Hermite/Laguerre-Gaussian modes &amp; lower bounds for quasimodes of semiclassical operators
In the first part of the dissertation, we are concerned with local lower bounds for (i) quasimodes of semiclassical Schrödinger operators on domains with boundary and for (ii) Bargmann transforms of certain functions. On domains with boundary, the main tool is a boundary Carleman estimate, essentially due to Lebeau and Robbiano. It is more elementary to prove lower bounds for Bargmann transforms, since Bargmann transforms map to weighted spaces of holomorphic functions.
In the second part of the dissertation, we study the manipulation of Hermite-Gaussian modes and Laguerre-Gaussian modes for use in laser physics, building on the work of Calvo and Picón. Specifically, we classify the self-adjoint extensions of Calvo and Picón's operators, and we study the associated unitary propagators using methods from semiclassical analysis.