"Weakly" Elliptic Gorenstein Singularities of Surfaces

The main message of the paper is that for Gorenstein singularities, whose (real) link is rational homology sphere, the Artin--Laufer program can be continued. Here we give the complete answer in the case of elliptic singularities. The main result of the paper says that in the case of an elliptic Gorenstein singularity whose link is rational homology sphere, the geometric genus is a topological invariant. Actually, it is exactly the length of the elliptic sequence in the minimal resolution (or, equivalently, in S. S.-T. Yau's terminology: these singularities are maximally elliptic). In the paper we characterize the singularities with this property, and we compute their Hilbert-Samuel function from their resolution graph (generalizing some results of Laufer and Yau). The obstruction for a normal surface singularity to be maximally elliptic can be connected with the torsion part of some Picard groups, this is the new idea of the paper.Comment: 21 pages, latex. To appear in Inventiones mat

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