The spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions

We consider the operator $L = - (d/dx)^2 + x^2 y + w(x) y , y \in L^2(\mathbb{R})$, where $w(x) = s [ δ(x - b) - δ(x + b)], b \neq 0,$ real, $s \in \mathbb{C}$. This operator has a discrete spectrum: eventually the eigenvalues are simple and $λ_n = (2n + 1) + s^2 (κ(n) / n) + ρ(n)$, where $κ(n) = \frac{1}{2π} [(-1)^{n + 1} \sin ( 2 b \sqrt{2n} ) - \frac{1}{2} \sin ( 4 b \sqrt{2n} ) ]$ and $|ρ(n) | \leq C (\log n) / (n^{3/2})$ If $s = i γ$, $γ$ real, the number $T(γ)$ of non-real eigenvalues is finite, and $T(γ) \leq [ C (1 + | γ|) \log (e + | γ|)]^2.$ The analogue of the above equations is given in the case of any two-point interaction perturbation $w(x) = c_+ δ(x - b) + c_- δ(x + b), c_+, c_- \in \mathbb{C}.$

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