Timeline for Uniform boundedness Schrödinger operator eigenfunctions with Dirichlet conditions

8 events

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May 16, 2022 at 17:28	comment	added	Giorgio Metafune		@ChristianRemling Thank you, right!
May 16, 2022 at 17:22	comment	added	Christian Remling		@GiorgioMetafune: I think one way of doing this is to take $V=0$ on $(0,a)$ and $V=N\to\infty$ on $(a,L)$. In the limit $N\to\infty$, this will simulate a Dirichlet boundary condition at $x=a$, and if $a$ is small, then $\\|\phi_n\\|_{\infty}/\\|\phi_n\\|_2$ will become large.
May 16, 2022 at 17:10	comment	added	Giorgio Metafune		@Christian Remling Christian, do you know examples where $\sup_n \\|\phi_n\\|_\infty$ really depends on $\\|V\\|_1$ ($V \geq 0$)?
May 16, 2022 at 14:51	history	edited	Christian Remling	CC BY-SA 4.0	added 5 characters in body
May 15, 2022 at 20:46	vote	accept	Paolo Bernuzzi
May 15, 2022 at 20:25	comment	added	Christian Remling		The book by Poschel and Trubowitz discusses asymptotics in some detail (though with an $L^2$ assumption on $V$, which isn't necessary; you can adapt the proofs if required) and it's what I usually quote when I use these things, but there are many other references, so keep searching if you don't like this (Marchenko's book is another option).
May 15, 2022 at 20:18	comment	added	Paolo Bernuzzi		Thank you for your answer. Your assumptions are correct. I understand what you mean and numerical simulations point that way but I could not manage to find precise proof of that. Could you suggest any reference?
May 15, 2022 at 20:13	history	answered	Christian Remling	CC BY-SA 4.0