Consider $LX=-X''$ on $\mathbb{R}^+:=\{x:\, x>0\}$ with boundary condition $X'(0)-\alpha X(0)=0$.
a. Find values $\alpha\in \mathbb{R}$ such that there are eigenfunctions. Find corresponding eigenvalues.
b. Find generalized eigenfunctions and the corresponding continuous spectrum.
Remark.
a. Eigenfunctions must belong to $L^2(\mathbb{R}^+)$, that means $\int_0^\infty |X(x)|^2\,dx <\infty$.
b. Generalized eigenfunctions cannot grow exponentially as $x\to +\infty$ but they do not belong $L^2(\mathbb{R}^+)$.
