Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace–Beltrami operator

It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary m∈N one can construct a periodic non-compact Riemannian manifold M with at least m gaps in the spectrum of the corresponding Laplace–Beltrami operator −ΔM. In this work we want not only to produce a new type of periodic manifolds with spectral gaps but also to control the edges of these gaps. The main result of the paper is as follows: for arbitrary pairwise disjoint intervals (αj,βj)⊂[0,∞), j=1,…,m (m∈N), for an arbitrarily small δ>0 and for an arbitrarily large L>0 we construct a periodic non-compact Riemannian manifold M with at least m gaps in the spectrum of the operator −ΔM, moreover the edges of the first m gaps belong to δ-neighbourhoods of the edges of the intervals (αj,βj), while the remaining gaps (if any) are located outside the interval [0,L].