Generalized Kronig-Penney Hamiltonians (Q1269061)

The article is devoted to study the class of periodic models with point interactions for one-dimensional case. Generalized point interactions are obtained on the base of the new class of selfadjoint extensions of operator \(A=-\Delta\mid C_0^\infty(\mathbb{R} \setminus \{0\}).\) These interactions have been introduced in \textit{P. Chernoff} and \textit{R. Hughes} [J. Funct. Anal. 111, No. 1, 97-117 (1993; Zbl 0790.47039)]. The \(\delta\)- and \(\delta'\)-potentials are included as special cases. In the article, these results are extended to the periodic case. Consideration is analogous to the classical Kronig-Penney model. In particular, the direct integral decomposition can be implemented and generalized Kronig-Penney relation can be obtained. The models under consideration have purely absolutely continuous spectrum. Several examples are examined in detail.