By Paul Bracken

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As for (1) we indeed observe both random fluctuation of energy levels as well as many 38 Advances in Quantum Mechanics 16 Advances in Quantum Mechanics avoided crossings. As for (2), we have picked the bouncing-type periodic orbit PO22 as an example. The predicted peak position (29) of its contribution to DOS varies with the change of γ as shown by a red (almost straight) curve. 6. We have investigated how the cross-shaped scar travels in the spectrum space suffering many avoiding crossings. 65 258th γ 266th Figure 8.

Most interesting is the test using the scar strength function. 2), we can quantitatively observe that prominent periodic orbits systematically contribute to the quantum theory endowed with random energy spectrum. In section 3 we investigate that how the scaring phenomena are affected by the variation of the anisotropy parameter. It is well known that the energy levels show successive avoiding crossings. On the other hand, in the periodic orbit formula, each term in the series for the density of states (DOS) consists of a contribution of an unstable PO with a pole (with an imaginary part given by the Lyapunov exponent of the PO) at the Bohr-Sommerfeld-type energy; thus each term smoothly varies with the anisotropy.

Matrix diagonalization in Sturmian basis We here summarize WMB method for efficient matrix diagonalization. Firstly, in the Sturmian basis 26 4 Advances in Quantum Mechanics Advances in Quantum Mechanics r |nℓm = 1 r λr n! e− 2 (λr )ℓ+1 L2nℓ+1 (λr )Yℓm (θ, ϕ) (2ℓ + n + 1) ! (4) with a scaling parameter λ, the Schrödinger equation of the AKP becomes a matrix equation: ←→ ← → 2 ←→ → ∂ E← 1 − λ ∆ (3) + ( 1 − γ ) λ − 2 Ψ = Id Ψ r λ ∂z2 (5) ←→ 2 ←→ ← → ← → ∂ 2 M Ψ ≡ −∆(3) + (1 − γ) 2 − ǫ Id Ψ = Ψ.