Quantum knots

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?).