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We consider the singular semibounded self-adjoint Schrödinger operator acting in $L_2 ([0,+\infty))$ formally defined as

$H=-\frac{\,d^2}{\,dx^2}+\sum_{k=1}^{+\infty} a_k \delta_{x_k},\\ D(H)=\left\{u \in W_2^2([0,+\infty)\setminus \{x_k, k \in \mathbb{N}\}) \cap C([0,+\infty)) \ : \ u(0)=0 \right\},$

where $a_k \in \mathbb{R}$, $x_k$ is an increasing sequence of positive real numbers and $\delta_{y}$ denotes the Dirac delta function supported at $y$.

We prove constructively tha for every closed semi-bounded set $S \subset \mathbb{R}$ one can always choose the values of parameters $a_k$ and $x_k$ such that the essential spectrum of the operator $H$ coincides with the set $S$.

We will also show the same approach can also be applied to the operator in $L_2 ([0, +\infty))$ of the form

$L=-\frac{\,d^2}{\,dx^2}+\sum_{k=1}^{+\infty}a_k \chi_{[x_{k-1}, x_k]},\\ D(L)=\left\{u \in W_2^2 ([0,+\infty)) \ : \ u(0)=0 \right\},$

(where $\chi_A$ stands for the characteristic function of the set $A$).

In both cases the boundary condition at $0$ can be choosen Neumann instead of Dirichlet without affecting the main result.