Unstable points of the spectrum of a periodic difference operator
Let T be an integral lattice in \({\mathbb{Z}}^ m\), \(m\geq 2\), and let \(\Gamma\) be its dual lattice. By the symbol \(F_ T({\mathbb{Z}}^ m)={\mathbb{Z}}^ m/T\) and \(F_{\Gamma}({\mathbb{R}}^ m)={\mathbb{R}}^ m/\Gamma\) we denote the fundamental set of the lattice T in \({\mathbb{Z}}^ m\) and of the lattice \(\Gamma\) in \({\mathbb{R}}^ m\), respectively. Let \(U_{\tau}: {\mathfrak l}_ 2({\mathbb{Z}}^ m)\to {\mathfrak l}_ 2({\mathbb{Z}}^ m)\) be the unitary operator defined by the formula \((U_{\tau}f)(n)=f(n+\tau)\), \(\tau\in T\), \(f\in {\mathfrak l}_ 2({\mathbb{Z}}^ m).\) A bounded operator H: \({\mathfrak l}_ 2({\mathbb{Z}}^ m)\to {\mathfrak l}_ 2({\mathbb{Z}}^ m)\) is said to be periodic with lattice of periods T if for any vector \(\tau\), \(\tau\in T\), we have the relation \(U_{\tau}H=HU_{\tau}\). In this paper we consider in \(\ell_ 2({\mathbb{Z}}^ m)\) the operator defined by the formula \(H=-\Delta +Q\), where Q is an operator of small norm, while \(\Delta\) is the Laplace difference operator, i.e. \[ (\Delta u)(n)=\sum_{\ell:| n-\ell | =1}(u(\ell)-u(n)),\quad u(n)\in \ell_ 2({\mathbb{Z}}^ m). \]
unstable points of the spectrum of a periodic difference operator
scientific article; zbMATH DE number 4192471