Applications of semiclassical analysis to partial differential equations
Semiclassical analysis and other types of asymptotic analysis are important tools in the study of partial differential equations. We demonstrate the versatility of these methods by proving two types of results.
First we study the dynamics of solitary waves for two different nonlinear Schr\"odinger equations. The first is the Hartree equation the second is a general nonlinear Schr\"odinger equation with an $L^2$-subcritical power nonlinearity. For both of these equations, we show that a solution found using initial conditions within $\eps \le \sqrt h$ of a soliton (in $H^1$) evolves according to the equations of motion given by the effective Hamitonian up to time $\sim |\log h|/h$, with errors up to size $\eps +h^2$. We achieve this result using the methods of Holmer-Zworski and the spectral results of Lenzmann and Weinstein.
Next we study the spectral properties of $n$-dimensional semiclassical radial Schr\"odinger operators. We prove that their spectrum determines the potential within a large class of potentials for which we assume no symmetry or analyticity. Our proof is analogous to the spectral determination of the sphere and uses the first two semiclassical trace invariants and the isoperimetric inequality.