  The formulation of the eigenvalue problem for the Schr\"odinger equation is
studied, for the numerical solution a new approach is applied. With the usual
exponentially rising free-state asymptotical behavior, and also with a first
order correction to it, the lower half of the $k$ plane is systematically
explored. Note that no other method, including the complex rotation one, is
suitable for calculating virtual bound states far from the origin. Various
phenomena are studied about how the bound and the virtual bound states are
organized into a system. Even for short range potentials, the free state
asymptotical condition proves to be inadequate at some distance from the real
$k$ axis; the first order correction doesn't solve this problem.
  Therefore, the structure of the space spanned by the mathematically possible
solutions is studied and the notion of minimal solutions is introduced. Such
solutions provide the natural boundary condition for bound states, their
analytical continuation to the lower half plane does the same for resonances
and virtual bound states. The eigenvalues coincide with the poles of the
S-matrix. The possibility of continuation based on numerical data is also
demonstrated. The proposed scheme is suitable for the definition of the in- and
outgoing solutions for positive energies; therefore, also for the calculation
of the so called distorted S-matrix in case of long range interactions.
