Generalized Fredholm operator equation and its solution based on intermediate coordinate-momentum representation

摘要

Based on the intermediate coordinate-momentum representations, we construct the appropriate generalized Fredholm operator equation and then find its solution. The application of this equation for deriving Hermite-polynomial operator formulae is demonstrated. The connection of the generalized Wigner operator to the form of the Gaussian bivariate distribution of random variables in statistics is shown, which is useful for studying the tomogram of quantum states.