Scattering Resonances of Convex Obstacles for General Boundary Conditions

Abstract

Abstract. We study the distribution of resonances for smooth strictly convex obstacles under general boundary conditions. We show that under a pinched curvature condition for the boundary of the obstacle, the resonances are separated into cubic bands and the distribution in each bands satisfies Weyl's law.

On the multilevel internal structure of the asymptotic distribution of resonances

“…Reading the works on the case of one strictly convex obstacle O, one could guess in [47,48,49] (see also references therein and [30]) a program on the study of the structure of Σ(H O ) in regions adjacent to R. One of the instruments in this program is the counting function in various shaped strips N ϕ (R) = #{k ∈ Σ(H) : −ϕ(| Re k|) ≤ Im k, |k| ≤ R}, (7.1) where the function ϕ : [0, +∞) → R describes the shape of the strip {−ϕ(| Re k|) ≤ Im k ≤ 0} (see [48]). In particular, [48,49] where an additional parameter α > 0 takes into account the possible polynomial growth [41,53,47,49], we can infer from the results of [25,48,49] that, in the case of a strictly convex obstacle O ⊂ R m satisfying additional pinching conditions of [49] on curvatures of the boundary of O, the support of the measure dAd cub m−1 (µ) is separated from 0 and there exists a partition 0 = µ 0 < µ 1 < µ 2 < · · · < µ 2n+1 < µ 2n+2 = +∞ such that supp dAd cub m−1 (µ) ∩ [µ 2j−1 , µ 2j ] = ∅ and supp dAd cub m−1 (µ) ∩ [µ 2j , µ 2j+1 ] = ∅ , j = 0, .…”

“…On the other side, the existence of infinitely many resonances in a certain horizontal strip S and asymptotic sequences in S were studied for the case of two convex obstacles in connection with trapped broken characteristic rays (see [29,18,28] and the reviews in [55,28]). Trapping effects for obstacle scattering have motivated also the investigation of asymptotics of N S H (R) for different types of shaped strips S, including semi-logarithmic strips {−C 1 ln(| Re z| + 1) ≤ Im z ≤ 0} and cubic strips [47,48,49,28,30] and references therein; here C 0 , C 1 , C 2 > 0).…”

“…Reading the works on the case of one strictly convex obstacle O, one could guess in [47,48,49] (see also references therein and [30]) a program on the study of the structure of Σ(H O ) in regions adjacent to R. One of the instruments in this program is the counting function in various shaped strips…”

“…Acknowledgements. The authors are grateful to Vladimir Lotoreichik for a stimulating discussion on the paper [40], to Olaf Post for an interesting discussion of spectral problems on graph-like structures, and to Plamen Stefanov for informing us about the program of [47,48,49,30]. The second author (IK) is grateful to Herbert Koch for the hospitality of the University of Bonn, to Jürgen Prestin for the hospitality of the University of Lübeck, and to Dirk Langemann for the hospitality of TU Braunschweig.…”

A sharp lower bound for a resonance-counting function in even dimensions

“…In either even or odd dimensions, Stefanov [33,Section 4] proved lower bounds on n odd (r) and n −1 (r) proportional to r d under certain trapping assumptions on the geometry of R d \ O. On the other hand, again in either even or odd dimensions, for a class of strictly convex O asymptotics of the number of resonances (of order r d−1 ) in certain regions are known, [18,32]. In odd dimensions asymptotics of the resonance counting function have been proved in the special case of O equal to a ball [34,40].…”