The set \(E= [0,\infty )\setminus \bigcup_{j\geq 1} (a_j, b_j)\) with \(\sum_{j\geq 1} (b_j- a_j) 0\) \(\forall \lambda\in E\) \(\forall \delta>0\) \(|(\lambda- \delta, \lambda+ \delta)\cap E|\geq \varepsilon \delta\). \(\mathbb{Q} (E)\) denotes the class of all bounded real potentials \(q\) continuous on \(\mathbb{R}\), such that the spectrum of the Sturm-Liouville operator \({\mathcal L}[q]=- {d^2\over dx^2}+ q\) in \({\mathcal L}^2 (\mathbb{R})\) coincides with \(E\) and the corresponding Weyl functions \(m_\pm (z)\) \((\text{Im } z\neq 0)\) satisfy the condition \(m_+ (\lambda+ i0)= m_- (\lambda- i0)\) for a.e. \(\lambda\in E\). With \(\Omega= \mathbb{C}\setminus E\), \(\pi (\Omega)\) denotes the discrete fundamental group of \(\Omega\), \(\pi^* (\Omega)\) is the compact abelian group of characters of \(\pi (\Omega)\). The main theorem: there exists a homeomorphism between the compacts \(\mathbb{Q} (E)\) and \(\pi^* (\Omega)\) conjugating the shift of the potential \(q(x) \mapsto q(x+ t)\) and the motion \(\alpha\mapsto \alpha+ \delta t\) on \(\pi^* (\Omega)\), where \(\delta\) is a fixed character. Therefore every \(q\in \mathbb{Q} (E)\) is a uniform almost periodic function. With \(E^{(N)}= [0, \infty)\setminus \bigcup^N_{j=1} (a_j, b_j)\), \(N=1, 2, \dots\), the approximation of \(\mathbb{Q}(E)\) by \(\mathbb{Q} (E^{(N)})\) is investigated.