Perturbation of the scattering resonances of an open cavity by small particles. Part I: the transverse magnetic polarization case

Perturbations of the scattering resonances of an open cavity by small particles: Part II—the transverse electric polarization case

“…These exceptional resonances are due to the non-Hermitian character of the operator T ω D , see [12,22]. For simplicity and in view of the Jordan-type decomposition of the operator T ω D established in [12], we assume that, for ω near ω 0 , G(x, y; ω) behaves like G(x, y; ω) = Γ m (x, y; ω) + c 1 (ω) h (1) (x; ω)h (1) (y; ω) ω − ω 0 + c 2 (ω) h (2) (x; ω)h (2) (y; ω) (ω − ω 0 ) 2 + R(x, y; ω), (18) for two functions h (1) and h (2) in L 2 (D). Here, the functions ω → c j (ω), j = 1, 2 and ω → R(x, y; ω) are all holomorphic in a neighborhood of ω 0 , and (x, y) → R(x, y; ω) is smooth.…”

“…(v, ∇h (1) ) ω − ω 0 ∇h (1) − c 2 (ω) (v, ∇h (2) ) (ω − ω 0 ) 2 ∇h (2) = 0, or equivalently, v − c 1 (ω) (v, ∇h (1) ) ω − ω 0 L −1 [∇h (1) ] − c 2 (ω) (v, ∇h (2) ) (ω − ω 0 ) 2 L −1 [∇h (2) ] = 0.…”

“…By multiplying the above equations by ∇h (1) and ∇h (2) , respectively, and integrating by parts over D, we obtain the following system of equations: (1) ], ∇h (1) ) ω − ω 0 = c 2 (ω)(v, ∇h (2) ) (L −1 [∇h (2) ], ∇h (1) ) (ω − ω 0 ) 2 , (v, ∇h (2) ) 1 − c 2 (ω) (L −1 [∇h (2) ], ∇h (2) ) (ω − ω 0 ) 2 = c 1 (ω)(v, ∇h (1) ) (L −1 [∇h (1) ], ∇h (2) ) ω − ω 0 .…”

“…In this paper, which is a follow-up of [1], we consider dielectric radiating cavities [13,15,25] and rigorously obtain asymptotic formulas for the shifts in the scattering resonances that are due to a small particle of arbitrary shape. Our formula shows that the perturbations of the scattering resonances are proportional to the polarization tensor of the small particle.…”

“…The new technique introduced in this paper cannot be easily extended to the transverse magnetic case considered in [1] due to the hyper-singular character of the associated volume-integral operator.…”