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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Pointwise bounds on eigenfunctions and wave packets in $ N$-body quantum systems. III


Author: Barry Simon
Journal: Trans. Amer. Math. Soc. 208 (1975), 317-329
MSC: Primary 35P99; Secondary 81.47
MathSciNet review: 0417597
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Abstract: We provide a number of bounds of the form $ \vert\psi \vert \leqslant O(\exp ( - \alpha \vert x{\vert^\alpha })),\alpha > 1$, for $ {L^2}$-eigenfunctions $ \psi $ of $ - \Delta + V$ with $ V \to \infty $ rapidly as $ \vert x\vert \to \infty $. Our strongest results assert that if $ \vert V(x)\vert \geqslant c{x^{2m}}$ near infinity, then $ \vert\psi (x)\vert \leqslant {D _\varepsilon }\exp ( - {(c - \varepsilon )^{1/2}}{(m + 1)^{ - 1}}{x^{m + 1}})$, and if $ \vert V(x)\vert \leqslant c{x^{2m}}$ neat infinity, then for the ground state eigenfunction, $ \Omega ,\Omega (x) \geqslant {E _\varepsilon }\exp ( - {(c + \varepsilon )^{1/2}}{(m + 1)^{ - 1}}{x^{m + 1}})$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0417597-8
PII: S 0002-9947(1975)0417597-8
Article copyright: © Copyright 1975 American Mathematical Society