Semiclassical form factor for spectral and matrix element fluctuations of multidimensional chaotic systems

We present a semiclassical calculation of the generalized form factor $K_{ab}\left(\tau \right)$ which characterizes the fluctuations of matrix elements of the operators $\widehat{a}$ and $\widehat{b}$ in the eigenbasis of the Hamiltonian of a chaotic system. Our approach is based on some recently developed techniques for the spectral form factor of systems with hyperbolic and ergodic underlying classical dynamics and $f=2$ degrees of freedom, that allow us to go beyond the diagonal approximation. First we extend these techniques to systems with $f>2$. Then we use these results to calculate $K_{ab}\left(\tau \right)$. We show that the dependence on the rescaled time $\tau$ (time in units of the Heisenberg time) is universal for both the spectral and the generalized form factor. Furthermore, we derive a relation between $K_{ab}\left(\tau \right)$ and the classical time-correlation function of the Weyl symbols of $\widehat{a}$ and $\widehat{b}$.

Figure 1

Representation of a periodic orbit $\gamma$ (solid line) in phase space with a close encounter together with its time-reverse version ${\gamma }^i$ (dashed line) and its partner orbit ${\gamma }^p$ (dotted line). The orbit $\gamma$ is characterized by two stretches which are almost time reverse of one another. One of these stretches is situated between the two Poincaré surface of sections (PSS) perpendicular to the orbit shown by the grey squares. The time-reverse map $\mathcal{T}$ is the reflection with respect to the plane $p=0$. The picture should be thought of as a projection of the whole $2f$-dimensional phase space on a subspace formed by one momentum and two position coordinates.

Figure 2

Configuration space representation of a periodic orbit $\gamma$ with a close encounter (solid line) together with its partner orbits ${\gamma }^p$ (dotted line) for a system with two degrees of freedom.

Figure 3

Schematic drawing of the encounter region in phase space. The Poincaré surface of section (PSS) is a $(2f-2)$-dimensional surface defined at ${\mathbf{x}}_t^{\gamma }$ in the $2f$-dimensional phase space by the perpendicular coordinates $(\delta {\mathbf{q}}^{\perp },\delta {\mathbf{p}}^{\perp })$. The original orbit $\gamma$ is represented by the solid line. Also drawn are two segments of the time-reversed orbit ${\gamma }^i$ (dashed lines) which yield the two closest intersection points. The corresponding “loop times” $t_{\mathrm{l}}$ are denoted by $t_{\mathrm{l}1}$ and $t_{\mathrm{l}2}$. The displacement vector $\delta \vec{y}=\delta \vec{y}({\mathbf{x}}_t^{\gamma },t_{\mathrm{l}})$ points from the original orbit to the intersection points.

Figure 4

PSS at ${\mathbf{x}}_t$ right after part $\mathcal{L}$ and before $\mathcal{R}$. The displacement vector $\delta \vec{y}$ points from the orbit $\gamma$ to its time-reversed version ${\gamma }^i$. The deviation of the partner orbit ${\gamma }^p$ from the original orbit $\gamma$ is described by the vector $\delta {\vec{x}}_{R,i}$. Shown is only a two-dimensional projection of the $(2f-2)$-dimensional PSS.

Figure 5

Sketch of the PSS as it is shifted along the periodic orbit $\gamma$ (solid line). Three pieces of the time-reversed orbit ${\gamma }^i$ are represented by dashed lines. If the PSS moves with the flow in phase space from ${\mathbf{x}}_{t1}^{\gamma }$ to ${\mathbf{x}}_{t2}^{\gamma }$ all intersection points of the PSS with ${\gamma }^i$ change their positions according to the linearized equations of motion (20). Note that not only ${\gamma }^i$ but also $\gamma$ itself could come close to ${\mathbf{x}}_{t1}^{\gamma }$ at a later time. However, we have not include this in the sketch above.

Figure 6

Schematic drawing of a projection of the Poincaré surface of section (PSS). The flow of intersection points (black filled circles) is represented by the thin arrows. For long enough time the unstable component $u_j$ grows while the stable component $s_j$ shrinks with $S_j=S_{\mathrm{cl}}{s}_j{u}_j$ being constant. There are two ways to count the intersection points. Either the flux through the $u_j=c$ surface (dotted line) is considered, as in Eq. (44), or one counts the number of points in the volume of the encounter region (dashed area) and normalizes that by the time each point spends in there, as in Eq. (45).