具有渐近预分配谱的周期椭圆算子
Periodic elliptic operators with asymptotically preassigned spectrum
摘要：
我们处理的运营商在$\mathbb{R}^n$的形式$$\mathbf{A}=-{1\over \mathbf{b}(x)}\sum\limits_{k=1}^n\ds{\partial\over\partial x_k}(\mathbf{a}(x){\partial \over\partial x_k})$$其中$\mathbf{a}(x),\mathbf{b}(x)$是积极的，有界的和周期的功能。我们用$\mathbf{L}_{\mathrm{per}}$表示这样的算子的集合。本文的主要结果如下：对于任意的$L>0$和任意的两两不相交区间$(α_j,β_j)\subset[0,L]$，$j=1,...,m$（$m\in\mathbb{N}$），我们构造了算子族$\{\mathbf{A}^\varepsilon\in \mathbf{L}_{\mathrm{per}}\}_{\varepsilon}$，使得当$\varepsilon$足够小时，$\mathbf{A}^\varepsilon$的谱在$[0,L]$中恰好有$m$间隙，并且这些间隙趋向于$(α_j,β_j)$作为$\varepsilon\to 0$ .构造${\mathbf{A}\e}_\eps$族的思想是基于均匀化理论的方法。
英文摘要：
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 $(α_j,β_j)\subset[0,L]$, $j=1,...,m$ ($m\in\mathbb{N}$) 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 $(α_j,β_j)$ as $\varepsilon\to 0$. The idea how to construct the family ${\mathbf{A}\e}_\eps$ is based on methods of the homogenization theory.
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