Rapid Scheme of Producing Generalized Fourier Expansion of Matrix Functions and its Application to Physical Problems

Application of orthogonal polynomial expansion to quantum simulations is formulated on a general footing, implementing the regulation technique by Sota and Itoh for treating for the Gibbs oscillation. It is an alternative to the kernel polynomial method using Tchebyshev polynomial, but is simpler to handle and makes it possible to use all the popular orthogonal polynomials, covering finite, semi-infinite and infinite intervals of the eigenvalue spectrum. The accuracy can be made equivalent to direct diagonalization, with the resolution being homogeneous in the whole range of the spectrum. The target quantities can be as diverse as including eigenvectors, as well as all sorts of one-particle properties and correlation functions, involving thermal average and quantum time evolution. It can also be used as a handy tool for solving linear algebraic equations.