An improved stability result for resonances

Abstract:

We prove stability of shape resonances for the sequence of Schrödinger equations $( - {d^2}/d{x^2} + U(x) + {W_n}(x))\psi (x) = E\psi (x),0 \leqslant x < \infty$, in the limit $n \to \infty$ where the barrier potentials ${W_n}(x)$ are integrable, nonnegative, supported in the interval $[1,a]\;(1 < a < \infty )$, and approach infinity pointwise a.e. for $x \in [1,a]$ as $n \to \infty$. In the course of our investigation we prove that for suitable complex initial conditions the solution to the Riccati equation $S’(x) = 1 - ({W_n}(x) - E){[S(x)]^2}$ goes to $0$ as $n \to \infty$ uniformly on compact subsets of $[1,a]$. Our approach is via ordinary differential equations using outgoing wave boundary conditions to define resonances. Our stability result extends a similar result of Ashbaugh and Harrell, who use an argument based on asymptotics and the implicit function theorem to study the above problem with $\lambda V(x)$ replacing ${W_n}(x)$. Our approach is to use the Riccati equation analysis mentioned above and an application of Hurwitz’s Theorem from complex variable theory.