Patterns and structure in systems governed by linear second-order differential equations (Q1089815)

This article represents a survey of transmutation ideas and their interaction with typical physical problems. For linear second order differential operators \(P\) and \(Q\) one deals with canonical connections \(B:P\to Q\) (transmutations) satisfying \(QB=BP\) and the related transport of ''structure'' between the theories of \(P\) and \(Q\). One can study in an intrinsic manner, e.g., Parseval formulas, eigenfunction expansions, integral transform, special functions, inverse problems, integral equations, and related stochastic filtering and estimation problems, etc. There are applications in virtually any area where such operators arise.