Magnetic localization of an electron on a spherocylindrical nanotube: Analytical solution via Heun functions

We present an analytical study of the quantum behavior of a spinless electron confined to the surface of a spherocylindrical nanotube under the influence of a uniform magnetic field aligned with the tube’s axis. By applying the thin-layer quantization method, we derive the effective Schrödinger equation on the curved surface and obtain exact solutions in terms of confluent Heun functions. The resulting wave functions exhibit distinct parity and are matched across the hemispherical and cylindrical regions using appropriate boundary conditions. We identify critical values of the magnetic field strength, above which the electron becomes strongly localized on the hemispherical caps of the nanotube. The transition from delocalized to localized regimes is characterized by changes in the probability density and energy spectrum, depending on the magnetic field strength, azimuthal quantum number, and geometry of the system. Our findings provide analytical insight into magnetic localization in curved nanostructures and highlight the interplay between geometry, boundary conditions, and external fields in quantum confinement.