  Inspired by the generalized uncertainty principle (GUP), which adds
gravitational effects to the standard description of quantum uncertainty, we
extend the exact uncertainty principle (EUP) approach by Hall and Reginatto [J.
Phys. A: Math. Gen. (2002) 35 3289], and obtain a (quasi)nonlinear
Schr\"odinger equation. This quantum evolution equation of unusual form, enjoys
several desired properties like separation of non-interacting subsystems or
planewave solutions for free particles. Starting with the harmonic oscillator
example, we show that every solution of this equation respects the
gravitationally-induced minimal position uncertainty proportional to the Planck
length. Quite surprisingly, our result successfully merges the core of
classical physics with non-relativistic quantum mechanics in its extremal form.
We predict that the commonly accepted phenomenon, namely a modification of a
free-particle dispersion relation due to quantum gravity might not occur in
reality.
