Discreteness of spectrum and positivity criteria for Schr\"odinger operators

We provide a class of necessary and sufficient conditions for the discreteness of spectrum of Schr\"odinger operators with scalar potentials which are semibounded below. The classical discreteness of spectrum criterion by A.M.Molchanov (1953) uses a notion of negligible set in a cube as a set whose Wiener's capacity is less than a small constant times the capacity of the cube. We prove that this constant can be taken arbitrarily between 0 and 1. This solves a problem formulated by I.M.Gelfand in 1953. Moreover, we extend the notion of negligibility by allowing the constant to depend on the size of the cube. We give a complete description of all negligibility conditions of this kind. The a priori equivalence of our conditions involving different negligibility classes is a non-trivial property of the capacity. We also establish similar strict positivity criteria for the Schr\"odinger operators with non-negative potentials.Comment: 24 pages, final version, some minor misprints correcte

Spectral gaps for periodic Schr\"odinger operators with strong magnetic fields

We consider Schr\"odinger operators $H^h = (ih d+{\bf A})^* (ih d+{\bf A})$ with the periodic magnetic field ${\bf B}=d{\bf A}$ on covering spaces of compact manifolds. Under some assumptions on $\bf B$, we prove that there are arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of strong magnetic field $h\to 0$.Comment: 14 pages, LaTeX2e, xypic, no figure

Proposal of the Readout Electronics for the WCDA in LHAASO Experiment

The LHAASO (Large High Altitude Air Shower Observatory) experiment is proposed for very high energy gamma ray source survey, in which the WCDA (Water Cherenkov Detector Array) is the one of the major components. In the WCDA, a total of 3600 PMTs are placed under water in four ponds, each with a size of 150 m x 150 m. Precise time and charge measurement is required for the PMT signals, over a large signal amplitude range from single P.E. (photo electron) to 4000 P.E. To fulfill the high requirement of signal measurement in so many front end nodes scattered in a large area, special techniques are developed, such as multiple gain readout, hybrid transmission of clocks, commands, and data, precise clock phase alignment, and new trigger electronics. We present the readout electronics architecture for the WCDA and several prototype modules, which are now under test in the laboratory.Comment: 8 pages, 8 figure