Math 6410 • ODE

Consider the differential operator acting on \(L^2(\R)\),
\[
  L=-\frac{d^2}{dx^2},\qquad 0\le x<\infty
\]
with self-adjoint boundary conditions \(\psi(0)/\psi'(0)=\tan\theta\) for some fixed
angle \(\theta\).
\begin{enumerate}
\item[(a)] Show that when \(\tan\theta<0\) there is a single negative eigenvalue with
a normalizable eigenfunction \(\psi_0(x)\) localized near the origin, but none when
\(\tan\theta>0\).
\item[(b)] Show that there is a continuum of eigenvalues \(\lambda=k^2\) with
eigenfunctions \(\psi_k(x)=\sin(kx+\eta(k))\), where the phase shift \(\eta\) is found
from
\[
  e^{i\eta(k)}=\frac{1+ik\tan\theta}{\sqrt{1+k^2\tan^2\theta}}.
\]
\item[(c)] Evaluate the integral
\[
  I(x,x')=\frac{2}{\pi}\int_0^\infty
  \sin(kx+\eta(k))\sin(kx'+\eta(k))\dd k,
\]
and interpret the result with regards the relationship to the Dirac Delta function
and completeness, that is,
\(\delta(x-x')-I(x,x')=\psi_0(x)\psi_0(x')\). You will need the following standard
integral
\[
  \int_{-\infty}^{\infty} e^{ikx}\frac{1}{1+k^2t^2}\frac{\dd k}{2\pi}
  =\frac{1}{2\abs{t}}e^{-\abs{x/t}}.
\]
\end{enumerate}

HINT: you should monitor how the bound state contribution (for \(\tan\theta<0\))
switches on and off as \(\theta\) is varied. Keeping track of the modulus signs
\(\abs{\cdots}\) in the standard integral is crucial for this.