Quantum mechanics of one-dimensional time-independent system whose energy level statistics obeys the Gaussian ensemble is numerically studied. Recently the nano-size quantum dots and anti-dots made by the highly sophisticated fabrication process on the heterojunction structure of semiconductors often exhibit the anomalous physical behaviors. In order to understand them the study of the lower dimensional quantum electron transport from the viewpoint of quantum chaos is inevitably important. One-dimensional conserved systems are known to be integrable. However, at least numerically, it is also shown that we can construct the potential for the Schrödinger equation that reproduces a finite number of given energy levels of chaotic regime, e.g., the random matrix theory. In this work a potential is constructed numerically by the standard gradient method or by the inverse scattering method. The more energy levels of chaotic regime we take, the more complicated and finer the ripples of the potential become The potential has fractal structure at high energy limit. [DOI: 10.1380/ejssnt.2003.175]