# Functional analysis – Asymptotically constant sequence of the eigenfunctions of the Laplacian

I am not an expert in this kind of thing, so forgive me if I am naive.

I wonder if there is a sequence of functions $$f_t: B _ { mathbb {R} ^ 2} (0, t) to [0,infty]$$ (right here $$B (0, t)$$ means radius ball $$t$$ focused on $$0$$) such as, if $$triangle$$ is the Laplacian dish

1) $$triangle f_t = 4f_t$$ (or more generally $$triangle f_t = cf_t$$ for a fixed positive $$c$$)

2) $$f_t to 0$$ punctually as $$t to infty$$

3) $$f_t (x, y) / f_t (0) to 1$$ punctually as $$t to infty$$

In other words, I would like to know if there exists a family of eigenfunctions of the Laplacian, defined on a growing series of marbles, converging towards $$0$$ and such as the convergence rate, at least $$infty$$, does not depend on the point in question.

Here is a vague idea of ​​why, in my opinion, such a sequence may exist: I can imagine a proper function of the Laplacian on the unit disk whose norm goes up and down slightly from a small amount in small disks . The domain stretch forces the oscillation to go down and finally converge towards $$0$$. Maybe I'm looking for a Bessel function.

Thank you.