For an unknown potential , the Schrödinger equation
with boundary conditions
and initial condition
gives rise to a response operator (or, in control terms, input-output map)
for each control .
The problem considered here is to determine from . If denotes the solution, for a given control , at time and , it is shown that the ‘connection’ operator
where , satisfies The spectral data of the operator given by
can be found from (and hence from ) by the usual variational principle. Finally, the unknown potential is determined by using the boundary controllability of the associated wave equation. This is based on the fact that the above spectral data completely determine the connecting operator for the associated wave equation.