We analyze the Hamiltonian proposed by Smilansky to describe irreversible dynamics in quantum graphs and studied further by Solomyak and others. We derive a weak-coupling asymptotics of the ground state and add new insights by finding the discrete spectrum numerically in the subcritical case. Furthermore, we show that the model then has a rich resonance structure.

We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.

We analyze spectral properties of the operator <i>H</i> = <i></i>²/<i>x</i>² <i></i>²/<i>y</i>² + <i></i>²<i>y</i>² <i>y</i>²<i>V</i>(<i>xy</i>) in <i>L</i>²(²), where <i></i> 0 and <i>V </i> 0 is a compactly supported and sufficiently regular potential. It is known that the spectrum of <i>H</i> depends on the one-dimensional Schrdinger operator <i>L</i> = <i>d</i>²/<i>dx</i>² + <i></i>² <i>V</i>(<i>x</i>) and it changes substantially as inf(<i>L</i>) switches sign. We prove that in the critical case, inf(<i>L</i>) = 0, the spectrum of <i>H</i> is purely essential and covers the interval [0, ). In the subcritical case, inf <i></i>(<i>L</i>) &gt; 0, the essential spectrum starts from <i></i> and there is a nonvoid discrete spectrum in the interval [0, <i></i>). We also derive a bound on the corresponding eigenvalue moments.