  It is known that besides the usual unitary mappings $\Omega =
1/\Omega^\dagger$ between the equivalent representations of the physical
Hilbert space of Quantum Mechanics (often, Fourier transformations), the
generalized non-unitary maps $\Omega \neq 1/\Omega^\dagger$ can also help to
simplify the analysis. We adapt the standard Dirac's notation and recollect the
Buslaev's and Grecchi's repulsive quartic oscillator Hamiltonian as an example.
Then we propose the whole new class of the models of the similar type,
characterized by a complexification of the path ${\cal C}$ of the (obviously,
not observable!) "coordinates". An exactly solvable potentialless
Schr\"{o}dinger equation is finally chosen for illustration. In it, the
dynamical (i.e., in our example, confining) role of the traditional potentials
$V(x)$ is shown to be taken over by the mere topologically nontrivial shape of
${\cal C}$. Our construction evokes several new open questions in physics
(${\cal PT}-$symmetric wave packets at a single energy?) as well as in
mathematics (a three-Hilbert-space generalized formulation of Quantum
Mechanics?).
