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A real $2n\times 2n$ Hamiltonian matrix has the general form

$$H=\begin{pmatrix} A & B \cr C & -A^T \end{pmatrix}$$

where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are symmetric. Are there any results regarding the eigenvalue distribution of an ensemble of such matrices? For example, the above condition is equivalent to the symmetry of $JH$ with

$$J=\begin{pmatrix} 0 & 1 \cr -1 & 0 \end{pmatrix},$$

so it would be natural to consider the Gaussian orthogonal ensemble for $JH$. Generically, the eigenvalues come in quadruples: $\lambda, -\lambda, \bar\lambda, -\bar\lambda$

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# Eigenvalues of random Hamiltonian matrices

A real $2n\times 2n$ Hamiltonian matrix has the general form

$$H=\begin{pmatrix} A & B \cr C & -A^T \end{pmatrix}$$

where $A$, $B$ and $C$ are $n\times n$ matrices. Are there any results regarding the eigenvalue distribution of an ensemble of such matrices? For example, the above condition is equivalent to the symmetry of $JH$ with

$$J=\begin{pmatrix} 0 & 1 \cr -1 & 0 \end{pmatrix},$$

so it would be natural to consider the Gaussian orthogonal ensemble for $JH$. Generically, the eigenvalues come in quadruples: $\lambda, -\lambda, \bar\lambda, -\bar\lambda$