We consider the dynamics on a quantum graph as the limit of the dynamics generated by a one-particle Hamiltonian in R^2 with a potential having a deep strict minimum on the graph, when the width of the well shrinks to zero. For a generic graph we prove convergence outside the vertices to the free dynamics on the edges. For a simple model of a graph with two edges and one vertex, we prove convergence of the dynamics to the one generated by the Laplacian with Dirichlet boundary conditions in the vertex.Comment: 28 pages, 3 figure

We consider a system of N non-relativistic spinless quantum particles ("electrons") interacting with a quantized scalar Bose field (whose excitations we call "photons"). We examine the case when the velocity v of the electrons is small with respect to the one of the photons, denoted by c (v/c = ε ≪ 1). We show that dressed particle states exist (particles surrounded by "virtual photons"), which, up to terms of order (v/c) 3 , follow Hamiltonian dynamics. The effective Nparticle Hamiltonian contains the kinetic energies of the particles and Coulomb-like pair potentials at order (v/c) 0 and the velocity dependent Darwin interaction and a mass renormalization at order (v/c) 2 . Beyond that order the effective dynamics are expected to be dissipative.The main mathematical tool we use is adiabatic perturbation theory. However, in the present case there is no eigenvalue which is separated by a gap from the rest of the spectrum, but its role is taken by the bottom of the absolutely continuous spectrum, which is not an eigenvalue. Nevertheless we construct approximate dressed electrons subspaces, which are adiabatically invariant for the dynamics up to order (v/c) ln[(v/c) −1 ]. We also give an explicit expression for the non adiabatic transitions corresponding to emission of free photons. For the radiated energy we obtain the quantum analogue of the Larmor formula of classical electrodynamics.

Abstract. We study the dynamics of a quantum particle in R n+m constrained by a strong potential force to stay within a distance of order (in suitable units) from a smooth n−dimensional submanifold M . We prove that in the semiclassical limit the evolution of the wave function is approximated in norm, up to terms of order 1/2 , by the evolution of a semiclassical wave packet centred on the trajectory of the corresponding classical constrained system.

We study special vanishing horizon limit of 'boosted' black D3-branes having a compact light-cone direction. The type IIB solution obtained by taking such a zero temperature limit is found to describe a nonrelativistic system with dynamical exponent 3. We discuss about such limits in M2-branes case also.