Long-time deviations from exponential decay for inverse-square potentials

Quantal systems are predicted to show a changeover from exponential to a slower decay rate at very long times. Asymptotically most models predict power-law decay with integer exponents. However, the postexponential decay of a trapped particle from a potential can occur with a continuous range of power-law exponents. We show that this happens when the outer part of the potential is repulsive and decreases with the inverse square of the distance. We demonstrate a simple relation between the strength of the long-range tail and the power-law exponent. We also give explicit forms for the postexponential decay laws when the outer part of the potential is weakly attractive. These systems are amenable to experimental scrutiny.

Figure 1

Potential $v\left(r\right)$ for one illustrative example: $\hslash =2m=1$, $v_0=0.5$, $v_b=1.8$, $r_a=3.0$, $r_d=3.4$, and $\beta =1.$

Figure 2

Energy densities $\omega \left(E\right)$ vs energy for $\beta =-0.4$, $-0.1$, 0.3, and 0.7, showing that peak heights increase with $\beta$, while the threshold increase becomes slower.

Figure 3

Survival probabilities $P\left(t\right)$: exact (solid lines) and asymptotic approximation (dashed lines) of Eq. (37). From top to bottom, $\beta =-0.1$, 0.3, and 0.7.

Figure 4

Survival probability $P\left(t\right)$ for $\beta =-0.4$ $\left(\nu =0.1\right)$. Continuous line, exact; dashed line, single-term asymptotic approximation Eq. (37); crosses, the expansion of Eq. (42) truncated at four terms.

Figure 5

Effective ${\mu }_f$ vs $\beta$. Continuous line with filled circles, standard choice of parameters for the inner part of the potential. Dotted line with crosses, standard set except $v_b=1.6$. Dashed line with filled diamonds, standard set except $r_d=3.5$. The straight dashed line is $\mu =2\beta +3$.

Figure 6

Survival probability $P\left(t\right)$ for ${\nu }^{\prime }=0.05$. Continuous line, exact; dashed line, steepest-descent method, Eq. (44).