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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An improved stability result for resonances

Authors: Mark S. Ashbaugh and Carl Sundberg
Journal: Trans. Amer. Math. Soc. 281 (1984), 347-360
MSC: Primary 81C12; Secondary 34B25
MathSciNet review: 719675
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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^{\prime}(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.

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Keywords: Quantum mechanical resonances, stability of shape resonances, Riccati equations, outgoing wave boundary condition
Article copyright: © Copyright 1984 American Mathematical Society

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