Multiplicity of Solutions for Linear Partial Di?fferential Equations Using (Generalized) Energy Operators

J. P. Montillet

Published 2015-09-09Version 1

Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, the author has introduced the notion of energy spaces included in the Schwartz space $\mathbf{S}^-(\mathbb{R})$. In this model, the key is to look at which ones of these subspaces are reduced to {0} with the help of energy operators (and generalized energy operators). It leads to define additional solutions for a nominated PDE. Beyond that, this work intends to develop the concept of multiplicity of solutions for a linear PDE through the study of these energy spaces (i.e. emptiness). The main concept is that the PDE is viewed as a generator of solutions rather than the classical way of solving the given equation with a known form of the solutions together with boundary conditions. The theory is applied to the wave equation with the special case of the evanescent waves. The work ends with a discussion on another concept, the duplication of solutions. The discussion takes place for the special case of waves trapped in an electromagnetic chamber (i.e. a closed tapered waveguide)