Now we go to detailed analysis of these operators.

Two-point interaction potentials {#sec:2pt}
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 {#subsect:genpointpot}

Now we apply general constructions of Sections \[sec:prelims\], \[sec:evalperturb\] to the case of the two-point interaction potentials $$w(x) = c^+ {\delta \left( x - b \right) } + c^- {\delta \left( x + b \right) }, \quad b > 0 \label{eq:pointmass1}$$ and particular cases of an odd potential $$\begin{aligned}
s v^o(x), &\quad s \in {\mathbb{C}}\quad \text{ where }&& v^o(x) = {\delta \left( x - b \right) } - {\delta \left( x + b \right) } \label{eq:pointodd}\\
\intertext{and an even potential}
t v^e(x), &\quad t \in {\mathbb{C}}&& v^e(x) = {\delta \left( x - b \right) } + {\delta \left( x + b \right) } \label{eq:pointeven}\end{aligned}$$ — see [@HCW] and [@FassRin].

Of course, for any *odd* potential or , not just for $v^0 \in$ , the matrix elements $w_{jk}$ have the property . Indeed, with parity $$\begin{aligned}
w_{jk} &= {\left\langle w(x) h_j(x),\,  h_k(x) \right\rangle} = \label{eq:woddprop1}\\
&=  {\left\langle w(-x) h_j(-x),\,  h_k(-x) \right\rangle} = -(-1)^{j + k} w_{jk}\label{eq:woddprop2}\end{aligned}$$ so $$w_{jk} = 0 \text{ if } j + k \text{ even.} \label{eq:orthooddeven} $$ If, however, $w$ in or is *even* then we conclude $$w_{jk} = 0 \text{ if } 