This work extends monotonicity-based methods in inverse problems
to the case of the Helmholtz (or stationary Schrödinger) equation
in a bounded domain for fixed
nonresonance frequency
and real-valued scattering coefficient function
.
We show a monotonicity relation between the scattering coefficient
and
the local Neumann-to-Dirichlet operator that holds up to finitely many eigenvalues.
Combining this with the method of localized potentials, or Runge approximation,
adapted to the case where finitely many constraints are present, we derive a
constructive monotonicity-based characterization of scatterers from partial boundary
data. We also obtain the local uniqueness result that two coefficient functions
and
can be
distinguished by partial boundary data if there is a neighborhood of the boundary part
where
and
.
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