THE BEHAVIOR OF REAL RESONANCES UNDER PERTURBATION IN A SEMI-STRIP

We study the large time asymptotics of solutions u(x, t) of the wave e quation with time-harmonic force density f(x)e(-i infinity t), omega g reater than or equal to 0, in the semi-strip Omega = (0, infinity) x ( 0, 1) for a given f is an element of C-0(infinity) (Omega). We assume that u satisfies the initial condition u = (partial derivative/partial derivative t) u = 0 for t = 0 and the boundary conditions u = 0 for x (2) = 0 and x(2) = 1, and (partial derivative/partial derivative x(1)) u = alpha u for x(1) = 0, with given alpha, - pi less than or equal t o alpha < infinity. Let D-alpha be the self-adjoint realization of - D elta in Omega with this boundary condition. For - pi less than or equa l to alpha < 0, D-alpha has eigenvalues lambda(j) = pi(2)j(2) - alpha( 2), j = 1, 2, ... For j greater than or equal to 2 these eigenvalues a re embedded in the continuous spectrum of D-alpha, sigma(c) (D-alpha) = [pi(2), infinity). For alpha greater than or equal to 0, D-alpha has no eigenvalues. We consider the asymptotic behaviour of u(x, t), t -- > infinity, as a function of alpha. In the case alpha = 0 resonances o f order root t at omega = pi j, j = 1, 2, ..., were found in Reference s 5 and 10. We prove that for alpha = - pi there is a resonance of ord er t(2) for omega = 0 and resonances of order t for every omega > 0 (n ote that 0 is an eigenvalue of D-pi). Moreover, for - pi < alpha < 0 t here are resonances of order t at omega = root lambda(j). The resonanc e frequencies are continuous functions of alpha for - pi < alpha < 0 a nd tend to pi j, j = 1, 2, ... as alpha goes to zero. On the contrary in the case alpha > 0 there are no real resonances in the sense that t he solution remains bounded in time as t --> infinity. Actually in thi s case, the limit amplitude principle is valid for all frequencies ome ga greater than or equal to 0. This rather striking behaviour of the r esonances is explained in terms of the extension of the resolvent R(ka ppa) = (D-alpha - kappa(2))(-1) as a meromorphic function of kappa int o an appropriate Riemann surface. We find that as a crosses zero the r eal poles of R(kappa) associated with the eigenvalues remain real, but go into a second sheet of the Riemann surface. This behaviour under p erturbation is rather different from the case of complex resonances wh ich has been extensively studied in the theory of many-body Schrodinge r operators where the (real) eigenvalues embedded in the continuous sp ectrum turn under a small perturbation into complex poles of the merom orphic extension of the resolvent, as a function of the spectral param eter kappa(2). (C) 1998 by B. G. Teubner Stuttgart-John Wiley & Sons L td.