#### Milestone Year

### 2000

#### Spectral theory on metric graphs

Metric graph is a system consisting of a family of points (vertices)
and a family of finite or infinite line segments (edges).
The endpoints of each finite edge are vertices.
An infinite edge has only one endpoint which also is one of the vertices.
Function on a metric graph is a family of functions defined on its edges.
Differential operator on such graph is a family of differential operators
on edges, complemented by the so-called matching conditions at the vertices.
The most important operator is the Laplacian,
which is defined as the operator of second derivative on each edge,
with the Kirchhoff conditions at each vertex
(motivated by the theory of electricity).
Spectral theory on a metric graph consists of a study of
the eigenvalue behavior of a differential operator
and also of the structure of its continuous spectrum.

Among the general metric graphs those which enjoy an important property
of "radial symmetry" play a special role.
Scientists of WIS found a special orthogonal decomposition of
the Laplacian on such graphs, which reduces the study of the Laplacian
to the analysis of much simpler operators acting on the half-line.
It plays the same role as the Fourier decomposition in the standard setting,
which refers, for instance, to solving the Poisson equation in the rectangle
(with appropriate boundary conditions).
Later this decomposition was applied (by the specialists in
mathematical physics) for the spectral analysis
of many problems originated in Physics.

Another topic is the spectral analysis of the so-called "Smilansky model".
This model leads to a family of operators on metric graphs.
The operators depend on a non-negative parameter,
and the main problem is how does the spectrum depend on this parameter.
This problem was investigated in detail in WIS,
both for the discrete part of the spectrum
and for its absolute continuous part.