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Abstract We prove a Morse index theorem for action functionals on paths that are allowed to reflect at a hypersurface (either in the interior or at the boundary of a manifold). Both fixed and periodic boundary conditions are treated. 

Abstract In this article, we study the propagation of defect measures for Schrödinger operators$$-h^2\Delta _g+V$$ $-h^2{\Delta }_g+V$on a Riemannian manifold (M, g) of dimensionnwithVhaving conormal singularities along a hypersurfaceYin the sense that derivatives along vector fields tangential toYpreserve the regularity ofV. We show that the standard propagation theorem holds for bicharacteristics travelling transversally to the surfaceYwhenever the potential is absolutely continuous. Furthermore, even when bicharacteristics are tangential toYat exactly first order, as long as the potential has an absolutely continuous first derivative, standard propagation continues to hold. 

We prove explicit asymptotics for the location of semiclassical scattering resonances in the setting of $h$-dependent delta-function potentials on $\mathbb{R}$. In the cases of two or three delta poles, we are able to show that resonances occur along specific lines of the form $\mathrm{Im}<\#comment/>z\sim <\#comment/>-<\#comment/>\mathrm{\gamma <\#comment/>}h\log <\#comment/>\left(1/h\right).$More generally, we use the method of Newton polygons to show that resonances near the real axis may only occur along a finite collection of such lines, and we bound the possible number of values of the parameter $\mathrm{\gamma <\#comment/>}.$We present numerical evidence of the existence of more and more possible values of $\mathrm{\gamma <\#comment/>}$for larger numbers of delta poles. 

Maas ( J. Fluid Mech. , vol. 684, 2011, pp. 5–24) showed that, for an oscillating two-dimensional barotropic tide flowing over sub-critical topography of compact support, some topographic forms existed that produced non-radiating baroclinic disturbances. The problem is related to ‘stealth’ and ‘cloaking’ problems. Here Maas's result is derived using a simpler approach, not involving complicated mappings, but formally restricted to perturbation topography. Wider results come from the discussion of nearly compact support topographic disturbances provided by Schwartz functions with weak high-wavenumber radiation and by exploiting both a known functional equation formulation and Fourier methods. The problem is extended to disturbances on uniform slopes. A variety of non-radiating topographies can be found, although they are mathematically delicate and unlikely to be found in nature. Topography with weak radiation at high wavenumber is a much wider class of structures. Application of these solutions would lie with the ability to estimate dissipation over and near the topography from motions observed at a distance.