QUANTUM FEW‐BODY SYSTEMS: Proceedings of the Joint Physics/Mathematics Workshop on Quantum Few‐Body Systems
998(2008); http://dx.doi.org/10.1063/1.2915634View Description Hide Description
Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of “fat graphs”—in other words, suitable families of shrinking manifolds—and discuss convergence of the spectra and resonances in such a setting.
998(2008); http://dx.doi.org/10.1063/1.2915632View Description Hide Description
In the frame of non‐relativistic quantum mechanics we discuss the systems of N particles, whose energy is close to that of the dissociation threshold. We show that in systems, where a long‐range repulsion acts between the dissociation fragments, there is a super‐size blocking, i.e. the halo structures in these systems do not appear. We discuss the connection between bound states at the threshold and spreading and derive the conditions on pair potentials, which guarantee the super‐size blocking. Under minor assumptions we prove that negative atomic ions have a bound state at the threshold when the charge of the nucleus is critical.
998(2008); http://dx.doi.org/10.1063/1.2915636View Description Hide Description
We consider a system of a hydrogen atom interacting with the quantized electromagnetic field. Instead of fixing the nucleus, we assume that the system is confined by its center of mass. This model is used in theoretical physics to explain the Lamb‐Dicke effect. After a brief review of the literature, we explain how to verify some properly chosen binding conditions which, by , lead to the existence of a ground state for our model, and for all values of the fine‐structure constant.
998(2008); http://dx.doi.org/10.1063/1.2915638View Description Hide Description
The physical relevance of the resonance wave function is discussed in view of the complex scaling theory. It is argued that although it is unphysical in the sense that it corresponds to a complex energy it is useful when we want to understand and compute several physical observables. We first review our work on the influence of resonances on a scattering cross sections. We then discuss the partial widths concept as presented by Peshkin, Moiseyev and Lefebvre [J. Chem. Phys. 92 2902 (1990)]. Finally we use this formalism to suggest a way to define a root mean square radius of a resonant state.
998(2008); http://dx.doi.org/10.1063/1.2915640View Description Hide Description
Feshbach resonances are quasi‐bound states that are represented, in first approximation, as bound states in energetically closed channels. Owing to coupling with open channels they can spontaneously break up into two or more fragments. I will review those states that separate into two fragments within the framework of multichannel effective range theory. Magnetic tuning of scattering lengths is described within this context. Such tuning has opened up many opportunities for fundamental studies of few‐body systems. By using multichannel zero‐range potentials, conventionally called pseudo‐potentials, it is possible to employ the model when the few‐body systems involve three or more fragments. The emergence of Efimov states for critical values of the tuning parameters is discussed.
Local and global properties of eigenfunctions and one‐electron densities of Coulombic Schrödinger operators998(2008); http://dx.doi.org/10.1063/1.2915642View Description Hide Description
We review recent results by the authors on the regularity of molecular eigenfunctions ψ and their corresponding one‐electron densities ρ, as well as of the spherically averaged one‐electron atomic density Furthermore, we prove an exponentially decreasing lower bound for in the case when the eigenvalue is below the essential spectrum. This result also holds when the Hamiltonian is restricted to symmetry subspaces.
998(2008); http://dx.doi.org/10.1063/1.2915644View Description Hide Description
We calculate energies, condensate fractions and two‐body correlation functions for a system of N identical bosons in a trap. Both attractive and repulsive finite‐range interactions with a large range of positive scattering lengths are used. At small scattering lengths the system is model independent. When the scattering length is comparable to the trap length the properties of the system depend on the details of the interaction.
Uniform asymptotics of eigenfunctions for the three‐body Schrödinger operator in one‐dimensional case998(2008); http://dx.doi.org/10.1063/1.2915630View Description Hide Description
The three‐body scattering problem with finite pair potentials for one‐dimensional case is investigated. The asymptotic function which satisfies the three‐body Schrödinger equation in whole configuration space outside of compact domain Ω, where the supports of all three pair potentials cross each other, has been presented in a mathematically rigorous way. For large distances the function determines the asymptotics of the solution up to the circular wave with smooth coefficient in whole configuration space. The method is based on analogies between few‐body scattering problem and diffraction one of the plane wave on the system of half‐transparent infinite screens. Presented here formalism are believed to be useful also for the few‐body scattering problem of higher dimensions.