Predefined exponential basis set for half-bounded multi domain spectral method

A fast and efficient multi-domain spectral method (MDSM) based on wide range non-orthogonal predefined exponential basis set is presented. This method works extremely well for semi-infinite differential problems. It spans wide range of exponential decay rates with multi scaling and does not suffer from zero crossing. These two conditions are necessary for many physical problems. For comparison, the method is used to approximate different exponentially decaying functions and compared with Laguerre basis method. Also for comparison purpose, it is used to analyze arbitrary quantum wells (QW) and optical waveguides. The comparisons exhibit the accuracy and the efficiency of the presented method. In the analytical limit, the relative error in the quantized energy level of the studied QW is in the order of 10^-12 with very small number of bases.