arXiv subjects
Publications and source records attributed to Isaac Alvarez-Romero.
We prove that a solution of the Schrödinger-type equation $\mathrm{i}\partial_t u= Hu$, where $H$ is a Jacobi operator with asymptotically constant coefficients, cannot decay too fast at two different times unless it is trivial.
We prove that a solution of the Toda lattice cannot decay too fast at two different times unless it is trivial. In fact, we establish this result for the entire Toda and Kac-van Moerbeke hierarchies.
We study direct and inverse scattering problem for systems of interacting particles, having web-like structure. Such systems consist of a finite number of semi-infinite chains attached to the central part formed by a finite number of particles. We assume that the semi-infinite channels are homogeneous at infinity, but the limit values of the coefficients may vary from one chain to another.
We consider the Schrödinger evolution on graph, i.e. solution to the equation $\partial_tu(t,α)=i\sum_{β\in\mathcal{A}}L(α,β)u(t,β)$, here $\mathcal{A}$ is the set of vertices of the graph and the matrix $(L(α,β))_{α,β\in\mathcal{A}}$ describes interaction between the vertices, in particular two vertices $α$ and $β$ are connected if $L(α,β)\neq0$. We assume that the graph has a "web-like" structure, i.e, it consists of an inner part, formed by a finite number of vertices, and some threads attach to it. We prove that such solution $u(t,α)$ cannot decay too fast along one thread at two different times, unless it vanishes at this thread. We also give a characterization of the dimension of the vector space formed by all the solutions of $\partial_tu(t,α)=i\sum_{β\in\mathcal{A}}L(α,β)u(t,β)$ when $\mathcal{A}$ is a finite set, in terms of the number of the different eigenvalues of the matrix $L(\cdot,\cdot)$