arXiv subjects
Publications and source records attributed to Ivan Ventura.
We prove that the spectrum of an n-dimensional semiclassical radial Schrödinger operator determines the potential within a large class of potentials for which we assume no symmetry or analyticity. Our proof is based on the first two semiclassical trace invariants and on the isoperimetric inequality.
We study the Hartree equation with a slowly varying smooth potential, $V(x) = W(hx)$, and with an initial condition which is $ε\le \sqrt h$ away in $H^1$ from a soliton. We show that up to time $|\log h|/h$ and errors of size $ε+ h^2$ in $H^1$, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian. This result is based on methods of Holmer-Zworski, who prove a similar theorem for the Gross-Pitaevskii equation, and on spectral estimates for the linearized Hartree operator recently obtained by Lenzmann. We also provide an extension of the result of Holmer-Zworski to more general inital conditions.