  We investigate recovery of the (Schr\"odinger) potential function from many
boundary measurements at a large wavenumber. By considering such a linearized
form, we obtain a H\"older type stability which is a big improvement over a
logarithmic stability in low wavenumbers. Furthermore we extend the discussion
to the linearized inverse Schr\"odinger potential problem with attenuation,
where an exponential dependence of the attenuation constant is traced in the
stability estimate. Based on the linearized problem, a reconstruction algorithm
is proposed aiming at the recovery of the Fourier modes of the potential
function. By choosing the large wavenumber appropriately, we verify the
efficiency of the proposed algorithm by several numerical examples.
