Bound states in tubular quantum waveguides with torsion

We consider a quantum particle constrained to move within a tube of central radius d embedded in 3-space, subject to Dirichlet boundary conditions. Taking the central axis of the tube as a reference curve and setting up a locally cylindrical polar coordinate system around this curve, we derive an exact expression for the effective potential introduced by the imposition of curvature and torsion. We then employ a minimax method to ascertain a sufficient requirement for the curvature and torsion to guarantee the existence of a bound state.