P. Werner

Resonance phenomena in cylindrical waveguides

In the following we study aperiodic wave processes, generated by periodic forces, in infinite tubes. Consider the n-dimensional cylinder Q = Sz’ x ( co, co), where Q’ is a bounded (n 1 )-dimensional domain with smooth boundary XT. Sections 24 of this paper are devoted to the initial and boundary value problem (af-A)u=fe-i’~~’ in Q x (0, a), (1.1) u=o on an, (1.2) 44 0) = u,(x), a,t+, 0) = U,(X) for xE52, (1.3) with prescribed data f, u0 and u1 belonging to C;(Q). We are mainly interested in the asymptotic behavior of the solution u(x, t) as t -+ co. Our results are of the following type: If w2 does not coincide with one of the eigenvalues A,, A*,... of the Dirichlet problem (a:+ ... +a:p,)v+w=o in Q’, v=o on a521 (1.4)

A distribution-theoretical approach to certain Lebesgue and Sobolev spaces☆

Abstract The main object of this paper is to discuss some basic operations in the Hilbert space L2 and in related Sobolev spaces by using only concepts of distribution theory without referring to Lebesgue integration. Special emphasis will be put on those aspects of the theory which seem to be relevant for applications to partial differential equations and spectral analysis of differential operators.

Resonance phenomena in local perturbations of parallel-plane waveguides

We study initial and boundary value problems for the wave equation ∂ t 2 u - Δu = fe -iωt with Dirichlet or Neumann boundary data in smooth domains Ω, which coincide with Ω o = R 2 x (0,1) outside a sufficiently large sphere. The concept of a standing wave, introduced in [7], seems to be of special relevance. A standing wave of frequency ω is defined as a non-trivial solution U of the equation ΔU + ω 2 U = 0 in Ω, which satisfies on ∂Ω the prescribed boundary condition U = 0 or ∂U/∂n = 0, respectively, and a suitable condition at infinity. For instance, U(x) = sin πkx 3 is a standing wave of frequency πk in the unperturbed domain Ω o with Dirichlet boundary data. As shown in a series of joint papers with K. Morgenrother, u(x,t) is bounded as t → ∞ if Ω does not admit standing waves of frequency ω. The main purpose of the present paper is the proof of the converse statement: If standing waves of frequency ω exist in Ω, then u(x,t) is unbounded as t → ∞ for suitably chosen f ∈ C 0 ∞(Ω). Thus the appearance of resonances is closely related to the presence of standing waves. The leading term of the asymptotic expansion of u(x, t) as t → ∞ will be specified. In particular, it turns out that the resonance rate is either t or In t. The rate t appears if and only if ω 2 is an eigenvalue of the spatial operator, while the rate In t can only occur if ω = πk. In the case of Neumann boundary data, U = 1 is a standing wave of frequency 0 for every local perturbation Ω of Ω 0 . The corresponding resonance has already been studied in [21].

Einige Ausnahmen zur Rosserschen Regel in der Theorie der Riemannschen Zetafunktion

ZusammenfassungEs wird eine Methode beschrieben die es ermöglicht, mit relativ geringem numerischen Aufwand Ausnahmen zur Rosserschen Regel für die Riemannsche Zetafunktion zu finden. Vor kurzem von R. P. Brent durchgeführte systematische Rechnungen zeigen, daß durch unsere Methode im gemeinsam untersuchten Bereich alle Ausnahmen erfaßt werden.AbstractWe describe a method which enables us to find failures of Rossers rule with moderate numerical effort. Systematic, computations recently performed by R. P. Brent show that our method yields all failures in the intersection of the intervals considered by him and by us.

The behaviour of real resonances under perturbation in a semi-strip

We study the large time asymptotics of solutions u(x,t) of the wave equation with time-harmonic force density f(x)e -lcot , ω ≥ 0, in the semi-strip Ω = (0, ∞) x (0,1) for a given f ∈ C 0 ∞(Ω). We assume that u satisfies the initial condition u = (∂/∂t) u = 0 for t = 0 and the boundary conditions u = 0 for x 2 = 0 and x 2 = 1, and (∂/∂x 1 ) u = αu for x 1 = 0, with given α, - π ≤ α 0 (note that 0 is an eigenvalue of D -π ). Moreover, for - π 0 there are no real resonances in the sense that the solution remains bounded in time as t → ∞. Actually in this case, the limit amplitude principle is valid for all frequencies ω ≥ 0. This rather striking behaviour of the resonances is explained in terms of the extension of the resolvent R(κ) = (D α - κ 2 ) -1 as a meromorphic function of K into an appropriate Riemann surface. We find that as α crosses zero the real poles of R(K) associated with the eigenvalues remain real, but go into a second sheet of the Riemann surface. This behaviour under perturbation is rather different from the case of complex resonances which has been extensively studied in the theory of many-body Schrodinger operators where the (real) eigenvalues embedded in the continuous spectrum turn under a small perturbation into complex poles of the meromorphic extension of the resolvent, as a function of the spectral parameter K 2 .