cond-mat0502362.tex
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\begin{center}
Fluctuations of Quantum Statistical Two-Dimensional Systems of Electrons.
\end{center}
\begin{center}
Maciej M. Duras
\end{center}
\begin{center}
Institute of Physics, Cracow University of Technology, 
ulica Podchor\c{a}\.zych 1, PL-30084 Cracow, Poland
\end{center}

\begin{center}
Email: mduras @ riad.usk.pk.edu.pl
\end{center}

%\begin{center}
%"Nanoscale Fluctuations in Magnetic and Superconducting Systems";
%May 10, 2005 - May 14, 2005;
%Max Planck Institute for the Physics of Complex Systems,
%Dresden, Germany (2005).
%\end{center}

\begin{center}
AD 2005 February 15
\end{center}

\section{Abstract}
\label{sec-Abstract}
The random matrix ensembles (RME) of quantum statistical Hamiltonian operators, 
{\em e.g.} Gaussian random matrix ensembles
(GRME) and Ginibre random matrix ensembles (Ginibre RME), are applied 
to following quantum statistical 
systems: nuclear systems, molecular systems, 
and two-dimensional electron systems (Wigner-Dyson electrostatic analogy). 
Measures of quantum chaos and quantum integrability 
with respect to eigenergies of quantum systems are defined and calculated.
Quantum statistical information functional is defined as negentropy 
(either opposite of entropy or minus entropy).
The distribution function for the random matrix ensembles
is derived from the maximum entropy principle.


%The random matrix ensembles are applied to the quantum
%statistical two-dimensional systems of electrons.
%The quantum systems are studied using the finite
%dimensional real, complex and quaternion Hilbert spaces of the
%eigenfunctions.
%The linear operators describing the systems act on these Hilbert spaces
%and they are treated as random matrices in generic bases of the
%eigenfunctions.
%The random eigenproblems are presented and solved.
%Examples of random operators
%are presented with connection to physical problems.

%The Ginibre ensemble of nonhermitean random Hamiltonian
%matrices $K$ is considered.
%Each quantum system described by $K$ is a dissipative system 
%and the eigenenergies $Z_{i}$
%of the Hamiltonian are complex-valued random variables.
%The second difference of complex eigenenergies
%is viewed as discrete analog of Hessian with respect to labelling index.
%The results are considered in view of Wigner and Dyson's
%electrostatic analogy.
%An extension of space of dynamics of random magnitudes
%is performed by introduction of discrete space of labeling
%indices.
%The comparison with the Gaussian ensembles of random
%hermitean Hamiltonian matrices $H$ is performed. 

\section{Introduction}
\label{sec-introduction}

Random Matrix Theory RMT studies
quantum Hamiltonian operators $H$
which are random matrix variables.
Their matrix elements
$H_{ij}$ are independent random scalar variables
\cite{Haake 1990,Guhr 1998,Mehta 1990 0,Reichl 1992,Bohigas 1991,Porter 1965,Brody 1981,Beenakker 1997}.
There were studied among others the following
Gaussian Random Matrix ensembles GRME:
orthogonal GOE, unitary GUE, symplectic GSE,
as well as circular ensembles: orthogonal COE,
unitary CUE, and symplectic CSE.
The choice of ensemble is based on quantum symmetries
ascribed to the Hamiltonian $H$. The Hamiltonian $H$
acts on quantum space $V$ of eigenfunctions.
It is assumed that $V$ is $N$-dimensional Hilbert space
$V={\bf F}^{N}$, where the real, complex, or quaternion
field ${\bf F}={\bf R, C, H}$,
corresponds to GOE, GUE, or GSE, respectively.
If the Hamiltonian matrix $H$ is hermitean $H=H^{\dag}$,
then the probability density function of $H$ reads:
\begin{eqnarray}
& & f_{{\cal H}}(H)={\cal C}_{H \beta} 
\exp{[-\beta \cdot \frac{1}{2} \cdot {\rm Tr} (H^{2})]},
\label{pdf-GOE-GUE-GSE} \\
& & {\cal C}_{H \beta}=(\frac{\beta}{2 \pi})^{{\cal N}_{H \beta}/2}, 
\nonumber \\
& & {\cal N}_{H \beta}=N+\frac{1}{2}N(N-1)\beta, \nonumber \\
& & \int f_{{\cal H}}(H) dH=1,
\nonumber \\
& & dH=\prod_{i=1}^{N} \prod_{j \geq i}^{N} 
\prod_{\gamma=0}^{D-1} dH_{ij}^{(\gamma)}, \nonumber \\
& & H_{ij}=(H_{ij}^{(0)}, ..., H_{ij}^{(D-1)}) \in {\bf F}, \nonumber
\end{eqnarray}
where the parameter $\beta$ assume values 
$\beta=1,2,4,$  for GOE($N$), GUE($N$), GSE($N$), respectively,
and ${\cal N}_{H \beta}$ is number of independent matrix elements
of hermitean Hamiltonian $H$.
The Hamiltonian $H$ belongs to Lie group of hermitean $N \times N {\bf F}$-matrices,
and the matrix Haar's measure $dH$ is invariant under
transformations from the unitary group U($N$, {\bf F}).
The eigenenergies $E_{i}, i=1, ..., N$, of $H$, are real-valued
random variables $E_{i}=E_{i}^{\star}$.
It was Eugene Wigner who firstly dealt with eigenenergy level repulsion
phenomenon studying nuclear spectra \cite{Haake 1990,Guhr 1998,Mehta 1990 0}.
RMT is applicable now in many branches of physics:
nuclear physics (slow neutron resonances, highly excited complex nuclei),
condensed phase physics (fine metallic particles,  
random Ising model [spin glasses]),
quantum chaos (quantum billiards, quantum dots), 
disordered mesoscopic systems (transport phenomena),
quantum chromodynamics, quantum gravity, field theory.

\section{The Ginibre ensembles}
\label{sec-ginibre-ensembles}

Jean Ginibre considered another example of GRME
dropping the assumption of hermiticity of Hamiltonians
thus defining generic ${\bf F}$-valued Hamiltonian $K$
\cite{Haake 1990,Guhr 1998,Ginibre 1965,Mehta 1990 1}.
Hence, $K$ belong to general linear Lie group GL($N$, {\bf F}),
and the matrix Haar's measure $dK$ is invariant under
transformations form that group.
The distribution of $K$ is given by:
\begin{eqnarray}
& & f_{{\cal K}}(K)={\cal C}_{K \beta} 
\exp{[-\beta \cdot \frac{1}{2} \cdot {\rm Tr} (K^{\dag}K)]},
\label{pdf-Ginibre} \\
& & {\cal C}_{K \beta}=(\frac{\beta}{2 \pi})^{{\cal N}_{K \beta}/2}, 
\nonumber \\
& & {\cal N}_{K \beta}=N^{2}\beta, \nonumber \\
& & \int f_{{\cal K}}(K) dK=1,
\nonumber \\
& & dK=\prod_{i=1}^{N} \prod_{j=1}^{N} 
\prod_{\gamma=0}^{D-1} dK_{ij}^{(\gamma)}, \nonumber \\
& & K_{ij}=(K_{ij}^{(0)}, ..., K_{ij}^{(D-1)}) \in {\bf F}, \nonumber
\end{eqnarray}
where $\beta=1,2,4$, stands for real, complex, and quaternion
Ginibre ensembles, respectively.
Therefore, the eigenenergies $Z_{i}$ of quantum system 
ascribed to Ginibre ensemble are complex-valued random variables.
The eigenenergies $Z_{i}, i=1, ..., N$,
of nonhermitean Hamiltonian $K$ are not real-valued random variables
$Z_{i} \neq Z_{i}^{\star}$.
Jean Ginibre postulated the following
joint probability density function 
of random vector of complex eigenvalues $Z_{1}, ..., Z_{N}$
for $N \times N$ Hamiltonian matrices $K$ for $\beta=2$
\cite{Haake 1990,Guhr 1998,Ginibre 1965,Mehta 1990 1}:
\begin{eqnarray}
& & P(z_{1}, ..., z_{N})=
\label{Ginibre-joint-pdf-eigenvalues} \\
& & =\prod _{j=1}^{N} \frac{1}{\pi \cdot j!} \cdot
\prod _{i<j}^{N} \vert z_{i} - z_{j} \vert^{2} \cdot
\exp (- \sum _{j=1}^{N} \vert z_{j}\vert^{2}),
\nonumber
\end{eqnarray}
where $z_{i}$ are complex-valued sample points
($z_{i} \in {\bf C}$).
 
We emphasize here Wigner and Dyson's electrostatic analogy.
A Coulomb gas of $N$ unit charges moving on complex plane (Gauss's plane)
{\bf C} is considered. The vectors of positions
of charges are $z_{i}$ and potential energy of the system is:
\begin{equation}
U(z_{1}, ...,z_{N})=
- \sum_{i<j} \ln \vert z_{i} - z_{j} \vert
+ \frac{1}{2} \sum_{i} \vert z_{i}^{2} \vert. 
\label{Coulomb-potential-energy}
\end{equation}
If gas is in thermodynamical equilibrium at temperature
$T= \frac{1}{2 k_{B}}$ 
($\beta= \frac{1}{k_{B}T}=2$, $k_{B}$ is Boltzmann's constant),
then probability density function of vectors of positions is 
$P(z_{1}, ..., z_{N})$ Eq. (\ref{Ginibre-joint-pdf-eigenvalues}).
Therefore, complex eigenenergies $Z_{i}$ of quantum system 
are analogous to vectors of positions of charges of Coulomb gas.
Moreover, complex-valued spacings $\Delta^{1} Z_{i}$
of complex eigenenergies of quantum system:
\begin{equation}
\Delta^{1} Z_{i}=Z_{i+1}-Z_{i}, i=1, ..., (N-1),
\label{first-diff-def}
\end{equation}
are analogous to vectors of relative positions of electric charges.
Finally, complex-valued
second differences $\Delta^{2} Z_{i}$ of complex eigenenergies:
\begin{equation}
\Delta ^{2} Z_{i}=Z_{i+2} - 2Z_{i+1} + Z_{i}, i=1, ..., (N-2),
\label{Ginibre-second-difference-def}
\end{equation}
are analogous to
vectors of relative positions of vectors
of relative positions of electric charges.

The eigenenergies $Z_{i}=Z(i)$ can be treated as values of function $Z$
of discrete parameter $i=1, ..., N$.
The "Jacobian" of $Z_{i}$ reads:
\begin{equation}
{\rm Jac} Z_{i}= \frac{\partial Z_{i}}{\partial i}
\simeq \frac{\Delta^{1} Z_{i}}{\Delta^{1} i}=\Delta^{1} Z_{i}.
\label{jacobian-Z}
\end{equation}
We readily have, that the spacing is an discrete analog of Jacobian,
since the indexing parameter $i$ belongs to discrete space
of indices $i \in I=\{1, ..., N \}$. Therefore, the first derivative
with respect to $i$ reduces to the first differential quotient.
The Hessian is a Jacobian applied to Jacobian.
We immediately have the formula for discrete "Hessian" for the eigenenergies $Z_{i}$:
\begin{equation}
{\rm Hess} Z_{i}= \frac{\partial ^{2} Z_{i}}{\partial i^{2}}
\simeq \frac{\Delta^{2} Z_{i}}{\Delta^{1} i^{2}}=\Delta^{2} Z_{i}.
\label{hessian-Z}
\end{equation}
Thus, the second difference of $Z$ is discrete analog of Hessian of $Z$.
One emphasizes that both "Jacobian" and "Hessian"
work on discrete index space $I$ of indices $i$.
The spacing is also a discrete analog of energy slope
whereas the second difference corresponds to
energy curvature with respect to external parameter $\lambda$
describing parametric ``evolution'' of energy levels
\cite{Zakrzewski 1,Zakrzewski 2}.
The finite differences of order higher than two
are discrete analogs of compositions of "Jacobians" with "Hessians" of $Z$.


The eigenenergies $E_{i}, i \in I$, of the hermitean Hamiltonian $H$
are ordered increasingly real-valued random variables.
They are values of discrete function $E_{i}=E(i)$.
The first difference of adjacent eigenenergies is:
\begin{equation}
\Delta^{1} E_{i}=E_{i+1}-E_{i}, i=1, ..., (N-1),
\label{first-diff-def-GRME}
\end{equation}
are analogous to vectors of relative positions of electric charges
of one-dimensional Coulomb gas. It is simply the spacing of two adjacent
energies.
Real-valued
second differences $\Delta^{2} E_{i}$ of eigenenergies:
\begin{equation}
\Delta ^{2} E_{i}=E_{i+2} - 2E_{i+1} + E_{i}, i=1, ..., (N-2),
\label{Ginibre-second-difference-def-GRME}
\end{equation}
are analogous to vectors of relative positions 
of vectors of relative positions of charges of one-dimensional
Coulomb gas.
The $\Delta ^{2} Z_{i}$ have their real parts
${\rm Re} \Delta ^{2} Z_{i}$,
and imaginary parts
${\rm Im} \Delta ^{2} Z_{i}$, 
as well as radii (moduli)
$\vert \Delta ^{2} Z_{i} \vert$,
and main arguments (angles) ${\rm Arg} \Delta ^{2} Z_{i}$.
$\Delta ^{2} Z_{i}$ are extensions of real-valued second differences:
\begin{equation}
\Delta^{2} E_{i}=E_{i+2}-2E_{i+1}+E_{i}, i=1, ..., (N-2),
\label{second-diff-def}
\end{equation}
of adjacent ordered increasingly real-valued eigenenergies $E_{i}$
of Hamiltonian $H$ defined for
GOE, GUE, GSE, and Poisson ensemble PE
(where Poisson ensemble is composed of uncorrelated
randomly distributed eigenenergies)
\cite{Duras 1996 PRE,Duras 1996 thesis,Duras 1999 Phys,Duras 1999 Nap,Duras 2000 JOptB,Duras 2001 Vaxjo,Duras 2001
Pamplona, Duras 2003 Spie03, Duras 2004 Spie04}.
The Jacobian and Hessian operators of energy function $E(i)=E_{i}$
for these ensembles read:
\begin{equation}
{\rm Jac} E_{i}= \frac{\partial E_{i}}{\partial i}
\simeq \frac{\Delta^{1} E_{i}}{\Delta^{1} i}=\Delta^{1} E_{i},
\label{jacobian-E}
\end{equation}
and
\begin{equation}
{\rm Hess} E_{i}= \frac{\partial ^{2} E_{i}}{\partial i^{2}}
\simeq \frac{\Delta^{2} E_{i}}{\Delta^{1} i^{2}}=\Delta^{2} E_{i}.
\label{hessian-E}
\end{equation}
The treatment of first and second differences of eigenenergies
as discrete analogs of Jacobians and Hessians
allows one to consider these eigenenergies as a magnitudes 
with statistical properties studied in discrete space of indices.
The labelling index $i$ of the eigenenergies is
an additional variable of "motion", hence the space of indices $I$
augments the space of dynamics of random magnitudes.

One may also study the finite expressions of random eigenenergies
and their distributions. The finite expressions are more general than finite
difference quotients and they represent the derivatives of eigenenergies
with respect to labelling index $i$ more accurately
\cite{Collatz 1955, Collatz 1960}.  
 
\section{The Maximum Entropy Principle}
\label{sec-maximal-entropy}
In order to derive the probability distribution
in matrix space  we apply
the maximum entropy principle:
\begin{equation}
{\rm max} \{S_{\beta}(f_{{\cal X}}): \left< 1 \right>=1, 
\left< {\cal H}_{{\cal X}} \right>=U_{\beta} \},
\label{maximal-entropy-problem}
\end{equation}
which yields:
\begin{equation}
{\rm max} \{S_{\beta}(f_{{\cal X}}): 
\int f_{{\cal X}}(X) d X=1,
\int {\cal H}_{{\cal X}}(X) f_{{\cal X}}(X) d X=U_{\beta} \},
\label{maximal-entropy-problem-equivalent}
\end{equation}
where $X=H$ or $X=K$ for Gaussian or Ginibre ensembles, respectively, 
and ${\cal H}_{{\cal X}}(X)= \frac{1}{2} {\rm Tr} (X^{\dag}X)$. 
The maximization of entropy 
$S_{\beta}(f_{{\cal X}})=\int (- k_{B} \ln f_{{\cal X}}(X)) f_{{\cal X}}(X) d X$ 
under two additional
constraints of normalization of the probability density function,
and of equality of its first momentum and intrinsic energy,
is equivalent to the minimization of the following functional
${\cal F}(f_{{\cal X}})$ with the use of Lagrange multipliers 
$\alpha_{1}, \beta_{1}$: 
\begin{eqnarray}
& & {\rm min} \{ {\cal F} (f_{{\cal X}}) \},
\label{maximal-entropy-problem-Lagrange} \\
& & {\cal F} (f_{{\cal X}}) 
= \int ( k_{B} \ln f_{{\cal X}}(X)) f_{{\cal X}}(X) d X
+\alpha_{1} \int f_{{\cal X}}(X) d X \nonumber \\
& & + \beta_{1} \int {\cal H}_{{\cal X}}(X) f_{{\cal X}}(X) d X. \nonumber
\end{eqnarray}
It follows, that the first variational derivative of ${\cal F}(f_{{\cal X}})$
must vanish:
\begin{equation}
\frac{\delta {\cal F} (f_{{\cal X}})}{\delta f_{{\cal X}}}=0,
\label{Lagrange-first-derivative}
\end{equation}
which produces:
\begin{equation}
k_{B} (\ln f_{{\cal X}}(X) + 1) 
+\alpha_{1} + \beta_{1} {\cal H}_{{\cal X}}(X)=0,
\label{Lagrange-integrand}
\end{equation}
and equivalently:
\begin{eqnarray}
& & f_{{\cal X}}(X)={\cal C}_{X \beta}  \cdot
\exp{[-\beta \cdot {\cal H}_{{\cal X}}(X)]}
\label{pdf-GOE-GUE-GSE-PH-partition-function-Lagrange} \\
& & {\cal C}_{X \beta}= \exp[ -(\alpha_{1}+1) \cdot k_{B}^{-1}],
\beta=\beta_{1} \cdot k_{B}^{-1}.
\nonumber
\end{eqnarray}
The variational principle of maximum entropy does not
force additional condition on functional form 
of ${\cal H}_{{\cal X}}(X)$. 
The quantum statistical information functional $I_{\beta}$ 
is the opposite of entropy:
\begin{equation}
I_{\beta}(f_{{\cal X}})=-S_{\beta}(f_{{\cal X}})
= \int ( + k_{B} \ln f_{{\cal X}}(X)) f_{{\cal X}}(X) d X.
\label{information-entropy}
\end{equation}
Information is negentropy, and entropy is neginformation.
The maximum entropy principle is equivalent to
minimum information principle. 

\begin{thebibliography}{99}
\bibitem{Haake 1990}
 F. Haake, {\em Quantum Signatures of Chaos} 
 (Springer-Verlag, Berlin Heidelberg New York 1990), 
 Chapters 1, 3, 4, 8, pp. 1-11, 33-77, 202-213.
\bibitem{Guhr 1998} 
 T. Guhr, A. M\"uller-Groeling, H. A. Weidenm\"uller: 
 {\em Phys. Rept.} {\bf 299}, 189-425 (1998) (cond-mat/9707301). 
\bibitem{Mehta 1990 0}
 M. L. Mehta, {\em Random matrices} 
 (Academic Press, Boston 1990), Chapters 1, 2, 9, pp. 1-54, 182-193.
\bibitem{Reichl 1992}
 L. E. Reichl, {\em The Transition to Chaos In Conservative Classical 
 Systems: Quantum Manifestations} (Springer-Verlag, New York, 1992), 
 Chapter 6, pp. 248-286.
\bibitem{Bohigas 1991}
 O. Bohigas, in: {\em Proceedings of the Les Houches Summer School on Chaos and
 Quantum Physics, Session LII, 1 - 31 August 1989 
 [LES HOUCHES \'ECOLE D'\'ET\'E DE PHYSIQUE TH\'EORIQUE, 
 SESSION LII, 1 - 31 AO\^UT 1989]; 
 M. - J. Giannoni, A. Voros, J. Zinn-Justin, Eds.} 
 (North-Holland, Amsterdam, 1991), pp. 87-199.
\bibitem{Porter 1965}
 C. E. Porter, {\em Statistical Theories of Spectra: Fluctuations} 
 (Academic Press, New York, 1965).  
\bibitem{Brody 1981}
 T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, S. S. M. Wong, 
 {\em Rev. Mod. Phys.} {\bf 53}, 385-479 (1981).
\bibitem{Beenakker 1997}
 C. W. J. Beenakker, 
 {\em Rev. Mod. Phys.} {\bf 69}, 731-808 (1997).  
\bibitem{Ginibre 1965} 
 J. Ginibre,  {\em J. Math. Phys.} {\bf 6}, 440-449 (1965). 
\bibitem{Mehta 1990 1}
 M. L. Mehta, {\em Random matrices} 
 (Academic Press, Boston, 1990), Chapter 15, pp. 294-310.
\bibitem{Zakrzewski 1}
 J. Zakrzewski, D. Delande, 
 {\em Phys. Rev.} E {\bf 47}, 1650-1664 (1993).
\bibitem{Zakrzewski 2}
 J. Zakrzewski, D. Delande, M. Ku\'s, 
 {\em Phys. Rev.} E {\bf 47}, 1665-1676 (1993). 
\bibitem{Duras 1996 PRE}
 M. M. Duras, K. Sokalski, 
 {\em Phys. Rev.} E {\bf 54}, 3142-3148 (1996). 
\bibitem{Duras 1996 thesis}
 M. M. Duras, {\em Finite difference and finite
 element distributions in statistical theory
 of energy levels in quantum systems}
 (PhD thesis, Jagellonian University, Cracow, 1996). 
\bibitem{Duras 1999 Phys}
 M. M. Duras, K. Sokalski,
 {\em Physica} {\bf D125}, 260-274 (1999).
\bibitem{Duras 1999 Nap}
 M. M. Duras,
 {\em Description of Quantum Systems 
 by Random Matrix Ensembles of Large Dimensions}, 
 in: {\it Proceedings of the Sixth International 
 Conference on Squeezed States and Uncertainty Relations,
 24 May-29 May 1999, Naples, Italy} 
 (NASA, Greenbelt, Maryland, at press 2000) (cond-mat/0211676).
\bibitem{Duras 2000 JOptB}
 M. M. Duras,  
{\em J. Opt. B: Quantum Semiclass. Opt.} {\bf 2}, 287-291 (2000)
(cond-mat/0211667).   
\bibitem{Duras 2001 Vaxjo}
 M. M. Duras,
 {\em Discrete Hessians in study of Quantum Statistical Systems: Complex 
 Ginibre Ensemble}, 
 in: {\em QP-PQ: Quantum Probability and White Noise Analysis - Volume 13, 
 Foundations of Probability and Physics, Proceedings of the Conference, 
 V\"axj\"o, Sweden, 25 November - 1 December 2000; A. Khrennikov, Ed.}  
 (World Scientific Publishers, Singapore, 2001), 
 Vol. {\bf 13}, pp. 115-120 [ISBN 981-02-4846-6] (cond-mat/0212453). 
\bibitem{Duras 2001 Pamplona}
 M. M. Duras,
 {\em Complex-valued second difference as a measure of stabilization 
 of complex dissipative statistical systems: Girko ensemble}, 
 in: {\em Space-time chaos: Characterization, control and synchronization; 
 Proceedings of the International Interdisciplinary School, Pamplona, Spain, 
 June 19-23, 2000; S. Boccaletti, J. Burguete, W. Gonz\'alez-Vi\~nas, 
 D. L. Valladares, Eds.}  
 (World Scientific Publishers, Singapore, 2001), 
 pp. 45-52 [ISBN 981-02-4506-8] (cond-mat/0212455).
\bibitem{Duras 2003 Spie03}
 M. M. Duras,
 {\em Quantum fluctuations and random matrix theory}, 
 in: {\em Fluctuations and Noise in Photonics and Quantum Optics, 
 D. Abbott, J. H. Shapiro, Y. Yamamoto, Eds.}  
 (Proceedings of SPIE; Society of Photo-optical Instrumentation Engineers, 
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 Vol. {\bf 5111}, pp. 456-459 [ISBN 981-02-4506-8] (cond-mat/0212455).
\bibitem{Duras 2004 Spie04}
 M. M. Duras, 
 {\em Quantum statistical information functional for random matrix ensembles}, 
 in: {\em Noise in Complex Systems and Stochastic Dynamics II, 
 Z. Gingl, J. M. Sancho, L. Schimansky-Geier, J. Kertesz, Eds.}
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\bibitem{Collatz 1955}
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 (Springer-Verlag, Berlin, G\"ottingen, Heidelberg, 1955), 
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\bibitem{Collatz 1960}
 L. Collatz, {\it The Numerical Treatment of Differential Equations},
 (Springer-Verlag, Berlin, Heidelberg, New York, 1960), 
 Chapters III, V, pp. 160-173, 378-380, Appendix, Table III, pp. 537-539.
 
\end{thebibliography}

%}

\end{document}












cond-mat0502362.extracted_bib
\begin{thebibliography}{99}
\bibitem{Haake 1990}
 F. Haake, {\em Quantum Signatures of Chaos} 
 (Springer-Verlag, Berlin Heidelberg New York 1990), 
 Chapters 1, 3, 4, 8, pp. 1-11, 33-77, 202-213.
\bibitem{Guhr 1998} 
 T. Guhr, A. M\"uller-Groeling, H. A. Weidenm\"uller: 
 {\em Phys. Rept.} {\bf 299}, 189-425 (1998) (cond-mat/9707301). 
\bibitem{Mehta 1990 0}
 M. L. Mehta, {\em Random matrices} 
 (Academic Press, Boston 1990), Chapters 1, 2, 9, pp. 1-54, 182-193.
\bibitem{Reichl 1992}
 L. E. Reichl, {\em The Transition to Chaos In Conservative Classical 
 Systems: Quantum Manifestations} (Springer-Verlag, New York, 1992), 
 Chapter 6, pp. 248-286.
\bibitem{Bohigas 1991}
 O. Bohigas, in: {\em Proceedings of the Les Houches Summer School on Chaos and
 Quantum Physics, Session LII, 1 - 31 August 1989 
 [LES HOUCHES \'ECOLE D'\'ET\'E DE PHYSIQUE TH\'EORIQUE, 
 SESSION LII, 1 - 31 AO\^UT 1989]; 
 M. - J. Giannoni, A. Voros, J. Zinn-Justin, Eds.} 
 (North-Holland, Amsterdam, 1991), pp. 87-199.
\bibitem{Porter 1965}
 C. E. Porter, {\em Statistical Theories of Spectra: Fluctuations} 
 (Academic Press, New York, 1965).  
\bibitem{Brody 1981}
 T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, S. S. M. Wong, 
 {\em Rev. Mod. Phys.} {\bf 53}, 385-479 (1981).
\bibitem{Beenakker 1997}
 C. W. J. Beenakker, 
 {\em Rev. Mod. Phys.} {\bf 69}, 731-808 (1997).  
\bibitem{Ginibre 1965} 
 J. Ginibre,  {\em J. Math. Phys.} {\bf 6}, 440-449 (1965). 
\bibitem{Mehta 1990 1}
 M. L. Mehta, {\em Random matrices} 
 (Academic Press, Boston, 1990), Chapter 15, pp. 294-310.
\bibitem{Zakrzewski 1}
 J. Zakrzewski, D. Delande, 
 {\em Phys. Rev.} E {\bf 47}, 1650-1664 (1993).
\bibitem{Zakrzewski 2}
 J. Zakrzewski, D. Delande, M. Ku\'s, 
 {\em Phys. Rev.} E {\bf 47}, 1665-1676 (1993). 
\bibitem{Duras 1996 PRE}
 M. M. Duras, K. Sokalski, 
 {\em Phys. Rev.} E {\bf 54}, 3142-3148 (1996). 
\bibitem{Duras 1996 thesis}
 M. M. Duras, {\em Finite difference and finite
 element distributions in statistical theory
 of energy levels in quantum systems}
 (PhD thesis, Jagellonian University, Cracow, 1996). 
\bibitem{Duras 1999 Phys}
 M. M. Duras, K. Sokalski,
 {\em Physica} {\bf D125}, 260-274 (1999).
\bibitem{Duras 1999 Nap}
 M. M. Duras,
 {\em Description of Quantum Systems 
 by Random Matrix Ensembles of Large Dimensions}, 
 in: {\it Proceedings of the Sixth International 
 Conference on Squeezed States and Uncertainty Relations,
 24 May-29 May 1999, Naples, Italy} 
 (NASA, Greenbelt, Maryland, at press 2000) (cond-mat/0211676).
\bibitem{Duras 2000 JOptB}
 M. M. Duras,  
{\em J. Opt. B: Quantum Semiclass. Opt.} {\bf 2}, 287-291 (2000)
(cond-mat/0211667).   
\bibitem{Duras 2001 Vaxjo}
 M. M. Duras,
 {\em Discrete Hessians in study of Quantum Statistical Systems: Complex 
 Ginibre Ensemble}, 
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