On a mathematical model of irreversible quantum graphs

The ``irreversible quantum graph'' model, suggested by U. Smilansky, is considered. Mathematically, the problem is in the investigation of the spectrum of the operator $\mathbf A_\alpha$ determined by an infinite system of ordinary differential equations on a graph and by a system of boundary conditions, such as conditions on the jumps of derivatives. The operator depends on a parameter $\alpha\ge 0$ involved in the boundary conditions only.

In the paper, the point spectrum and the absolute continuous spectrum of the operator $\mathbf A_\alpha$ are studied in detail in their dependence on $\alpha$. Some special effects appear, the main one being a ``phase transition'' for some value $\alpha=\alpha_0$ that depends on the geometry of the graph: the spectral properties of the operator for $\alpha<\alpha_0$ and $\alpha>\alpha_0$ differ greatly.