We show theoretically that the shape of a beam of matter waves cold atoms emerging from one subwavelength aperture pierced in a film can be collimated within a few degrees. By means of an external laser field, a potential well for the atoms in the direction perpendicular to the surface is created. In this way, a running surface matter wave can be excited when atoms diffract from the aperture. I...

We derive analytical solutions of the Schrödinger wave equation for some q-deformed potentials. Most of these solutions are obtained in terms of Heun functions. However, for some particular cases the solutions are obtained in terms of geometric and hypergeometric functions. Different observations are realized for each potential. However, for all the cases which we study the oscillations are the...

We consider the interaction of Bose-Einstein condensates in a nonlinear lattice formed from the local minima of an externally imposed periodic potential. Within a one-dimensional geometry, we develop a perturbation expansion for the motion of a condensate within the confines of the local minima of its potential well, coupled to the near-neighbor interaction with the adjacent condensates. We sho...

Understanding how humans control unstable systems is central to many research problems, with applications ranging from quiet standing to aircraft landing. Increasingly much evidence appears in favor of event-driven control hypothesis: human operators are passive by default and only start actively controlling the system when the discrepancy between the current and desired system states becomes l...

Recently, one debate in the literature is whether the fractional Schródinger equation in an infinite potential well has the same eigenfunctions as those of its standard (non-fractional) counterpart. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schródinger equation analytically. In this talk, we numerically stu...

We consider mass-constrained minimizers for a class of non-convex energy functionals involving a double-well potential. Based upon global quadratic lower bounds to the energy, we introduce a simple strategy to find sufficient conditions on a given critical point (metastable state) to be a global minimizer. We show that this strategy works well for the one exact and known metastable state: the c...

We present an approach to dynamic decoherence control of finite-temperature Bose-Einstein condensates in a double-well potential. Due to the many-body interactions the standard “echo” control method becomes less effective. The approach described here takes advantage of the interaction-induced change of the spectrum, to obtain the optimal rate of flips of the relative phase between maximally dis...

Simple pair potentials are constructed which give hexagonal-or square-lattice ground states in two dimensions, depending on the value of a single parameter controlling the width of the potential well. Molecular dynamics calculations for free clusters of a few hundred particles are used to examine a structural transition from the square lattice, at low temperatures, to the hexagonal lattice at h...

We study the equilibrium measure for a logarithmic potential in the presence of an external field V∗(ξ) + t p(ξ), where t is a parameter, V∗(ξ) is a smooth function and p(ξ) a monic polynomial. When p(ξ) is of an odd degree, the equilibrium measure is shown to be supported on a single interval as |t| is sufficiently large. When p(ξ) is of an even degree, the equilibrium measure is supported on ...

We consider an Allen-Cahn type equation of the form ut = ∆u + εf(x, t, u), where ε is a small parameter and f(x, t, u) = f(u) − εg(x, t, u) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u0 that is independent of ε, we perform a rigorous analysis of both the generation and the motion of interface....