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

Abstract

We study the large time asymptotics of solutions u(x, t) of the wave equation with time-harmonic force density f(x)e-iωt, ω⩾0, in the semi-strip Ω= (0, ∞)×(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 x2=0 and x2=1, and (∂/∂x1)u=αu for x1=0, with given α, −π⩽α<∞. Let Dα be the self-adjoint realization of −Δ in Ω with this boundary condition. For −π⩽α<0, Dα has eigenvalues λj=π2j2−α2, j=1, 2, … For j⩾2 these eigenvalues are embedded in the continuous spectrum of Dα, σc(Dα)=[π2, ∞]. For α⩾0, Dα has no eigenvalues. We consider the asymptotic behaviour of u(x, t), t→∞, as a function of α. In the case α=0 resonances of order √t at ω=πj, j=1, 2, …, were found in References 5 and 10. We prove that for α=−π there is a resonance of order t2 for ω=0 and resonances of order t for every ω>0 (note that 0 is an eigenvalue of D-π). Moreover, for −π<α<0 there are resonances of order t at ω=√λj. The resonance frequencies are continuous functions of α for −π<α<0 and tend to πj, j=1, 2, … as α goes to zero. On the contrary in the case α>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)=(Dα−κ2)−1 as a meromorphic function of κ into an appropriate Riemann surface. We find that as α crosses zero the real poles of R(κ) 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 Schrödinger 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 κ2. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.