Band Gap of the Schrödinger Operator with a Strong δ-Interaction on a Periodic Curve

In this paper we study the operator \( H_\beta = - \Delta - \beta\delta(\cdot - \Gamma) \) in¶\( L^2(\mathbb{R}^2) \), where \( \Gamma \) is a smooth periodic curve in \( \mathbb{R}^2 \). We obtain the asymptotic form of the band spectrum of \( H_\beta \) as \( \beta \) tends to infinity. Furthermore, we prove the existence of the band gap of \( \sigma(H_\beta) \) for sufficiently large \( \beta > 0 \). Finally, we also derive the spectral behaviour for \( \beta \rightarrow \infty \) in the case when \( \Gamma \) is non-periodic and asymptotically straight.

Authors and Affiliations

1. Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 25068 Řež,¶ Czech Republic, , , , , , CZ

   P. Exner

2. Graduate School of Mathematics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan,¶ e-mail: yositomi@math.kyushu-u.ac.jp, , , , , , JP

   K. Yoshitomi

Submitted 23/06/01, accepted 18/08/01

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