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Families of energy operators and generalized energy operators have recentlybeen introduced in the definition of the solutions of linear PartialDifferential Equations (PDEs) with a particular application to the waveequation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, theauthor has introduced the notion of energy spaces included in the Schwartzspace $\mathbf{S}^-(\mathbb{R})$. In this model, the key is to look at whichones of these subspaces are reduced to {0} with the help of energy operators(and generalized energy operators). It leads to define additional solutions fora nominated PDE. Beyond that, this work intends to develop the concept ofmultiplicity of solutions for a linear PDE through the study of these energyspaces (i.e. emptiness). The main concept is that the PDE is viewed as agenerator of solutions rather than the classical way of solving the givenequation with a known form of the solutions together with boundary conditions.The theory is applied to the wave equation with the special case of theevanescent waves. The work ends with a discussion on another concept, theduplication of solutions and some applications in a closed cavity.

Multiplicity of Solutions for Linear Partial Differential Equations Using (Generalized) Energy Operators