Resonances for random highly oscillatory potentials

Abstract

We study discrete spectral quantities associated to Schrödinger operators of the form −∆ R d + V N , d odd. The potential V N models a highly disordered crystal; it varies randomly at scale N −1 1. We use perturbation analysis to obtain almost sure convergence of the eigenvalues and scattering resonances of −∆ R d + V N as N → ∞. We identify a stochastic and a deterministic regime for the speed of convergence. The type of regime depends whether the low frequencies effects due to large deviations overcome the (deterministic) constructive interference between highly oscillatory terms. see for instance the proof of [DZ16, Theorem 2.8].Theorem 1. For any R > 0 such that q 0 has no resonance on ∂D(0, R), there exist C, c > 0 such that with probability 1 − Ce −cN γ ,(1.4)Conversely, if λ ∈ Res(q 0 ) ∩ D(0, R) has multiplicity m λ , then with probability 1 −An application of this theorem concerns local exponential decay for waves scattered by V N . Assume that q 0 and q are real-valued and that Res(q 0 ) is contained in {Im λ < −A} for some A > 0 (this is satisfied for instance if q 0 ≥ 0 and q 0 ≡ 0). Let R > 0 such that for any N , Res(V N ) ∩ {Im λ ≥ −A} ⊂ D(0, R).

Defect modes for dislocated periodic media

“…As a byproduct, we derive full expansions of the eigenpairs in powers of δ. The self-contained proof relies on (a) resolvent estimates for the bulk operators; (b) scattering theory for highly oscillatory potentials [Dr18b,Dr18a,Dr18c]. This work significantly advances the understanding of the topological stability of certain defect states, particularly the bulk-edge correspondence for continuous dislocated systems.…”

Bound States for Rapidly Oscillatory Schrödinger Operators in Dimension 2

“…Motivated by [DVW14] and by Christiansen [Ch06], we used a different approach to study in [Dr15] the resonances and eigenvalues of −∆ R d + W 0 + V ε , in any odd dimension d. When W 0 = 0, we proved that the resonances and eigenvalues of V ε escape all bounded regions as ε → 0 (except the one converging to 0 when d = 1). When W 0 = 0, we showed that the resonances and eigenvalues of W 0 + V ε converge to the one of W 0 , with a complete expansion in powers of ε.…”

“…To prove Theorem 1, we first follow [Si76, §3]: we use a modified Fredholm determinant to reduce the study of eigenvalues of −∆ R 2 + V ε to an equation involving a certain trace. Then, we provide estimates on this trace following some ideas of [Dr15], ending the proof. We mention that the result of Theorem 1 still applies when W is not smooth but satisfies instead the weaker bound…”

Characterization of edge states in perturbed honeycomb structures

“…We are interested in the eigenvalues of P δ [ζ]. We previously studied eigenvalue problems in seemingly different situations [Dr18a,Dr18b,Dr18c] as well as in a one-dimensional analog [DFW18]. The proofs of these results rely on a cyclicity principle: if A and B are two matrices then the non-zero eigenvalues of AB and BA are equal (together with their multiplicity).…”