Short scale asymptotics in nonlocal spectral problems

We study the asymptotic behavior of eigenvalues of nonlocal convolution-type operators $$L_{\epsilon }\rho =-\frac{1}{{\epsilon }^d}{\int }_{\mathrm{\Omega }}J\left(\frac{\mathrm{\mid }x-y{\mathrm{\mid }}^2}{{\epsilon }^2}\right)\rho (y)dy+a(x)\rho (x),$$ (with a continuous sufficiently fast decaying at infinity function $J\ge 0$) when the scaling parameter $\epsilon >0$ tends to zero. Assuming that the minimum of the function $a(x)$ is attained at a point $x_0\in \bar{\mathrm{\Omega }}\subset {\mathbb{R}}^d$ and is non-degenerate, we establish several terms in the asymptotic expansion of the first eigenvalues ${\lambda }_{\epsilon }^{(k)}$ of $L_{\epsilon }$. If $x_0$ is an interior point of the domain $\mathrm{\Omega }$, we show that ${\lambda }_{\epsilon }^{(k)}=a(x_0)-1+\epsilon {\mu }^{(k)}+\bar{o}(\epsilon )$, where ${\mu }^{(k)}$ are eigenvalues of a harmonic oscillator (obtained by the blow up analysis), the corresponding eigenfunctions of ${\mathcal{L}}_{\epsilon }$ are concentrated near $x_0$ on the scale ${\epsilon }^{1\mathrm{/}2}$. If $x_0\in \mathrm{\partial }\mathrm{\Omega }$ then the asymptotic expansion of the eigenvalues of $L_{\epsilon }$ has the form ${\lambda }_{\epsilon }^{(k)}=a(x_0)-1+{\epsilon }^{2\mathrm{/}3}{\mathrm{\Lambda }}_0+\epsilon ({\mu }^{(k)}+\alpha p_0)+\bar{o}(\epsilon )$. The corresponding eigenfunctions are represented as the product of a function depending on the distance to $\mathrm{\partial }\mathrm{\Omega }$ and concentrated on the scale ${\epsilon }^{2\mathrm{/}3}$, and functions localized on the scale ${\epsilon }^{1\mathrm{/}2}$ and asymptotically independent of the distance to $\mathrm{\partial }\mathrm{\Omega }$. The latter functions converge (after rescaling) to eigenfunctions of a harmonic oscillator (in tangential variables), ${\mu }^{(k)}$ being their eigenvalues. The former function is given, to the leading term, by the scaled Airy function, and ${\mathrm{\Lambda }}_0$, $\alpha$ and $p_0$ depend on $J(\mathrm{\mid }x{\mathrm{\mid }}^2)$ and $a(x)$.