by Point Interactions
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Introduction
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The operator $$L^0 = - \, \frac{d^2}{dx^2} + x^2, \quad x \in {\mathbb{R}}^1,$$ is the one-dimensional harmonic oscillator; this is an unbounded self-adjoint operator acting in $L^2({\mathbb{R}})$. As one can see in any introductory book on quantum mechanics, $L^0$ has a discrete spectrum $\Lambda^0 = {\left\lbrace z_n \right\rbrace}_{n = 0}^{\infty}$, $$z_n = 2n + 1, \quad n = 0, 1, \dotsc$$ and a compact resolvent $$\label{eq:firstresolve}
R^0(z) = (z - L^0)^{-1} , \quad z \not\in \Lambda^0.$$ A normalized orthogonal system of eigenfunctions can be chosen as the Hermite functions $$\label{eq:hefcnintro}
h_n(x) = \left( \pi^{1/2} 2^n n! \right)^{-1/2} e^{-x^2/2} H_n(x), \quad n = 0, 1, \dotsc$$ where $$\label{eq:hepolyintro}
H_n(x) = e^{x^2/2} \left(e^{-x^2/2}\right)^{(n)}$$ are Hermite polynomials.

Spectral analysis of perturbed operators $$\label{eq:elldecompintro}
L = L^0 + W$$ with special $W$, in particular, the point interaction peturbations $$\label{eq:wformintro}
Wf = wf, \quad w(x) = \sum_{j = 1}^J c_j {\delta \left( x - b_j \right) }, \quad J \text{ finite}$$ was studied in many mathematical and physical papers.

S. Fassari, F. Rinaldi and G. Inglese series of papers [@FassIng94], [@FassIng97], [@FassRin] investigate the spectrum of $L \in$ when the perturbation $$\label{eq:wspecificintro}
W = - \tau \left( {\delta \left( 