\(H^ 1\) convergence of Fourier integrals (Q1346445)

The author studies the linear differential operator \({\mathcal L} : D_ \theta : H^ 1 \to H^ 1\), \(\theta \in [0, \pi]\) defined by the differential expression \(ly = - y'' + ky\), \(k > 0\) and by the boundary conditions \(\cos \theta y(0) - \sin \theta y'(0) =0\). The Hilbert space \(H^ 1\) is generated by the inner product \[ \langle y,z \rangle_{H^ 1} = \begin{cases} (\cot \theta) y(0) \overline {z(0)} + \int^ \infty_ 0 (y' \overline z' + ky \overline z) dx \quad&\text{(in case \(\theta \neq 0\))} \\ \int^ \infty_ 0 (y' \overline z' + ky \overline z) dx \quad&\text{(in case \(\theta = 0\))}\end{cases}. \]