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Anderson localisation for (discrete) Schrödinger operators with Bernoulli potentials on $l^2(\mathbb{Z})$ was proven in

https://link.springer.com/article/10.1007/BF01210702

I am wondering if there is a similar reference for the corresponding result on $l^2(\mathbb{N})$, assuming it is still true?

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All the technics to prove Anderson localisation on $l^2(\mathbb{Z})$ also work for $l^2(\mathbb{N})$. It even easier as you only have to prove the exponential decay in one direction.

I guess everything one should know about 1d Anderson localisation are in the book of Carmona and Lacroix "Spectral Theory of Random Schrodinger Operators"

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  • $\begingroup$ I agree that at least most of the techniques generalise, but is there a reference for this particular case? I think it is much cleaner to reference a paper than argue a "small" generalisation. $\endgroup$ – Mathmo Jan 14 at 9:27
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    $\begingroup$ I'm not sure it's that easy. What is straightforward (given the usual machinery) is localization for a typical realization for almost all boundary conditions at the left endpoint (and there's always an exceptional set of bc's with no point spectrum). This is basically the same technical problem that's responsible for Bernoulli models being harder than those with an ac component in the distribution. $\endgroup$ – Christian Remling Jan 14 at 17:45

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