Reconstruction of the Schrödinger operator with a singular potential on half-line by its prescribed essential spectrum

Singular Schrodinger operators on $L^2([0,+\infty))$ with the potential of the form $\sum_{k=1}^{+\infty}a_k\delta_{x_k}$, where $x_k~{>}~0$ and $a_k~{\in}~\mathbb{R}$, are considered. It is constructively proved that every closed semibounded set $S\subset\mathbb{R}$ can be the essential spectrum of such operator.