Formation of Singularities

Bernie G. Nickel

Andrei V. Rybin, G. G. Varzugin, JUSSI TIMONEN

A generic mechanism of collapse in the Gross-Pitaevskii equation with attractive interparticle interactions is gained by reformulating this equation as Newton's equation of motion for a system of particles with a constraint. 'Quantum pressure' effects give rise to formation of a potential barrier around the emerging singularity, which prevents a fraction of the particles from falling into the singularity. For reasonable initial widths of the condensate, the fraction of collapsing particles, which are thereby removed from the condensate, is found to be a 'universal' number $\simeq 0.7$.

Sebastian Krämer

Tree tensor networks such as the tensor train (TT) format are a common tool for high-dimensional problems. The associated multivariate rank and accordant tuples of singular values are based on different matricizations of the same tensor. While the behavior of such is as essential as in the matrix case, here questions about the feasibility of specific constellations arise: which prescribed tuples can be realized as singular values of a tensor, and what is this feasible set? We first show the equivalence of the tensor feasibility problem (TFP) to the quantum marginal problem (QMP). In higher dimensions, in case of the TT-format, the conditions for feasibility can be decoupled. By present results for three dimensions for the QMP, it then follows that the tuples of squared, feasible TT-singular values form polyhedral cones. We further establish a connection to eigenvalue relations of sums of Hermitian matrices, which in turn are described by sets of interlinked, so-called honeycombs, as they have been introduced by Knutson and Tao. Besides a large class of universal, necessary inequalities as well as the vertex description for a special, simpler instance, we present a linear programming algorithm to check feasibility and a simple, heuristic algorithm to construct representations of tensors with prescribed, feasible TT-singular values in parallel.

Gusein Sh. Guseinov

Takeya Tsurumi, Miki Wadati

We investigate the singularity of the magnetically trapped Bose-Einstein condensate with attractive interatomic interactions. By using a model equation, three-dimensional attractive nonlinear Schrödinger equation with anisotropic harmonic potential terms, we show the scaling laws for the system near the singularity. We also discuss the self-similar behavior of the wavefunction in detail.

R. Brout