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The experimental techniques have evolved to a stage where various examples ofnanostructures with non-trivial shapes have been synthesized, turning thedynamics of a constrained particle and the link with geometry into a realisticand important topic of research. Some decades ago, a formalism to deduce ameaningful Hamiltonian for the confinement was devised, showing that ageometry-induced potential (GIP) acts upon the dynamics. In this work we studythe problem of prescribed GIP for curves and surfaces in Euclidean space$\mathbb{R}^3$, i.e., how to find a curved region with a potential given {\it apriori}. The problem for curves is easily solved by integrating Frenetequations, while the problem for surfaces involves a non-linear 2nd orderpartial differential equation (PDE). Here, we explore the GIP for surfacesinvariant by a 1-parameter group of isometries of $\mathbb{R}^3$, which turnsthe PDE into an ordinary differential equation (ODE) and leads to cylindrical,revolution, and helicoidal surfaces. Helicoidal surfaces are particularlyimportant, since they are natural candidates to establish a link betweenchirality and the GIP. Finally, for the family of helicoidal minimal surfaces,we prove the existence of geometry-induced bound and localized states and thepossibility of controlling the change in the distribution of the probabilitydensity when the surface is subjected to an extra charge.

Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential