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

Abstract

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 − Δ M . 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 ( α j , β j ) ⊂ [ 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 − Δ M , moreover the edges of the first m gaps belong to δ-neighbourhoods of the edges of the intervals ( α j , β j ) , while the remaining gaps (if any) are located outside the interval [ 0 , L ] .