Volterra series arising from the discrete Schrödinger wave equation in L2[a, b] space

The wave solution of the stationary Schrödinger wave equation has been studied extensively. However, after removing restriction on potential function, the problem of obtaining the wave solution becomes considerably difficult. Methods available in the literature for the closed-form wave solution are applicable to specific potential. For this reason, it is important to strive for the wave solution with a general class of potential functions. This paper introduces the Volterra series framework to address this limitation. The Volterra system is a natural extension of linear systems. Such systems take into account quadratic, cubic and more generally, polynomial kind of non-linearities in the systems. The Volterra system is a non-linear system with the structure of the Volterra series. Here, we decompose first a wave function in an appropriate function space and subsequently, the wave equation is discretized. As a result of this, we obtain a bilinear system, treating potential as input and wave function as output. Such an analysis is based on inherent bilinear character of the Schrödinger equation in which potential is bilinearly coupled to the wave function. This paper discusses state space methodology to recover the Volterra series from the bilinear systems. The Volterra series is attractive from the system-theoretic point of view, since it describes the output of a class of non-linear systems explicitly in terms of the input rather than involving coupling terms and allows substantial simplifications for the numerical simulation. A simulation result is introduced to demonstrate the efficacy of the proposed analysis.