Series Expansion of a Function's Expectation Matrix at the Zero Interval Length Limit

Abstract

Expectation matrix of a function serves us to evaluate the expectation value of an algebraic multiplication‐by‐a‐function type operator over a specified subspace of a given Hilbert space. It is also frequently called transition matrix due to quantum mechanical tradition. This work considers univariate functions on finite intervals. The elements of expectation matrix of such a given function are univariate integrals which can be expanded into powers of the interval length and the resulting series may converge in a disc with a certain radius located at the expansion point in the independent variable's complex plane. Convergence and practical applicability issues are also given.