A new route to finding bounds on the spectrum of many physical operators

Graeme W. Milton

Published 2018-03-10Version 1

Here we obtain bounds on the spectrum of that operator whose inverse, when it exists, gives the Green's function. We consider the wide of physical problems that can be cast in a form where a constitutive equation ${\bf J}({\bf x})={\bf L}({\bf x}){\bf E}({\bf x})-{\bf h}({\bf x})$ with a source term ${\bf h}({\bf x})$ holds for all ${\bf x}$ in some domain $\Omega$, and relates fields ${\bf E}$ and ${\bf J}$ that satisfy appropriate differential constraints, symbolized by ${\bf E}\in\cal{E}_\Omega$ and ${\bf J}\in\cal{J}_\Omega$ where $\cal{E}_\Omega$ and $\cal{J}_\Omega$ are orthogonal spaces that span the space $\cal{H}_\Omega$ of square-integrable fields in which ${\bf h}$ lies. Here we show that if the moduli ${\bf L}({\bf x})$ satisfy certain boundedness and coercivity conditions then there exists a unique ${\bf E}$ for any given ${\bf h}$, i.e., ${\bf E}={\bf G}_\Omega{\bf h}$ which then establishes the existence of the Green's function. For ${\bf L}({\bf x} )$ depending linearly on a vector of parameters ${\bf z}=(z_1, z_2,\ldots, z_n)$, we obtain constraints on ${\bf z}$ that ensure the Green's function exists, and hence which provide bounds on the spectrum.