緊密封閉的撞球檯在古典的和量子力學的領域裡有長服務的原型系統。 許多有趣的結果在能量統計數值和波函數的密度分佈上已經被獲得了。 我們研究量子撞球檯的特徵態, 把圓盤和體育場的撞球檯以特徵問題陳述。 為了使赫爾蒙茲(Helmholtz)方程式離散化, 我們用一半移動的格子點把非均勻的網格用於輻射狀方向。 我們用雅各大衛森(Jacobi—Davidson) 方法來解決這些特徵值問題。 為計算這些連續特徵值, 我們使用具有最新低等級的一個新穎清晰明確非等值(義)的壓縮技術。

Closed billiards have long served as prototype systems in the field of classical and quantum dynamics. Many interesting results on statistics of energy and the density distribution of the wave function have been obtained. We study eigenstates of quantum billiards in the eigenproblem representation for the disk and stadium billiard. To discretize the Helmholtz equation, we use non-uniform meshes with half-shifted grid points in the radial direction. We use Jacobi-Davidson to solve these eigenproblems. For computing the successive eigenvalues, we use a novel explicit non-equivalence deflation technique with low-rank updates.

1 Introduction 2
1.1 Quantum Chaology......................................2
1.2 Brief history of the quantum billiard eigenproblem....3
1.3 Definition of the billiard problem....................5
2 Disk-model problem 8
2.1 Finite difference discretization.......................8
2.2 Rational matrix........................................9
3 Stadium-model problem 11
3.1 Finite difference discretization.......................11
3.2 Rational matrix........................................12
4 Determining the smallest positive eigenvalue 15
5 Non-equivalence deflation 18
6 Numerical results 22
6.1 Disk model results.....................................24
6.2 Stadium model results..................................30
7 Conclusion 44
Appendix A 45
Appendix B 47
References 49

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