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The behaviour of real resonances under perturbation in a semi‐strip
Author(s) -
Weder Ricardo,
Werner Peter
Publication year - 1998
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19980110)21:1<1::aid-mma927>3.0.co;2-y
Subject(s) - eigenvalues and eigenvectors , mathematics , bounded function , order (exchange) , mathematical analysis , resonance (particle physics) , perturbation (astronomy) , amplitude , mathematical physics , zero (linguistics) , boundary (topology) , boundary value problem , combinatorics , physics , quantum mechanics , economics , linguistics , philosophy , finance
We study the large time asymptotics of solutions u ( x , t ) of the wave equation with time‐harmonic force density f ( x )e ‐iω t , 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 x 2 =0 and x 2 =1, and (∂/∂ x 1 ) u =α u for x 1 =0, with given α, −π⩽α<∞. Let D α be the self‐adjoint realization of −Δ in Ω with this boundary condition. For −π⩽α<0, D α has eigenvalues λ j =π 2 j 2 −α 2 , j =1, 2, … For j ⩾2 these eigenvalues are embedded in the continuous spectrum of D α , σ c ( D α )=[π 2 , ∞]. For α⩾0, D α 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=π j , j =1, 2, …, were found in References 5 and 10. We prove that for α=−π there is a resonance of order t 2 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=√λ j . The resonance frequencies are continuous functions of α for −π<α<0 and tend to π j , 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)=( D α −κ 2 ) −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 κ 2 . © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.