The Laplace operator in the Euclidean space has no L^2 eigenfunctions, but by perturbing the metric or adding a potential one can construct a plenty of examples of operators with L^2 integrable eigenfunctions.

Landis conjecture states that any non-zero solution to \Delta u + V u=0 in the Euclidean space with real bounded V cannot decay faster than exponentially near infinity.

If we are allowed to slightly perturb the coefficients of the Laplace operator (for instance taking a small smooth perturbation of the Euclidean metric and taking the Laplace operator for this metric ) how fast can we force an eigenfunction of the perturbed Laplace operator to be localized?

We will review known results, related open questions and recent constructions of Nazarov and AL, Filonov and Krymskii,  Pagano, AL and Krymskii of eigenfunctions to linear elliptic operators with smooth coefficients, which are localized much faster than exponentially.