Smoothed density of states of electrons and smoothed frequency spectrum of phonons for a mesoscopic system

For the finite systems of a sphere, a disk, a linear line, and a tube, the smoothed electron density of states (DOS) is derived directly from the distribution of eigenvalues of a one-particle model. The smoothed DOS is defined by setting one of the quantum numbers to a continuous variable. For a spherical particle, the DOS is expressed as the sum of three terms; a volume term, a surface term, and a circumference term. We clarify the changes in the DOS between a mesoscopic system and a macroscopic system. We apply the same method to phonons of a mesoscopic system and derive the smoothed frequency spectrum