Narrow Results

We investigate a class of generalized Schrödinger operators in L²(ℝ³) with a singular interaction supported by a smooth curve Γ. We find a strong-coupling asymptotic expansion of the discrete spectrum in the case when Γ is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schrödinger operator with a potential determined by the curvature of Γ. In the same way, we obtain asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if Γ is not a straight line and the singular interaction is strong enough.

We consider a Hamiltonian system given by a charged particle under the action of a constant electric field and interacting with a medium, which is described as a Vlasov fluid. We assume that the action of the charged particle on the fluid is negligible and that the latter has one-dimensional symmetry. We prove that if the singularity of the particle/medium interaction is integrable and the electric field intensity is large enough, then the particle escapes to infinity with a quasi-uniformly accelerated motion. A key tool in the proof is a new estimate on the growth in time of the fluid particle velocity for one-dimensional Vlasov fluids with bounded interactions.