= Solution
For a negative energy write $E=-\hbar^2\lambda^2/(2M)$, $\lambda>0$. Away from the delta interaction the decaying solutions are $Ce^{-\lambda x}$ for $x>0$ and $Ce^{\lambda x}$ for $x<0$, with the same $C$ by <continuity>. The derivative jump is $-2\lambda C=-C/\Delta$, giving $\lambda=1/(2\Delta)$. Thus the <bound state in a delta potential> has
$$
\boxed{E=-\frac{\hbar^2}{8M\Delta^2},\qquad\chi(x)=\frac1{\sqrt{2\Delta}}e^{-|x|/(2\Delta)}}.
$$
The normalization follows from $\int_{\mathbb R}e^{-2\lambda|x|}dx=1/\lambda$. This is a genuine square-integrable negative-energy state.
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