Boundary triples for Schr\"odinger operators with singular interactions on hypersurfaces

J. Behrndt ; M. Langer ; V. Lotoreichik

Nanosistemy: fizika, himiâ, matematika, Tome 7 (2016) no. 2, pp. 290-302

Résumé

The self-adjoint Schrödinger operator $A_{\delta,\alpha}$ with a $\delta$-interaction of constant strength $\alpha$ supported on a compact smooth hypersurface $\mathcal{C}$ is viewed as a self-adjoint extension of a natural underlying symmetric operator $S$ in $L^2(\mathbb{R}^n)$. The aim of this note is to construct a boundary triple for $S^*$ and a self-adjoint parameter $\Theta_{\delta,\alpha}$ in the boundary space $L^2(\mathcal{C})$ such that $A_{\delta,\alpha}$ corresponds to the boundary condition induced by $\Theta_{\delta,\alpha}$. As a consequence, the well-developed theory of boundary triples and their Weyl functions can be applied. This leads, in particular, to a Krein-type resolvent formula and a description of the spectrum of $A_{\delta,\alpha}$ in terms of the Weyl function and $\Theta_{\delta,\alpha}$.

Détail

Keywords: Boundary triple, Weyl function, Schrödinger operator, singular potential, $\delta$-interaction
Mots-clés : hypersurface.