117,133 research outputs found

    Is tame open?

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    Is tame open? No answer so far. One may pose the Tame-Open Conjecture: Tame is open. But how to support it? No effective way to date. In this note, the rank of a wild algebra is introduced. The Wild-Rank Conjecture, which implies the Tame-Open Conjecture, is formulated. The Wild-Rank Conjecture is improved to the Basic-Wild-Rank Conjecture. A covering criterion on the rank of a basic wild algebra is given, which can be effectively applied to verify the Basic-Wild-Rank Conjecture for concrete algebras. It makes all conjectures much reliable.Comment: 10 page

    Bose-Fermi Mapping and Multi-Branch Spin Chain Model for Strongly Interacting Quantum Gases in One-Dimension: Dynamics and Collective Excitations

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    We show that the wave function of a one dimensional spinor gas with contact ss-wave interaction, either bosonic or fermionic, can be mapped to the direct product of the wave function of a spinless Fermi gas with short-range pp-wave interaction and that of a spin system governed by spin parity projection operators. Applying this mapping to strongly interacting spinor gases, we obtain a generalized spin chain model that captures both the static and dynamics properties of the system. Using this spin chain model, we investigate the breathing mode frequency and the quench dynamics of strongly interacting harmonically trapped spinor gases.Comment: 5 pages of main text + 5 pages of Supplemental Materia

    Generic continuous spectrum for multi-dimensional quasi periodic Schr\"odinger operators with rough potentials

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    We study the multi-dimensional operator (Hxu)n=βˆ‘βˆ£mβˆ’n∣=1um+f(Tn(x))un(H_x u)_n=\sum_{|m-n|=1}u_{m}+f(T^n(x))u_n, where TT is the shift of the torus \T^d. When d=2d=2, we show the spectrum of HxH_x is almost surely purely continuous for a.e. Ξ±\alpha and generic continuous potentials. When dβ‰₯3d\geq 3, the same result holds for frequencies under an explicit arithmetic criterion. We also show that general multi-dimensional operators with measurable potentials do not have eigenvalue for generic Ξ±\alpha
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