Periodic elliptic operators with asymptotically preassigned spectrum

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 $\{{\mathbf{A}}^{\epsilon }\in {\mathbf{L}}_{\mathrm{per}}{\}}_{\epsilon }$ such that the spectrum of ${\mathbf{A}}^{\epsilon }$ has exactly $m$ gaps in $[0,L]$ when $\epsilon$ is small enough, and these gaps tend to the intervals $({\alpha }_j,{\beta }_j)$ as $\epsilon \to 0$. The idea how to construct the family ${\mathbf{A}\e}_\eps$ is based on methods of the homogenization theory.

28 pages, 2 figures