Mathematical Theory of Scattering Resonances

Attractors for Two‐Dimensional Waves with Homogeneous Hamiltonians of Degree 0

“…Hence, there is no choice for a self-adjoint extension. This follows from lemma E.45 in the book [11]. On a closed manifold, for k D 1 and s D 0, this lemma says the following: LEMMA B.1.…”

“…The first part works as follows: first, 1 .D/.H n D / H n D and hence A 1 .D/.H n D / H n : We can assume that W F 0 .A/ is included in the cone fd > 0g, so that there exists a smooth function identically equal to 1 on W F 0 .A/, and compactly supported in fd > 0g. We can thus define a pseudo-differential partition of unity .Id …/ C … with Id … smoothing near W F 0 .A/, i.e., such that W F 0 .… Id/ \ W F 0 .A/ D ¿ (see Proposition E.30 in [11]). Then, we define an elliptic self-adjoint pseudodifferential operator E of degree 1 by E WD …D… C .Id …/F .Id …/; with F globally elliptic (having a positive principal symbol), self-adjoint of degree 1.…”

“…In order to prove the third assertion, we use the following "division lemma" (see Proposition E.32 in the book [11]) : LEMMA C.2. If W F 0 .A/ Ell.E/, there exists a pseudo-differential operator Q so that A D QE modulo smoothing operators.…”

Stabilization Rates for the Damped Wave Equation with Hölder-Regular Damping

“…Take s ∈ C ∞ (T * S 1 ) with s = 0 on |ξ| < 1/2 and s = 1 on |ξ| > 1 and set S = Op(s), by a standard semiclassical elliptic estimate (c.f. Theorem E.32 [DZ19])…”

Generic nature of asymptotic completeness in dissipative scattering theory

“…In a first step, we use that resonances are generically simple. Namely, adapting the proof of [7,Theorem 3.14] in a straightforward way (see also [17]), one can show that there existsC 0 ∈ L ∞ Ω such that C 0 − C 0 ∞ ≤ ε 0 /2 and all spectral singularities of H V,C 0 are at most of order 1. Now, let λ 1 , .…”