| Summary: | In the present work we study electron and phonon transport in waveguides that locally are close to a periodical waveguide. We obtain that generally an electron moves along the waveguide for a long distance without evident reflection. Internal reflection occurs only in a vicinity of some points which are usually called turning points. The main goal of the work is to describe uniform asymptotic solutions in waveguides containing a single turning point or a pair of turning points. We also discuss the asymptotic properties of transition and reflection probabilities for the electron motion. In the case of four turning points the effect of resonant tunneling is observed. We show that the parts of the waveguide between two middle turning points can play the role of a resonator, which provides a series of resonances for the motion of electrons. This implies that if an electron energy is close to a resonance, then the transition probability for the electron motion is strongly increased. We use the same ideas to study the low temperature thermal conductance of dielectric nanofibers. We provide a simple method that allows to decrease thermal conductivity in nanofibers. For instance, such nanofibers can be used to isolate a thermal detector from a surface. The obtained methods can help in understanding the scattering in more general systems.
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