Fig. 1. (a) Two-dimensional harmonic potential (Eq. (2)), where is the character scale of the system, , and . To break the discrete symmetry, we added a potential at ; (b) On the potential given in Fig.(a), we added an additional Gaussian potential , where , , and . Thus the potential field forms two valleys and one peak, and the position of the bottom of the right valley is , with corresponding potential . There are two saddle points between the two valleys, as marked by the crosses, with corresponding potential values 0.591t and 0.976t.

Fig. 2. The Poincaré section of the motion of a particle moving in the potential field given by Fig. 1(b), e.g., when , vs. the angle of this point with respect to the bottom of the right valley ((a)–(c)). The total energy of the particle is (a), (b), and (c), respectively. (d) The six classes periodic orbits that will be discussed later.

Fig. 3. The representative eigen-wavefunctions of the billiard Fig. 1(a). Shown are the the square of the modulus of wavefunctions that are condensed on the Lissajous orbits. The ratio of the frequency in and directions are: (a)–(c) 1∶2, (d) 2∶3, (e) 3∶4, (f) 1∶3. For all case, , the range of is and the value of the harmonic potential at the boundary is .

Fig. 4. The quantum numbers along the trajectory vs. energy for bouncing ball states in the harmonic potential with a small perturbation. Crosses are the numbers of wavelengthes counted from the wavefunctions minus one, circles are derived from the semiclassical formulas: (a) Horizontal bouncing ball orbits; (b) vertical bouncing ball orbits. Insets show the difference between these two methods.

Fig. 5. The quantization condition for the four types of scars for the harmonic potential with a small perturbation. Crosses are the numbers of wavelengthes counted from the wavefunctions minus one, circles are the quantum numbers derived from the semiclassical formulas. Insets show the difference between these two methods.

Fig. 6. Two types of bouncing ball orbits in the potential shown in Fig. 1(b) and their projections on the zero energy surface. The potential function and its equipotential lines are also plotted. The first class of orbits (C1) corresponds to the center point of the two most significant KAM islands for , and in Fig. 2, and (C2) corresponds to the center points of the KAM islands for , and in Fig. 2(a).

Fig. 7. The quantum numbers along the trajectory vs. energy for bouncing ball states in the modified harmonic potential shown in Fig. 1(b) for C1 orbits (a) and C2 orbits (b). Crosses are the numbers of wavelengthes counted from the wavefunctions minus one, circles are derived from the semiclassical formulas. In each panel, the upper set of points are for , and the lower set of points are for . For C2 orbits, only when energy is small there are states. Insets show the difference (solid circles, left coordinates) between these two methods, and obtained from (empty circles, right coordinates), where the horizontal dashed line is the obtained from fitting to the data, and the corresponding energies are and for C1 and C2 orbits, respectively.

Fig. 8. The quantum numbers along the trajectory vs energy. for all cases. Crosses are the numbers of wavelengthes counted from the wavefunctions minus one, circles are derived from the semiclassical formulas. (a)–(d) correspond to C3-C6 orbits, with , , and , respectively. Insets show some typical scarring states and the corresponding classical orbits, and the difference between these two methods. Note that C3 orbits only appear for when C2 becomes unstable. C4 is the other unstable branch of C2, and becomes stable only for . C5 and C6 are orbits connecting the two potential valleys, only appear when higher energy is high enough.