  We consider the spectrum of Andreev bound states (ABSs) in an exemplary
4-terminal superconducting structure where 4 chaotic cavities are connected by
QPCs to the terminals and to each other forming a ring. Such a tunable device
can be realized in 2DEG-superconductor structures.
  We concentrate on the limit of a short structure and large conductance of the
QPCs where a quasi-continuous spectrum is formed. The energies can be tuned by
the superconducting phases. We observe the opening and closing of gaps in the
spectrum. This concerns the usual proximity gap that separates the levels from
zero energy as well as less usual "smile" gaps that split the levels of the
spectrum.
  We demonstrate a remarkable crossover in the overall spectrum that occurs
upon changing the ratio of conductance of the inner and outer QPCs. At big
values of the ratio, the levels exhibit a generic behavior expected for the
spectrum of a disordered system manifesting level repulsion and "Brownian
motion" upon changing the phases. At small values of the ratio, the levels are
squeezed into narrow bunches separated by wide smile gaps. Each bunch consists
of almost degenerate ABSs.
  We study in detail the properties of the spectrum in the limit of a small
ratio, paying special attention to the crossings of bunches. We distinguish two
types of crossings: i. with a regular phase dependence of the levels and ii.
crossings where the Brownian motion of the levels leads to an apparently
irregular phase-dependence. We work out a perturbation theory to explain the
observations.
  The unusual properties of the spectrum originate from unobvious topological
effects. Topology of the first kind is related to the winding of the
semiclassical Green's function. It is responsible for the proximity gaps.
Topology of the second kind comes about the discreteness of the number of modes
and is responsible for the smile gaps.
