Gap control by singular Schrodinger operators in a periodically structured metamaterial

We consider a family ${mathcal{H}^varepsilon}_{varepsilon>0}$ of $varepsilonmathbb{Z}^n$-periodic Schrodinger operators with $delta$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $minmathbb{N}$ surfaces. We show that in the limit when $varepsilonto 0$ and the interactions strengths are appropriately scaled, $mathcal{H}^varepsilon$ has at most $m$ gaps within finite intervals, and moreover, the limiting behavior of the first $m$ gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.