Leaky quantum graphs and Robin billiards: discrete spectrum, asymptotic expansions, and magnetic effects

The subject of this talk are properties of the discrete spectrum several operator classes appearing in models of various quantum systems. They include Schrödinger operators with an attractive singular ‘potential’, supported by a geometric complex of codimension one. The simplest of them can be formally written as $-\Delta-\alpha\delta(x-\Gamma)$ with $\alpha>0$, where $\Gamma$ is a curve in $\mathbb R^2$ or a surface in $\mathbb R^3$. Another class are Hamiltonians describing quantum motion in a region with attractive Robin boundary. We discuss the ways in which spectral properties of such systems are in uenced by the geometry of the interaction support. A particular attention is paid to situations when the coupling constant is large or the geometric perturbation is weak, and asymptotic expansions can be derived. We also discuss effects arising from the presence of a magnetic eld, in particular, sufficient conditions for existence of the discrete spectrum in planar wedges in presence of a homogeneous magnetic eld, and in uence of an Aharonov–Bohm flux on the so-called Welsh eigenvalues.