# Quantum Eigenstates of Curved and Varying Cross-Sectional Waveguides
## 1 Introduction  
With the vast possibilities to manufacture complex topological structures in quantum technology, it becomes increasingly important to address how physical properties change due to shape, size etc. Nanowire technology provides a rich platform to discover novel physical properties related to curvature effects such as flexible electronics [1], battery [2] and nanoelectromechanical sensors and generators [3; 4; 5; 6; 7]. A selection of recent nanoarchitecture work [8] includes the influence of a varying thickness of 2D WS\({}_{2}\) nanolayers [9], the role of topology of the thermopower of six-terminal Andreev interferometers [10], the voltage induced by moving vortices as a function of transport and magnetic fields for rolled-up nanostructured nanotubes [11], topological transitions in superconducting structures [12; 13], etc. Recently, the present authors [14] derived the first four-band spinless \(\mathbf{k}\cdot\mathbf{p}\) model in curved coordinates using [PERSON]'s model. In Ref. [15], a detailed discussion of the combined influence of curvature, torsion, and cross-sectional rotation of a nanostructurefor electron eigenstates and symmetry properties in nanowires was presented based on a one-dimensional effective mass equation. Evidently, the use of curved coordinates is effective to recast computationally intensive 3D wave problems of complex nanoarchitectures into lower-dimensional problems; examples include the well-known cases of a torus, a helix, and a Mobius structure [16; 17; 18; 19; 20; 21; 22; 23; 24].  
The use of analytical or numerical methods to determine 3D waveguide eigenstates and eigenfrequencies in acoustics and electromagnetics has a long history and is of high importance for applications. [PERSON]'s horn model in acoustics provides a simple, accurate one-dimensional approximation for the estimation of the sound field in waveguides of varying thickness that otherwise requires a full 3D analysis [25; 26]. [PERSON] [27; 28] derived a set of coupled ordinary differential equations for the varying thickness waveguide. In the latter case, even a small number of coupled equations was shown to provide an accurate description of the groundstate, the first excited states and their frequencies. Both methods [25; 27; 28], however, assumed a straight centerline (cylinder structures) and therefore cannot be applied to a more general class of nanowire geometries characterized by, simultaneously, a curved axis and a varying cross section.  
In this work, we present a new method using differential geometry and perturbation theory to determine eigenstates to the Schrodinger equation of a quantum-mechanical particle confined to a curved, varying-thickness waveguide with small cross-sectional dimensions relative to the waveguide length. The cross-sectional geometry can be arbitrary and may rotate along the waveguide axis. We examine the influence of all geometry parameters and compare with accurate full-wave numerical solutions. For the waveguide structures analyzed, excellent accuracy is demonstrated for the first several eigenstates. The developed model provides insight into the influence of waveguide curvature, cross-sectional thickness variation and frame rotation for eigenstate energies and symmetry properties. In particular, the present method sheds new light on resonance phenomena as a result of coupling between geometry parameters that is not attainable from a standard complex full-wave problem. Besides the insight that can be gained from the 1D equations, there is also significant computational advantages, as discussed in Appendix A.