  We formulate a simple condition for reconstructibility of a certain class of
Hamiltonians with real potentials from the knowledge of their complex-valued
eigenfunctions. This may be relevant to the question of preparability of
quantum states raised by W. Lamb in his 1969 paper on operational
interpretation of quantum mechanics. Of particular interest to engineering
applications are: (a) an exotic case of an upside-down harmonic-oscillator-type
potential (a variation of the inverted "Mexican hat" potential) whose
square-integrable complex eigenfunction describes a localized scattering state
similar to the "bound state in the continuum" of von Neumann and Wigner, and
(b) a spatially confined scattering state of a particle moving in an infinite
well with the properly shaped potential bottom.
