The fractional Schr\"{o}dinger operator and Toeplitz matrices

Confining a quantum particle in a compact subinterval of the real line with Dirichlet boundary conditions, we identify the connection of the one-dimensional fractional Schr\"odinger operator with the truncated Toeplitz matrices. We determine the asymptotic behaviour of the product of eigenvalues for the $\alpha$-stable symmetric laws by employing the Szeg\"o's strong limit theorem. The results of the present work can be applied to a recently proposed model for a particle hopping on a bounded interval in one dimension whose hopping probability is given a discrete representation of the fractional Laplacian.Comment: 10 pages, 2 figure

Three-dimensional Models of Core-collapse Supernovae From Low-mass Progenitors With Implications for Crab

We present 3D full-sphere supernova simulations of non-rotating low-mass (~9 Msun) progenitors, covering the entire evolution from core collapse through bounce and shock revival, through shock breakout from the stellar surface, until fallback is completed several days later. We obtain low-energy explosions [~(0.5-1.0)x 10^{50} erg] of iron-core progenitors at the low-mass end of the core-collapse supernova (LMCCSN) domain and compare to a super-AGB (sAGB) progenitor with an oxygen-neon-magnesium core that collapses and explodes as electron-capture supernova (ECSN). The onset of the explosion in the LMCCSN models is modelled self-consistently using the Vertex-Prometheus code, whereas the ECSN explosion is modelled using parametric neutrino transport in the Prometheus-HOTB code, choosing different explosion energies in the range of previous self-consistent models. The sAGB and LMCCSN progenitors that share structural similarities have almost spherical explosions with little metal mixing into the hydrogen envelope. A LMCCSN with less 2nd dredge-up results in a highly asymmetric explosion. It shows efficient mixing and dramatic shock deceleration in the extended hydrogen envelope. Both properties allow fast nickel plumes to catch up with the shock, leading to extreme shock deformation and aspherical shock breakout. Fallback masses of <~5x10^{-3} Msun have no significant effects on the neutron star (NS) masses and kicks. The anisotropic fallback carries considerable angular momentum, however, and determines the spin of the newly-born NS. The LMCCSNe model with less 2nd dredge-up results in a hydrodynamic and neutrino-induced NS kick of >40 km/s and a NS spin period of ~30 ms, both not largely different from those of the Crab pulsar at birth.Comment: 47 pages, 27 figures, 6 tables; minor revisions, accepted by MNRA

Spectra and Light Curves of GRB Afterglows

We performed accurate numerical calculations of angle-, time-, and frequency-dependent radiative transfer for the relativistic motion of matter in gamma-ray burst (GRB) models. Our technique for solving the transfer equation, which is based on the method of characteristics, can be applied to the motion of matter with a Lorentz factor up to 1000. The effect of synchrotron self-absorption is taken into account. We computed the spectra and light curves from electrons with a power-law energy distribution in an expanding relativistic shock and compare them with available analytic estimates. The behavior of the optical afterglows from GRB 990510 and GRB 000301c is discussed qualitatively.Comment: 8 pages, 7 figure

Three-dimensional models of core-collapse supernovae from low-mass progenitors with implications for Crab

We present 3D full-sphere supernova simulations of non-rotating low-mass (∼9 M_⊙) progenitors, covering the entire evolution from core collapse through bounce and shock revival, through shock breakout from the stellar surface, until fallback is completed several days later. We obtain low-energy explosions (∼0.5–1.0 × 10⁵⁰ erg) of iron-core progenitors at the low-mass end of the core-collapse supernova (LMCCSN) domain and compare to a super-AGB (sAGB) progenitor with an oxygen–neon–magnesium core that collapses and explodes as electron-capture supernova (ECSN). The onset of the explosion in the LMCCSN models is modelled self-consistently using the VERTEX-PROMETHEUS code, whereas the ECSN explosion is modelled using parametric neutrino transport in the PROMETHEUS-HOTB code, choosing different explosion energies in the range of previous self-consistent models. The sAGB and LMCCSN progenitors that share structural similarities have almost spherical explosions with little metal mixing into the hydrogen envelope. A LMCCSN with less second dredge-up results in a highly asymmetric explosion. It shows efficient mixing and dramatic shock deceleration in the extended hydrogen envelope. Both properties allow fast nickel plumes to catch up with the shock, leading to extreme shock deformation and aspherical shock breakout. Fallback masses of ≲ 5×10⁻³ M_⊙ have no significant effects on the neutron star (NS) masses and kicks. The anisotropic fallback carries considerable angular momentum, however, and determines the spin of the newly born NS. The LMCCSN model with less second dredge-up results in a hydrodynamic and neutrino-induced NS kick of >40 km s⁻¹ and a NS spin period of ∼30 ms, both not largely different from those of the Crab pulsar at birth