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We study asymptotical behaviour of resonances for a quantum graph consisting of a finite internal part and external leads placed into a magnetic field, in particular, the question whether their number follows the Weyl law. We prove that the presence of a magnetic field cannot change a non-Weyl asymptotics into a Weyl one and vice versa. On the other hand, we present examples demonstrating that for some non-Weyl graphs the effective size of the graph, and therefore the resonance asymptotics, can be affected by the magnetic field.

We consider a class of Hamiltonians in L2(R2) with attractive interaction supported by piecewise C2 smooth loops of a fixed length L, formally given by (x) with >0. It is shown that the ground state of this operator is locally maximized by a circular . We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of .

We analyse two-dimensional Schrdinger operators with the potential| xy| p (x 2+ y 2) p/(p+ 2) where p 1 and 0. We show that there is a critical value of such that the spectrum for < crit is bounded below and purely discrete, while for > crit it is unbounded from below. In the subcritical case, we prove upper and lower bounds for the eigenvalue sums.

A product formula is proved which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P. It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace Ran P.