The characteristic spectrum of the reduced wave equation with periodic coefficients in the three-dimensional space (Q2773589)

The authors study spectrum properties for the reduced wave equation \(\text{div}(a\nabla u) + bu = 0\) in \(\mathbb R^3\) with boundary conditions \([u] = 0\) and \([a \partial u/\partial n] = 0\). Here \([\cdot]\) denotes the jump of a function on a non-self-intersecting \(2\pi\)-periodic in \(z\) surface \(\Gamma\) which bounds a domain \(P\). The coefficients \(a\) and \(b\) are \(2\pi\)-periodic in \(z\) functions; the function \(a\) is positive and \(a\), \(b\) are equal to constants \(a_0\) and \(b_0\) outside \(\bar P\).NEWLINENEWLINENEWLINEIt is proven that the continuous spectrum \(\sigma_c\) coincides with the set NEWLINE\[NEWLINE \Sigma = \bigl\{i\mu + ik: k\in\mathbb Z, \;\mu\in \mathbb R, \;|\mu|< \sqrt{b_0/a_0}\bigr\}\cup\{\nu + ik:k\in\mathbb Z, \;\nu\in\mathbb R\}; NEWLINE\]NEWLINE the point spectrum represents a discrete set in \(\mathbb C\) without finite limit points.NEWLINENEWLINENEWLINEApplying the notion of indefinite scalar product for the reduced wave equation, the authors calculate a value of the indefinite scalar product of eigenfunctions corresponding to continuous spectrum for various values of spectrum parameter. In this case, the indefinite scalar product is orthonormal with respect to the spectrum parameter for the Dirac function. Moreover, an orthonormal system of eigenfunctions with respect to the spectrum parameter can be chosen for almost all \(\xi\) from the spectrum set.NEWLINENEWLINENEWLINEThe authors note that the spectrum properties obtained are applied for studying eigenfunctions of the so-called continuous optical fiber which can be described by Maxwell equations.