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We provide evidence that level repulsion in semiclassical spectrum is not just a feature of classically chaotic systems, but classically integrable systems as well. While in chaotic systems level repulsion develops on a scale of the mean level spacing, regardless of the location in the spectrum, in integrable systems it develops on a much longer scale — such as geometric mean of the mean level spacing and the running energy in the spectrum for hard wall billiards. We show that at this scale level correlations in integrable systems have a universal dependence on the level separation, as well as discuss their exact form at any scale. These correlations have dramatic consequences, including deviations from the Poissonian statistics in the nearest level spacing distribution and persistent oscillations of the level number variance over an energy interval as a function of the interval width. We illustrate our findings on two specific models — rectangular infinite well and a modified Kepler problem — that serve as generic types of a hard wall billiard and a potential problem without extra symmetries. Our theory and numerical work are based on the concept of parametric averaging that allows sampling of a statistical ensemble of integrable systems at a given spectral location (running energy).

We analyze two theoretical approaches to ensemble averaging for integrable systems in quantum chaos, spectral averaging (SA) and parametric averaging (PA). For SA, we introduce a new procedure, namely, rescaled spectral averaging (RSA). Unlike traditional SA, it can describe the correlation function of spectral staircase (CFSS) and produce persistent oscillations of the interval level number variance (IV). PA while not as accurate as RSA for the CFSS and IV, can also produce persistent oscillations of the global level number variance (GV) and better describes saturation level rigidity as a function of the running energy. Overall, it is the most reliable method for a wide range of statistics.

Some magnetic characteristics of Nd-Fe-B sintered magnets have been clarified, especially in the near vicinity within three times their length for cylindrical magnets. The flux densities as a function of distance (z) along the z-axis from the center of a single magnet were determined by utilizing Hall sensors. The repulsions between two magnets possessing identical shape were directly measured, also as a function of z, by using a home-made apparatus adopting piezoelectric device. The respective result has turned out well, coinciding with that of corresponding finite element method analysis and some analytical solutions. Also in this investigation, the correct formula directly applicable to the near H-field strength along the magnetic moment of a cylindrical magnet has been determined to be the exact solution that defies all confusing approximations or assumptions in theory, which were seen in textbooks or published in papers. Furthermore an analytical solution for the repulsion of magnet twins (cylindrical or hexagonal in shape), which can be handy in designing a variety of superconducting and/or electromagnetic devices, has been derived basically from Biot–Savart Law.

Casimir energies and forces have been calculated in various configurations and boundary conditions. The calculations indicated that the Casimir energy might change its sign depending not only on the boundary conditions but also on geometry and topology of the configuration. With the development of nanotechnology, it is known that repulsive Casimir force is very important in nanodevices. In this paper, we review some research results on repulsive Casimir force, and discuss whether it could be realizable theoretically and experimentally.