Potential: \(V(x)=-\alpha\,\delta(x)\), \(\alpha>0\). Bound state: \(E<0\).

For \(x\ne0\) (\(V=0\)): \[\frac{d^2\psi}{dx^2}-\gamma^2\psi=0,\qquad \gamma=\sqrt{\frac{-2mE}{\hbar^2}}.\]

Finite solutions: \(\psi=Ae^{\gamma x}\) (\(x\le0\)), \(\psi=Be^{-\gamma x}\) (\(x\ge0\)); continuity at \(0\) \(\Rightarrow A=B\)