The Schrodinger operator with delta potential $H$ is defined as follows:
$$\begin{cases} D(H) & = \{ u \in H^1 (\mathbb{R}) \cap H^2 (\mathbb{R} \setminus \{0\}); u'(0 +) - u'(0 -) = 2 q u (0)\},\\ Hu &= - \frac{1}{2} \frac{d^2}{dx^2} u, \hspace{1cm} u \in D(H). \end{cases}$$
Find the solution $f_{\pm}(x,\xi)$ for the following value problem:
$$\begin{cases} & Hf_{\pm} = \frac{1}{2} \xi^2 f_{\pm},\hspace{1cm} x \in \mathbb{R}, \xi \in \mathbb{R}\\ &\lim_{x \to \pm \infty}(f_{\pm}(x,\xi)) - e^{\pm i x \xi} ) =0. \end{cases}$$
I am studying this very new subject for me and I come cross this problem while I am studying the nonlinear Schrodinger PDE with $\delta$- potential, namely $V(x) = q \delta(x)$ where $\delta$ is Dirac function and $q$ is real number.
My try
I would the general solution for the IVP as
$$f_{\pm} (x,\xi) = A e^{\pm i x \xi} + B e^{\mp i x \xi }$$
I was not able to show that $f_{\pm}$ belongs to $D(H)$. Could you please enlighten my about how to solve this problem