Quasimodes in Integrable Systems and Semi-Classical Limit

Abstract

Quasimodes are long-living quantum states that are localized along classical orbits. They can be considered as resonances, whose wave functions display semi-classical features. In some integrable systems, they have been constructed mainly by the coherent state method, and their connection with the classical motion has been extensively studied, in particular as a tool to perform the semi-classical limit of a quantum system. In this work, we present a method to construct quasimodes in integrable systems. Although the method is based on elementary procedures, it is quite general. It is shown that the requirement of a long lifetime and strong localization implies that the quasimode must be localized around a closed classical orbit. At a fixed degree of localization, the lifetime of the quasimode can be made arbitrarily longer with respect to the classical period in the asymptotic limit of large quantum numbers. It turns out that the coherent state method is a particular case of this general scheme.

Appendix

In this appendix, we evaluate for a generic potential the dependence of the angle $\Delta \theta$ in Eq. (36) on the angular momentum l at a fixed energy E. In the expression of Eq. (35), it is convenient to introduce the new variable $y^{2}\, =\, l^{2}/2mEr^{2}$. In the new variable, one gets

$$\displaystyle{ \Delta \theta \, =\, 2\,\int _{y_{_{m}}}^{y_{_{M}} } \frac{dy} {\sqrt{1\, -\, [y^{2 } \, +\, V (r)/E]}} }$$

(37)

where

$$\displaystyle{ r\, =\, r(y)\, =\, \frac{l} {\sqrt{2mE}\,y} }$$

(38)

The limits of integration $y_{_{m}}$ and $y_{_{M}}$ are the values at which the square root vanishes. Under the assumption of a monotonically increasing potential at increasing r, there are only two values, corresponding to the extremes of the radial oscillations

$$\displaystyle{ \frac{1} {E}V (r(y_{_{0}}))\, +\, y_{_{0}}^{2}\, =\, 1 }$$

(39)

where $y_{_{0}}$ is either $y_{_{m}}$ or $y_{_{M}}$. If the potential is smooth and the radial motion has a non-zero amplitude (i.e. the trajectory is not exactly circular), y ₀ is a smooth function of l, which actually appears only in the combination $l/\sqrt{2mE}$. However, the integrand is singular at the integration limits, although the integral is of course converging. This does not allow to do any derivative with respect to l inside the integral to calculate the derivative of $\Delta \theta$. We have then to analyse the contribution to the integral from an interval close to the limits of integration. If the trajectory is not circular, at $y_{_{M}}$, the function R(y) inside the square root vanishes linearly, and the integrand can be written as

$$\displaystyle{ \frac{1} {\sqrt{1\, -\, R(y)}}\, =\, \frac{1} {\sqrt{R'(y_{_{M } } )(y_{_{M } } \, -\, y)}}\, +\, S(y) }$$

(40)

where R′ is the derivative of R and the remainder S(y) is a regular smooth function. The contribution to the integral from an interval $\,y_{_{1}}\, <\, y\, <\, y_{_{M}}$, with $y_{_{1}}$ some value close to $y_{_{M}}$, can then be written as

$$\displaystyle{ \int _{y_{_{ 1}}}^{y_{_{M}} } \frac{dy} {\sqrt{R(y)}}\, =\, \frac{2} {\sqrt{R'(y_{_{M } } )}}\sqrt{y_{_{M } } \, -\, y_{_{1 }}}\, +\,\int _{ y_{_{ 1}}}^{y_{_{M}} }dy\,S(y) }$$

(41)

Explicitly, the derivative R′ is

$$\displaystyle{ R'(y_{_{M}})\, =\, -\frac{1} {E}V '\left ( \frac{l} {\sqrt{2mE}\,y_{_{M}}}\right ) \frac{l} {\sqrt{2mE}\,y_{_{M}}^{2}}\, +\, 2y_{_{M}} }$$

(42)

where $\,\,V '(x)\, =\, dV/dx$. The same procedure can be followed for the lower limit $y_{_{m}}$. At fixed E, the integral is therefore a smooth function of $\,\,l/\sqrt{2mE}$.