The electronic spectrum of an infinite quasi-one-dimensional disorder-free system, of varying width, is considered. It is shown that as a result of the varying width, the spectrum includes extended states as well as states which are exponentially decaying. The effect of magnetic fields is considered and criteria for the weak and strong field regimes are derived. It is shown that the amplitude of the Aharonov-Bohm oscillations in multiply-connected systems can be exponentially small due to the spatial confinement of the probability density. Physical implications for mesoscopic samples are discussed.