Motivation: quantum kinematics of decays, with and without repeated measurements Zeno dynamics: existence, form of the generator Anti-Zeno effect: what is it and under which conditions it can occur? Solvable model: a caricature description of a system of a quantum wire and dots Comparison: relations between the stable and Zeno dynamics in this and other models

In this paper, we prove that the existence of absolutely continuous spectrum of the Kirchhoff Laplacian on a radial metric tree graph together with a finite complexity of the geometry of the tree implies that the tree is in fact eventually periodic. This complements the results by Breuer and Frank in (Rev Math Phys 21(7):929945, 2009) in the discrete case as well as for sparse trees in the metric case.

For an arbitrary nonempty, open set ^<i>n</i>, <i>n</i> of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian (-\Delta)^m|_{{C^\infty_0}(\Omega)}, <i>m</i> , and its Kreinvon Neumann extension <i>A</i>_{<i>K</i>,,<i>m</i>} in <i>L</i>²(). with <i>N</i>(, <i>A</i>_{<i>K</i>,,<i>m</i>)}, (-\Delta)^m|_{{C^\infty_0}(\Omega)}, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of <i>A</i>_{<i>K</i>,,<i>m</i>}, we derive the bound <i>N</i>(, <i>A</i>_{<i>K</i>,,<i>m</i>}) (2<i></i>)^-<i>n</i><i></i>_n||{1 + [2<i>m</i>/(2<i>m</i> + <i>n</i>)]}^{<i>n</i>/(2<i>m</i>)}^{<i>n</i>/(2<i>m</i>)}, &gt; 0, where <i>_n</i> <i></i>^<i>n</i>/2 /((<i>n</i> + 2)/2) denotes the (Euclidean) volume of the unit ball in ^<i>n</i>.

The aim of the paper is to derive spectral estimates into several classes of magnetic systems. They include three-dimensional regions with Dirichlet boundary as well as a particle in 3 confined by a local change of the magnetic field. We establish two-dimensional BerezinLiYau and LiebThirring-type bounds in the presence of magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians.

We analyze two-dimensional Schrdinger operators with the potential |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} where |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} and |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} which exhibit an abrupt change of spectral properties at a critical value of the coupling constant . We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for below the critical value the spectrum is purely discrete and we establish a LiebThirring-type bound on its moments. In the critical case where the essential spectrum covers the positive halfline while the negative spectrum can only be discrete, we demonstrate numerically the existence of a ground-state eigenvalue.