Normal and generalized Bose condensation in traps: One dimensional examples

We prove the following results. (i) One-dimensional Bose gases which interact via unscaled integrable pair interactions and are confined in an external potential increasing faster than quadratically undergo a complete generalized Bose-Einstein condensation (BEC) at any temperature, in the sense that a macroscopic number of particles are distributed on a o(N)number of one-particle states. (ii) In a one dimensional harmonic trap the replacement of the oscillator frequency \omega by \omega\ln N/N gives rise to a phase transition at a=\hbar\omega\beta=1 in the noninteracting gas. For a<1 the limit distribution of n_0/N^a is exponential and /N^a tends to 1. For a>1 there is BEC with a condensate density /N going to 1-1/a. For a>=1, (\ln N/N)(n_0-) is asymptotically distributed following Gumbel's law. For any a>0 the free energy is -(\pi^2/6a\beta)N/\ln N+o(N/\ln N), with no singularity at a=1. (iii) In Model (ii) both above and below the critical temperature the the gas undergoes a complete generalized BEC, thus providing a coexistence of ordinary and generalized condensates below the critical point. (iv) Adding an interaction =o(N\ln N) to Model (ii) we prove that a complete generalized BEC occurs at all temperatures.Comment: Published version with further improvement

Galilean invariance in confined quantum systems: Implications on spectral gaps, superfluid flow, and periodic order

Galilean invariance leaves its imprint on the energy spectrum and eigenstates of $N$ quantum particles, bosons or fermions, confined in a bounded domain. It endows the spectrum with a recurrent structure which in capillaries or elongated traps of length $L$ and cross-section area $s_\perp$ leads to spectral gaps $n^2h^2s_\perp\rho/(2mL)$ at wavenumbers $2n\pi s_\perp\rho$, where $\rho$ is the number density and $m$ is the particle mass. In zero temperature superfluids, in toroidal geometries, it causes the quantization of the flow velocity with the quantum $h/(mL)$ or that of the circulation along the toroid with the known quantum $h/m$. Adding a "friction" potential which breaks Galilean invariance, the Hamiltonian can have a superfluid ground state at low flow velocities but not above a critical velocity which may be different from the velocity of sound. In the limit of infinite $N$ and $L$, if $N/L=s_\perp\rho$ is kept fixed, translation invariance is broken, the center of mass has a periodic distribution, while superfluidity persists at low flow velocities. This conclusion holds for the Lieb-Liniger model.Comment: Improved, final version. Equation (22) is slightly more general than in the publication. The upper bound for the critical velocity on p. 4 is correcte

Unknowns and unknown unknowns: from dark sky to dark matter and dark energy

Answering well-known fundamental questions is usually regarded as the major goal of science. Discovery of other unknown and fundamental questions is, however, even more important. Recognition that "we didn't know anything" is the basic scientific driver for the next generation. Cosmology indeed enjoys such an exciting epoch. What is the composition of our universe? This is one of the well-known fundamental questions that philosophers, astronomers and physicists have tried to answer for centuries. Around the end of the last century, cosmologists finally recognized that "We didn't know anything". Except for atoms that comprise slightly less than 5% of the universe, our universe is apparently dominated by unknown components; 23% is the known unknown (dark matter), and 72% is the unknown unknown (dark energy). In the course of answering a known fundamental question, we have discovered an unknown, even more fundamental, question: "What is dark matter? What is dark energy?" There are a variety of realistic particle physics models for dark matter, and its experimental detection may be within reach. On the other hand, it is fair to say that there is no widely accepted theoretical framework to describe the nature of dark energy. This is exactly why astronomical observations will play a key role in unveiling its nature. I will review our current understanding of the "dark sky", and then present on-going Japanese project, SuMIRe, to discover even more unexpected questions.Comment: 11 pages, 7 figures, to appear in the proceedings of SPIE Astronomical Instrumentation "Observational frontiesr of astronomy for the new decade", based on a plenary talk on June 28, 201