Spectral asymptotics for the Schrödinger operator on the line with spreading and oscillating potentials

Duchêne, Vincent; Raymond, Nicolas

doi:10.48550/arxiv.1609.01990

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2018

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Cited by 2 publications

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“…Recently, Duchêne-Raymond [DR16] obtained homogenization results for potentials of the form ε −β V ε , β ∈ (0, 2), in dimension 1, using a normal form approach. Dimassi [Di16] applied ε-semiclassical calculus to show a trace formula and a Weyl law for potentials of the form ε −2 V ε , in any dimension d.…”

Section: Introductionmentioning

confidence: 99%

Bound States for Rapidly Oscillatory Schrödinger Operators in Dimension 2

Drouot¹

2018

SIAM J. Math. Anal.

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Section: Introductionmentioning

confidence: 99%

Bound States for Rapidly Oscillatory Schrödinger Operators in Dimension 2

Drouot¹

2018

SIAM J. Math. Anal.

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“…On a somewhat unrelated note, Duchêne-Raymond [DR16] obtained homogenization results for large HOPs in dimension one. Dimassi [Di16] and Dimassi-Duong [DD17] used the effective Hamiltonian method of Gérard-Martinez-Sjöstrand [GMS91] to count resonances and eigenvalues of semiclassical rescaled HOPs in any dimension d. They obtained a nice Weyl law in the semiclassical limit, related to papers of Klopp [Kl12,Kl16] and Phong [Ph15a,Ph15b].…”

Section: Introductionmentioning

confidence: 99%

Resonances for random highly oscillatory potentials

Drouot

2018

Journal of Mathematical Physics

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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).

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Spectral asymptotics for the Schrödinger operator on the line with spreading and oscillating potentials

Cited by 2 publications

References 23 publications

Bound States for Rapidly Oscillatory Schrödinger Operators in Dimension 2

Bound States for Rapidly Oscillatory Schrödinger Operators in Dimension 2

Resonances for random highly oscillatory potentials

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