R^0 + \sum_{j = 1}^{\infty} U_j, \\
U_0 = R^0 , \quad U_k = R^0 W U_{k - 1} = U_{k - 1} W R^0, \quad k \geq 0.\end{gathered}$$

Moreover, we essentially use the property of perturbations $W \in $ to have such a matrix $$w_{jk} = \left\langle W h_j, j_k \right\rangle, \quad j, k = 0, 1, \dotsc$$ that for some $\alpha > 0$ there exists $M > 0$ such that $$\label{eq:wcondintro}
\vert w_{jk} \vert \leq \frac{M}{(1 + j)^{\alpha} (1 + k)^{\alpha}}, \quad j, k = 0, 1, \dotsc ,$$ Detailed results on the spectrum and convergence of spectral decompositions of $L = L^0 + W$ for a general $W$ under the condition were given by B. Mityagin and P. Siegl [@MiSi]. In the case of the finite point interaction perturbations ${\displaystyle}\alpha = \frac{1}{4}$.

Preliminaries, Technical Introduction, Review the Results {#sec:prelims}
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Our main concern is the harmonic oscillator operator and its special perturbation $W$. We will focus on this case, although many constructions are very general and could be performed in analysis of other differential operators — see [@DM06B], [@AM12], [@MiSi], [@Elton03].

Let $L^0$ be an operator in $\ell^2({\mathbb{Z}}_+)$, $$L^0 e_k = z_k e_k, \quad z_k = (2k 