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

Simon, Barry

doi:10.1090/S0002-9947-1975-0417597-8

Pointwise bounds on eigenfunctions and wave packets in $N$-body quantum systems. III
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Trans. Amer. Math. Soc. 208 (1975), 317-329 Request permission

Abstract:

We provide a number of bounds of the form $|\psi | \leqslant O(\exp ( - \alpha |x{|^\alpha })),\alpha > 1$, for ${L^2}$-eigenfunctions $\psi$ of $- \Delta + V$ with $V \to \infty$ rapidly as $|x| \to \infty$. Our strongest results assert that if $|V(x)| \geqslant c{x^{2m}}$ near infinity, then $|\psi (x)| \leqslant {D _\varepsilon }\exp ( - {(c - \varepsilon )^{1/2}}{(m + 1)^{ - 1}}{x^{m + 1}})$, and if $|V(x)| \leqslant c{x^{2m}}$ 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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