Semiclassical accuracy for billiards
Semiclassical accuracy for billiards
复制标题
台球的半古典精度
DOI：
10.1088/0951-7715/7/2/010
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发表时间：
1994
期刊：
影响因子：
1.7
通讯作者：
P. A. Boasman
中科院分区：
文献类型：
--
作者：
P. A. Boasman
通过研究，在边界积分法 (BIM) 的背景下研究了半经典近似对台球谱的影响。分析地并且；在数值上，使用 BIM 核的渐近逼近时各个特征值的变化。推导了特征值偏移的一般公式，然后将其应用于圆形台球，其中精确谱和半经典谱之间的半经典偏移显示为接近常数。然后对混沌台球进行近似评估。再次显示了预期的半经典转变如何具有恒定的平均行为。但在这种情况下，周期性轨道的出现（在附录中）显示了围绕平均行为施加波动。 BIM 的数值应用将数值误差引入到所发现的特征值中。导出并显示了一个通用公式，可以简化为圆台球的正确结果。它还针对两个混沌台球进行了测试，尽管只能得出有限的结论。同样，理论和计算都意味着带有波动的恒定偏移。主要结论是半经典近似会导致 O(h(cross)2) 的误差，系数为 0.01 阶，因此它为台球的精确频谱提供了非常好的近似。
The effect of the semiclassical approximation on the spectra of billiards is investigated within the context of the boundary integral method (BIM) by studying. analytically and; numerically, the changes in individual eigenvalues when an asymptotic approximation to the kernel of the BIM is used. A general formula for the shift in an eigenvalue is derived and then applied to the circle billiard where the semiclassical shift between the exact and semiclassical spectra is shown to approach a constant. It is then evaluated approximately for chaotic billiards. again showing how the semiclassical shift to be expected has a constant average behaviour. In this case though, the appearance of periodic orbits is shown (in an appendix) to impose fluctuations around the average behaviour. The numerical application of the BIM introduces numerical errors into the eigenvalues found. A general formula is derived and shown to reduce to the correct result for the circle billiard. It is also tested against two chaotic billiards, although only limited conclusions can be made. Again, both theory and computations imply a constant off-set dressed with fluctuations. The principal conclusion is that the semiclassical approximation leads to errors of O(h(cross)2) with a coefficient of order 0.01, and hence it provides a very good approximation to the exact spectrum for a billiard.