  During the last two decades or so much effort has been devoted to the
discussion of quantum mechanics (QM) that in some way incorporates the notion
of a minimum length. This upsurge of research has been prompted by the modified
uncertainty relation brought about in the framework of string theory. In
general, the implementation of minimum length in QM can be done either by
modification of position and momentum operators or by restriction of their
domains. In the former case we have the so called soccer-ball problem when the
naive classical limit appears to be drastically different from the usual one.
Starting with the latter possibility, an alternative approach was suggested in
the form of a band-limited QM. However, applying momentum cutoff to the
wave-function, one faces the problem of incompatibility with the
Schr\"{o}dinger equation. One can overcome this problem in a natural fashion by
appropriately modifying Schr\"{o}dinger equation. But incompatibility takes
place for boundary conditions as well. Such wave-function cannot have any more
a finite support in the coordinate space as it simply follows from the
Paley-Wiener theorem. Treating, for instance, the simplest quantum-mechanical
problem of a particle in an infinite potential well, one can no longer impose
box boundary conditions. In such cases, further modification of the theory is
in order. We propose a non-local modification of QM, which has close ties to
the band-limited QM, but does not require a hard momentum cutoff. In the
framework of this model, one can easily work out the corrections to various
processes and discuss further the semi-classical limit of the theory.
