Bound States for Rapidly Oscillatory Schrödinger Operators in Dimension 2

Drouot, Alexis

doi:10.1137/16m1099352

Cited by 8 publications

(11 citation statements)

References 26 publications

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“…This phenomena was captured first for small perturbations of q 0 in a pioneering work of Simon [Si76], in dimension one and two. It was observed for highly oscillatory perturbations in [BG06,Bo07,DW11,DVW14,Dr15,Dr16a], again in dimension one. When q 0 is real-valued and λ 0 ∈ iR, we can pick f = g in (1.6) and we obtain a refinement of Theorem 3, which in particular implies that eigenvalues might emerge from the edge of the continuous spectrum in dimension one: Corollary 1.2.…”

Section: Introductionmentioning

confidence: 82%

“…This allows to apply the results of §2. Ideas related to [Dr15,Dr16a] quickly yield Theorems 1 and 2 in §4. Theorem 3 requires more attention.…”

Section: Introductionmentioning

confidence: 95%

“…It can be seen as an idealized version of the random case, because the oscillations are perfectly alternated. In this (deterministic) idealized case, the works of Borisov-Gadyl'Shin [BG06], Borisov [Bo07], Duchêne-Weinstein [DW11], Duchêne-Vukićević-Weinstein [DVW14] and ourselves [Dr15,Dr16a] investigate the behavior of scattering resonances of −∆ R d + V N for large N . The present work aims to provide an extension of these works to the random case.…”

Section: Introductionmentioning

confidence: 99%

“…The works of Borisov and Gadyl'shin [BG06, Bo07], Duchêne-Weinstein-Vukićević [DVW14] and ourselves [Dr15,Dr16a] show that the typical distance between resonances of deterministic highly oscillatory potentials and of their weak limit is of order N −2 . This is due to constructive interference between oscillatory terms.…”

Section: Introductionmentioning

confidence: 99%

“…They also obtained precise asymptotic for the transmission coefficient. We developed new techniques in [Dr15,Dr16a] to extend their results to higher dimensions. We obtained a full expansion for eigenvalues and resonances of HOPs, and a refined formula for the effective potential, and logarithmic resonance-free regions when the weak limit vanishes.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 82%

“…This allows to apply the results of §2. Ideas related to [Dr15,Dr16a] quickly yield Theorems 1 and 2 in §4. Theorem 3 requires more attention.…”

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Resonances for random highly oscillatory potentials

Drouot

2018

Journal of Mathematical Physics

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Defect Modes for Dislocated Periodic Media

Drouot

Fefferman

Weinstein

2020

Commun. Math. Phys.

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Dirac Operators and Domain Walls

Lu¹,

Watson²,

Weinstein³

2020

SIAM J. Math. Anal.

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We study the eigenvalue problem for a one-dimensional Dirac operator with a spatially varying "mass" term. It is well-known that when the mass function has the form of a kink, or domain wall, transitioning between strictly positive and strictly negative asymptotic mass, ±κ∞, at ±∞, the Dirac operator has a simple eigenvalue of zero energy (geometric multiplicity equal to one) within a gap in the continuous spectrum, with corresponding exponentially localized zero mode.We consider the eigenvalue problem for the one-dimensional Dirac operator with mass function defined by "glue-ing" together n domain wall-type transitions, assuming that the distance between transitions, 2δ, is sufficiently large, focusing on the illustrative cases n = 2 and 3. When n = 2 we prove that the Dirac operator has two real simple eigenvalues of opposite sign and of order e −2|κ∞|δ . The associated eigenfunctions are, up to L 2 error of order e −2|κ∞|δ , linear combinations of shifted copies of the single domain wall zero mode. For the case n = 3, we prove the Dirac operator has two non-zero simple eigenvalues as in the two domain wall case and a simple eigenvalue at energy zero. The associated eigenfunctions of these eigenvalues can again, up to small error, be expressed as linear combinations of shifted copies of the single domain wall zero mode. When n > 3 no new technical difficulty arises and the result is similar. Our methods are based on a Lyapunov-Schmidt reduction/Schur complement strategy, which maps the Dirac operator eigenvalue problem for eigenstates with near-zero energies to the problem of determining the kernel of an n × n matrix reduction, which depends nonlinearly on the eigenvalue parameter.The class of Dirac operators we consider controls the bifurcation of topologically protected "edge states" from Dirac points (linear band crossings) for classes of Schrödinger operators with domain-wall modulated periodic potentials in one and two space dimensions. The present results may be used to construct a rich class of defect modes in periodic structures modulated by multiple domain walls.

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Bound States for Rapidly Oscillatory Schrödinger Operators in Dimension 2

Abstract: We study the eigenvalues of Schrödinger operatorsWe show that for ε small enough, such operators have a unique negative eigenvalue, that is exponentially close to 0.

Cited by 8 publications

References 26 publications

Resonances for random highly oscillatory potentials

Resonances for random highly oscillatory potentials

Defect Modes for Dislocated Periodic Media

Dirac Operators and Domain Walls

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