Periodic elliptic operators with asymptotically preassigned spectrum

TL;DR

This paper constructs periodic elliptic operators with spectra that can be asymptotically preassigned to have specific gaps, using homogenization techniques, which advances control over spectral properties in mathematical physics.

Contribution

It introduces a method to design periodic elliptic operators with spectra that approximate any desired set of spectral gaps, expanding the tools for spectral engineering.

Findings

Operators with exactly m spectral gaps in [0,L] can be constructed.

Spectral gaps of these operators tend to preassigned intervals as epsilon approaches zero.

The construction relies on homogenization methods.

Abstract

We deal with operators in $\mathbb{R}^n$ of the form $$\mathbf{A}=-{1\over \mathbf{b}(x)}\sum\limits_{k=1}^n\ds{\partial\over\partial x_k}(\mathbf{a}(x){\partial \over\partial x_k})$$ where $\mathbf{a}(x),\mathbf{b}(x)$ are positive, bounded and periodic functions. We denote by $\mathbf{L}_{\mathrm{per}}$ the set of such operators. The main result of this work is as follows: for an arbitrary $L>0$ and for arbitrary pairwise disjoint intervals $(\alpha_j,\beta_j)\subset[0,L]$, $j=1,...,m$ ($m\in\mathbb{N}$) we construct the family of operators $\{\mathbf{A}^\varepsilon\in \mathbf{L}_{\mathrm{per}}\}_{\varepsilon}$ such that the spectrum of $\mathbf{A}^\varepsilon$ has exactly $m$ gaps in $[0,L]$ when $\varepsilon$ is small enough, and these gaps tend to the intervals $(\alpha_j,\beta_j)$ as $\varepsilon\to 0$. The idea how to construct the family ${\mathbf{A}\e}_\eps$ is based on methods of the homogenization theory.