Homogenization of a one-dimensional fourth-order periodic operator with a singular potential

A. A. Raev ; V. A. Sloushch ; T. A. Suslina

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 53, Tome 521 (2023), pp. 212-239

Résumé

In $L_2(\mathbb{R})$, we consider a fourth-order differential operator $B_{\varepsilon}$ of the form $B_{\varepsilon} = \frac{d^4}{dx^4} + \varepsilon^{-4} V({x}/\varepsilon)$, where $V(x)$ is a real-valued $1$-periodic function belonging to $L_{2, \operatorname{loc}}(\mathbb{R})$, and $\varepsilon >0$ is a small parameter. It is assumed that the point $\lambda_0 =0$ is the lower edge of the spectrum of the operator $B = \frac{d^4}{dx^4} + V({x})$ and the first band function $E_1(k)$ of the operator $B$ on the period $k \in [-\pi, \pi)$ reaches a minimum at exactly two points $\pm k_0$, $0 k_0 \pi$, and behaves like $g^{(1)}(k \mp k_0)^2$, $g^{(1)} >0$, near these points. The behavior of the resolvent $(B_{\varepsilon} + I)^{-1}$ for small $\varepsilon$ is studied. We obtain approximation for this resolvent in the operator norm with an error $O(\varepsilon^2)$. The approximation is described in terms of the spectral characteristics of the operator $B$ at the bottom of the spectrum.