Let $\Omega \subset \mathbb{R}^3$ be a sheared waveguide, i.e., $\Omega$ is built by translating a cross-section (an arbitrary bounded connected open set of $\mathbb{R}^2$ ) in a constant direction along an unbounded spatial curve. Consider $-\Delta_{\Omega}^D$ the Dirichlet Laplacian operator in $\Omega$. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-\Delta_{\Omega}^D$. After that, we state sufficient conditions that give rise to a non-empty discrete spectrum for $-\Delta_{\Omega}^D$. Finally, in case the cross section translates along a broken line in $\mathbb{R}^3$, we prove that the discrete spectrum of $-\Delta_{\Omega}^D$ is finite, furthermore, we show a particular geometry for $\Omega$ which implies that the total multiplicity of the discrete spectrum is equals 1.
By Alexander Sobolev
By Jussi Behrndt
By Albert Schwarz
By Albert Schwarz
By Albert Schwarz
By Giuseppe Savaré
By Stephen Coombes , Peter Liddle
By Victor Kleptsyn
By Jean Feydy
By Albert Schwarz
