Short scale asymptotics in nonlocal spectral problems
Abstract
We study the asymptotic behavior of eigenvalues of nonlocal convolution-type operators Lερ=−1/εd∫ΩJ((|x−y|2)/ε2)ρ(y)dy+a(x)ρ(x), (with a continuous sufficiently fast decaying at infinity function J≥ 0) when the scaling parameter ε>0 tends to zero. Assuming that the minimum of the function a(x) is attained at a point x0∈ Ω̅⊂ ℝd and is non-degenerate, we establish several terms in the asymptotic expansion of the first eigenvalues λε(k) of Lε. If x0 is an interior point of the domain Ω, we show that λε(k)=a(x0)−1+εμ(k)+o̅(ε), where μ(k) are eigenvalues of a harmonic oscillator (obtained by the blow up analysis), the corresponding eigenfunctions of ℒε are concentrated near x0 on the scale ε1/2. If x0∈∂ Ω then the asymptotic expansion of the eigenvalues of Lε has the form λε(k)=a(x0)−1+ε2/3Λ0+ε(μ(k)+αp0)+o̅(ε). The corresponding eigenfunctions are represented as the product of a function depending on the distance to ∂Ω and concentrated on the scale ε2/3, and functions localized on the scale ε1/2 and asymptotically independent of the distance to ∂Ω. The latter functions converge (after rescaling) to eigenfunctions of a harmonic oscillator (in tangential variables), μ(k) being their eigenvalues. The former function is given, to the leading term, by the scaled Airy function, and Λ0, α and p0 depend on J(|x|2) and a(x).
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BibTeX
@misc{rybalko2026short,
title = {{Short scale asymptotics in nonlocal spectral problems}},
author = {Volodymyr Rybalko and Antonina Rybalko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13513},
eprint = {2610.13513},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13513},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Short scale asymptotics in nonlocal spectral problems
AU - Volodymyr Rybalko
AU - Antonina Rybalko
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13513
UR - https://arxiv.org/abs/2610.13513
AB - We study the asymptotic behavior of eigenvalues of nonlocal convolution-type operators L_ερ=−1/ε^d∫_ΩJ((|x−y|^2)/ε^2)ρ(y)dy+a(x)ρ(x), (with a continuous sufficiently fast decaying at infinity function J≥ 0) when the scaling parameter ε>0 tends to zero. Assuming that the minimum of the function a(x) is attained at a point x_0∈ Ω̅⊂ ℝ^d and is non-degenerate, we establish several terms in the asymptotic expansion of the first eigenvalues λ_ε^(k) of L_ε. If x_0 is an interior point of the domain Ω, we show that λ_ε^(k)=a(x_0)−1+εμ^(k)+o̅(ε), where μ^(k) are eigenvalues of a harmonic oscillator (obtained by the blow up analysis), the corresponding eigenfunctions of ℒ_ε are concentrated near x_0 on the scale ε^1/2. If x_0∈∂ Ω then the asymptotic expansion of the eigenvalues of L_ε has the form λ_ε^(k)=a(x_0)−1+ε^2/3Λ_0+ε(μ^(k)+αp_0)+o̅(ε). The corresponding eigenfunctions are represented as the product of a function depending on the distance to ∂Ω and concentrated on the scale ε^2/3, and functions localized on the scale ε^1/2 and asymptotically independent of the distance to ∂Ω. The latter functions converge (after rescaling) to eigenfunctions of a harmonic oscillator (in tangential variables), μ^(k) being their eigenvalues. The former function is given, to the leading term, by the scaled Airy function, and Λ_0, α and p_0 depend on J(|x|^2) and a(x).
ER -