, the smoothness assumptions under which the known spectral results are true. The main results concern the asymptotics of the trace of the q-th power of the resolvent, where q>n/2m, in an angle of ellipticity with parameter. For example, for the Dirichlet problem these asymptotics are obtained in the case of bounded and measurable coefficients in A and continuous coefficients in the principal part of A, while the boundary is assumed to belong to the Hölder space C2m-1,1. The asymptotics of the moduli of the eigenvalues are investigated. The last section is devoted to indefinite spectral problems, with a real-valued multiplier changing the sign in front of the spectral parameter.
Convergence improvement for the infinite determinants of Hill systems
Die Floquet-Exponenten der matrizenwertigen Version der endlichen Hillschen Differentialgleichung k¨onnen als Nullstellen einer unendlichen Determinante berechnet werden. In dieser Arbeit wird die Konvergenz dieser Determinante durch Abspaltung geeigneter unendlicher Produkte verbessert. Die Definition dieser Produkte verwendet dabei die Kenntnis des asymptotischen Verhaltens der endlichen Abschnittsdeterminanten. Verschiedene Methoden der Konvergenzverbesserung werden sowohl für den symmetrischen als auch f¨ur den nicht-symmetrischen Fall der endlichen Hillschen Differentialgleichung vorgestellt. Numerische Beispiele belegen, daß diese Methoden zu einer effizienten Berechnung der unendlichen Determinante führen.