Math 6410 • ODE

The time-dependent Schr\"odinger equation can be written
\[
  i u_t=-\Delta u+Vu,
\]
where \(u(x,t)\), \(x\in\R^n\), \(t\in\R\), is the complex-valued ``wave function,'' and
\(V(x)\ge0\) is a given continuous real-valued potential function.
\begin{enumerate}
\item[(a)] Using complex exponential plane waves, describe the \textit{dispersion relation}
for this equation in case \(V\equiv1\). Is this problem dispersive?
\item[(b)] Assuming \(V\) is bounded, and the problem is formulated over the smooth bounded
connected spatial domain \(U\subset\R^n\), with homogeneous Dirichlet boundary conditions,
prove that the solutions to the time dependent problem can be expressed as a discrete
sum involving solutions to the \textit{time-independent} problem
\[
  E_k\phi_k=\Delta\phi_k+V\phi_k,\qquad \text{in }U
\]
where \(0<E_1\le E_2\le\cdots\).
\end{enumerate}