A gap in the essential spectrum of a cylindrical waveguide with a periodic perturbation of the surface (Q2786302)

Let \(\omega\) be a bounded subset of \({\mathbb R}^2\) with a simple closed \(C^{\infty}\)-smooth contour \(\partial \omega\), parameterized by \(n\), the oriented distance to \(\partial \omega\), and \(s\), the arc length on \(\partial \omega\) evaluated from a point of \(\partial \omega\); let \(\theta\) be a bounded nonempty domain in the half-space \(\{\xi \in {\mathbb R}^3: \xi_1 < 0 \}\); \(h, \; M, \; \lambda > 0\); \(w_h = (\omega \; \times \; ]-\frac12, \frac12[) \setminus T(h \theta)\), where \(T\) is the natural parameterization associated to the curvilinear system \((n,s)\) and the quote; \(w_{h,j} = \{ x \in {\mathbb R} ^3: (x_1, x_2, x_3-j) \in w_h \}\); let \(\Pi _h\) be a domain of \({\mathbb R} ^3\) which is the interior of \(\bigcup \{ \overline {w_{h,j}}: j \in \mathbb Z \}\); let \(\langle \;,\; \rangle\) be the scalar product of \(L^2(\Pi_h)\); let \(\{M_p: p \in {\mathbb Z}_+ \}\) be the increasing sequence of the eigenvalues of \(- \Delta V =M V\) in \(\omega\), \(V = 0\) on \(\partial \omega \).NEWLINENEWLINETheorem. We suppose \(M_2 -M_1 > \pi ^2\). Then there exist \(h_0, \; c_0, \; P > 0\) such that, for \(h \in \, ]0, h_0]\), the essential spectrum (which is a countable union of closed segments) of the problem \(\langle \nabla u, \nabla v \rangle = \lambda \langle u, v \rangle\) for all \(v \in H^{1,0} (\Pi _h)\), where \(u \in H^{1,0} (\Pi _h)\), has a gap of length \(l(h)\), with \(|l(h) - 2 P h^3| \leq c_0 h^{7/2}\), situated just after the first segment of the essential spectrum.