j + k \text{ odd.} \label{eq:orthoevenodd}$$ These observations lead to information on complex eigenvalues of $L = L^0 + W$.

\[prop:imagw\] Let the potential $$\begin{aligned}
w(x) \in L^p, \, \, 1 \leq p < \infty, &\quad \nu = {\left\Vert w \right\Vert}_p, \text{ or} \label{eq:wtype1}\\
w(x) = \sum_{k = 1}^{\infty} c_k {\delta \left( x - b_k \right) }, &\quad \nu = \sum {\left\vert c_k \right\vert} < \infty, \label{eq:wtype2}\end{aligned}$$ be real and odd. Then the operator $$L = L^0 + i W = - \frac{d^2}{dx^2} + x^2 + i w \label{eq:newell}$$ has at most finitely many non-real eigenvalues, if any, and their number does not exceed $N^*$, where

$$\begin{aligned}
N^* &= D\left (\nu \log (1 + \nu) \right)^{2p}, && p > 2 \label{eq:newnstar.plarge}\\
N^* &= D^* \left (\nu \log^2 (1 + \nu) \right)^{4}, && p = 2 \label{eq:newnstar.pmid}\\
N^* &= D \left (\nu \log (1 + \nu) \right)^{\frac{3}{\left(1 - \frac{1}{2p} \right)}}, && 1 \leq p < 2 \label{eq:newnstar.psmall}\\
N^* &= D_*\left( \nu \log (1 + \nu) \right)^{6}, && \text{ in the case \eqref{eq:wtype2}} \label{eq:newnstar.2}\end{aligned}$$

\[eq:newnstar\]

and $D^*$, $D_*$ are absolute constants although $D = D(p)$ does not depend on the norm $\nu$.

By and estimates in Corollary  and in we can use 