= Solvability of a decaying Robin strip
{title2=$g_n=0\text{ for }\lambda_n\leq0$}
Let $\lambda_n$ be the transverse <eigenvalues> of a <Sturm-Liouville problem> for the two <Robin boundary conditions>. For a harmonic strip solution decaying in the longitudinal direction, projection of the prescribed end <Neumann boundary condition> onto every nonpositive eigenvalue must vanish. Positive modes give $q_n(x)=-g_ne^{-\sqrt{\lambda_n}x}/\sqrt{\lambda_n}$. Zero modes are affine and negative modes are oscillatory, so neither can contribute to a decaying solution. The condition also establishes uniqueness within the decay class.
Back to article page
