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

TL;DR

This paper develops a method to construct periodic non-compact Riemannian manifolds with a prescribed number of spectral gaps in the Laplace-Beltrami operator's spectrum, controlling the location of these gaps' edges.

Contribution

It introduces a novel approach to design periodic manifolds with spectral gaps having edges near specified intervals, extending prior work by allowing precise control over gap edges.

Findings

Constructed manifolds with at least m spectral gaps.

Controlled the edges of the first m gaps near specified intervals.

Gaps can be placed outside a large interval [0, L].

Abstract

It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\in\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator $-\Delta_M$. 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 $(\a_j,\b_j)\subset[0,\infty)$, $j=1,...,m$ ($m\in\mathbb{N}$), for an arbitrarily small $\delta>0$ 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 $-\Delta_{M}$, moreover the edges of the first $m$ gaps belong to $\delta$-neighbourhoods of the edges of the intervals $(\a_j,\b_j)$, while the remaining gaps (if any) are located outside the interval $[0,L]$.