# Localization of Eigenfunctions

 For a given domain, we consider the eigenfunctions um(x) of the Laplace operator with Dirichlet or Neumann boundary condition. If the domain is the shape of a drum, the eigenfunctions describe its vibration modes, while the associated eigenvalues give the squared frequencies (as well as typical length scales of oscillations). The square of an eigenfunction can be interpreted as the spatial distribution over the drum of the potential energy stored in this vibration mode. Dirichlet boundary condition describes a drum with a fixed boundary (vibrations vanish here), while Neumann boundary condition represents a membrane with freely moving boundary (in this case, the normal derivative of the eigenfunctions vanishes at the boundary).

For the unit interval, the eigenfunctions are simply sine or cosine functions, depending on the boundary condition. Larger eigenvalues correspond to higher oscillations. The important feature of the sine and cosine functions is that any of these functions is extended over the whole interval, i.e., there is no specific region (subinterval) carrying the most potential energy of the eigenfunction. For instance, it is easy to check that for any sequence {um(x)} of eigenfunctions with increasing up to infinity eigenvalues, the integrals of their squares over any subinterval converge to the length of this subinterval. This means that in the limit of high spatial oscillations, any subinterval supports the amount of the potential energy that is proportional to its length. Figuratively speaking, the potential energy of highly oscillating eigenfunction is distributed "uniformly". Similar property holds for a disk and a sphere for which the Laplacian eigenfunctions are explicitly known (Bessel and spherical Bessel functions, respectively).

Quite surprisingly, the validity of the above statement is not known for other, even simple, domains. For instance, it is believed (but not proven!) that this statement is false for a stadium, composed of two semicircles and a rectangle. In other words, there may exist a sequence of the eigenfunctions and a subinterval that ... This phenomenon is known as scarring.

A stronger version of the above phenomenon is called localization when there may exist a sequence of the eigenfunctions with increasing eigenvalues such that the interval over a small ball is greater than some threshold value (e.g., 1/2). This statement is often reformulated in a simpler form that there exist eigenfunctions supported by small regions of the domain, i.e., localized in these regions. Bernard Sapoval studied such eigenfunctions in different domains and pointed out their importance for physical applications (e.g., for designing efficient noise-protective walls).

In what follows, we give several examples illustrating how the localized eigenfunctions appear. This study was suggested and conducted in collaboration with Peter Jones. The basic domain is formed by two squares of different sizes, one lying over the other (see Fig.?). Let us start with completely separated squares. The eigenfunctions of this domain are separated as well, being composed of the eigenfunctions for each square. Some of them "live" on bigger square, the others on the smaller one. These latter ones are obviously localized. If now one gradually erases the frontier between two squares, the corresponding eigenfunctions start interact between them. Since the "interaction" (or energy exchange) is only possible in regions of the associated spatial frequency, it is natural to think that some eigenfunctions remain localized on smaller square when the opening is still small. The following three movies illustrate this statement.
 Shape 1 Shape 2 Shape 3
movie.

We fixed the size of the smaller square to be 1, while the size of the larger square was varied by taking...