h_k \right\Vert}_{2q} &\sim k^{-\frac{1}{2} \left( \frac{1}{2} -\frac{1}{2q} \right)} = k^{-\frac{1}{4p}} , && p > 2.\label{eq:hefcnactlargep}\\
\intertext{For $p = 2$ we have $2q = 4$ and}
{\left\Vert h_k \right\Vert}_{4} &\sim k^{-\frac{1}{8} } \log k, && p = 2. \label{eq:hefcnactmidp}\\
\intertext{Finally, if $1 \leq p < 2$ then $2q > 4$ so  }
{\left\Vert h_k \right\Vert}_{2q} &\sim k^{-\frac{1}{6} \left( \frac{1}{2q} + \frac{1}{2} \right)} = k^{-\frac{1}{12} \left(2 - \frac{1}{p} \right)}, && 1 \leq p <2 & \qquad \label{eq:hefcnactsmallp}\end{aligned}$$ All these estimates are used in Proposition \[prop:imagw\] , — see Section \[subsect:genpointpot\] below.

Of course, shows that $\delta$-potentials $$w(x) = \sum_{k = 1}^m c_k \delta (x - b_k), \quad m \text{ finite,} \label{eq:wdelt}$$ are good for us as well; in this case, $$\begin{aligned}
{\left\langle W h_j, h_i \right\rangle} &= \sum_{k = 1}^m c_k h_j(b_k) h_i(b_l) \label{eq:wpairs}
\intertext{and}
{\left\vert W_{ji} \right\vert} &\leq CM j^{-1/12} i^{-1/12}, \quad i, j \geq 1, \quad M = \sum_{k = 1}^m \vert c_k \vert. \label{eq:wpairbds}\end{aligned}$$ But with more information on asymptotics of Hermite polynomials and Hermite functions we can be accurate in analysis of point-interaction potentials and spectra of operators $L^0 + W$, $L^0 \in $ , or — equivalently — . This is the main goal of this paper. 