A Semiclassical Condition for Chaos Based on Pesin Theorem

Abstract

A semiclassical method to determine if the classical limit of a quantum system shows a chaotic behavior or not based on Pesin theorem, is presented. The method is applied to a phenomenological Gamow–type model and it is concluded that in the classical limit the dynamics exhibited by its effective Hamiltonian is chaotic.

Appendices

Appendix A: The Classical Quantity μ(B(k ₀,k ₁,...,k _n )) Expressed as a Quantum Mean Value

In order to evaluate the KS entropy, we have to generate the following partition

$$\begin{array}{@{}rcl@{}} B(-n)= \bigvee_{j=0}^{n} T^{-j} Q = \left\{ \bigcap_{j=0}^{n} T^{-j} A_{k_{j}} : A_{k_{j}} \in Q \right\}, \end{array} $$

If $B(k_{0},k_{1},...,k_{n})=\bigcap _{j=0}^{n} T^{-j} A_{k_{j}}$ is a generic element of B(−n), then the measure of B(k ₀,k ₁,...,k _n ) is

$$\begin{array}{@{}rcl@{}} &\mu(B(k_{0},k_{1},...,k_{n}))=\mu\left(\bigcap_{j=0}^{n} T^{-j} A_{k_{j}}\right)={\int}_{\!\!\bigcap_{j=0}^{n} T^{-j} A_{k_{j}}}d^{2(N+1)}\phi= {\int}_{\!\!{\Gamma}} I_{\bigcap_{j=0}^{n} T^{-j} A_{k_{j}}}(\phi) d^{2(N+1)}\phi\\ &={\int}_{\!\!{\Gamma}}{\prod}_{j=0}^{n} I_{A_{k_{j}}}(T^{j}\phi) d^{2(N+1)}\phi= \left\langle{\prod}_{j=0}^{n}I_{A_{k_{j}}}\circ T^{j}(\phi),I(\phi)\right\rangle= \left\langle symb\left({\prod}_{j=0}^{n}\widehat{I_{A_{k_{j}}}\circ T^{j}}\right),symb(\hat{I})\right\rangle \\ &=\left({\prod}_{j=0}^{n}\widehat{I_{A_{k_{j}}}\circ T^{j}}|\hat{I}\right)=\left({\prod}_{j=0}^{n}\hat{I}_{A_{k_{j}}}(j) |\hat{I}\right), \end{array} $$

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where we have used the following properties:

• The characteristic function of an intersection of sets is the product of the characteristic functions of each set.

• If T is bijective, then $I_{T^{-j}A_{k_{j}}}(\phi )= I_{A_{k_{j}}}(T^{j}\phi )$.

• If $\hbar \approx 0$, then $symb\left ({{\prod }_{j}^{n}}\hat {f}_{j}\right )(\phi ){\simeq {\prod }_{j}^{n}}f_{j}(\phi )$, where we have neglected terms of order $\mathcal {O}(\hbar )$). This property is the generalization of (5) for a product of n functions f _i .

• $\widehat {I_{A_{k_{j}}}\circ T^{j}}=\hat {I}_{A_{k_{j}}}(j)=\hat {U}(j)\hat {I}_{A_{k_{j}}}(0)\hat {U}(j)^{\dag }$, where $\hat {U}(j)=e^{-\frac {i}{\hbar }\hat {H}\alpha j}$ is the evolution operator and α is a real parameter which defines the time steps. This property is a consequence of the formula (6).

Appendix B: An Expansion for Operators $\hat {I}_{A_{k_{j}}}$

We consider a Hamiltonian of the form

$$\begin{array}{@{}rcl@{}} \hat{H}=\sum\limits_{r} z_{r} \vert r\rangle\langle\widetilde{r}|, \end{array} $$

where z _r =R e(z _r )+i I m(z _r ) are complex eigenvalues and $\{\vert r\rangle \},\{\langle \widetilde {s}|\}$ are its two sets of eigenvectors, left and righ respectively [33]. Then we have

$$\begin{array}{@{}rcl@{}} \hat{I}_{A_{k_{j}}}(0)=\sum\limits_{r,s}\alpha_{A_{k_{j}}}(r,s)\vert r\rangle\langle \widetilde{s}|. \end{array} $$

Therefore,

$$\begin{array}{@{}rcl@{}} &\hat{I}_{A_{k_{j}}}(j)=e^{-\frac{i}{\hbar}\hat{H} \alpha j} \left({\sum}_{r,s}\alpha_{A_{k_{j}}}(r,s) \vert r \rangle\langle \tilde {s} \vert\right)e^{\frac{i}{\hbar}\hat{H}^{\dag} \alpha j}= \\ &e^{-(\frac{i}{\hbar} {\sum}_{p} z_{p} \vert p \rangle\langle \widetilde{p}|)\alpha j}\left({\sum}_{r,s}\alpha_{A_{k_{j}}}(r,s) \vert r \rangle\langle \widetilde{s}| \right) e^{(\frac{i}{\hbar}{\sum}_{q} z_{q}^{\ast} \vert q \rangle\langle \widetilde{q}|)\alpha j} \\ &={\sum}_{p}{\sum}_{q}\alpha_{A_{k_{j}}}(p,q)e^{(-\frac{i}{\hbar}z_{p})\alpha j}e^{(\frac{i}{\hbar}z_{q}^{\ast})\alpha j}\vert p\rangle\langle\widetilde{q}|, \end{array} $$

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where we have used the exponential of an operator ($e^{\hat {A}}={\sum }_{k=0}^{\infty } \frac {\hat {A}^{k}}{k!}$) and the orthogonal relations of the projectors $\vert r \rangle \langle \widetilde {s} \vert$, that is

$$\begin{array}{@{}rcl@{}} &(\vert r \rangle\langle \widetilde {r} \vert)^{k}=\vert r \rangle\langle \widetilde {r} \vert , \,\ and \\ &\langle \widetilde {s} |r\rangle=0 \,\,\,\,\, if \,\,\,\,\, r\neq s. \end{array} $$

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