Integration over angular variables for two coupled matrices (Q2759653)

The formula NEWLINE\[NEWLINE \int dU\exp\biggl(-\frac{1}{2t}\text{tr}(A-UA'U^{-1})^2\biggr)= NEWLINE\]NEWLINE NEWLINE\[NEWLINE =t^{n(n-1)/2}\biggl(\prod_{j=0}^{n-1}j!\biggr)(\Delta(x)\Delta(x'))^{-1} \det\left(\left[\exp\biggl(-\frac{1}{2t}(x_j-x_k')^2\biggr)\right]_{j,k=1,\dots ,n}\right) NEWLINE\]NEWLINE has been known for the last two decades [see \textit{C. Itzykson} and \textit{J. B. Zubr}, J. Math. Phys. 21, No. 3, 411-421 (1980)]. Here \(A\) and \(A'\) are \(n\times n\) complex Hermitian matrices having eigenvalues \(x=\{x_1,\dots ,x_n\}\) and \(x'=\{x'_1,\dots ,x'_n\}\) respectively, integration is over the \(n\times n\) complex unitary matrices \(U\) with the invariant Haar measure \(dU\) normalized as \(\int dU=1\). The function \(\Delta(x)\) is the product of differences of the \(x_j\). The authors generalize this formula to \(n\times n\) real symmetric or quaternion self-dual matrices. The integration is carried out over \(n\times n\) real orthogonal or quaternion symplectic matrices. They also show that this formula is related to the Calogero Hamiltonian. The integral can be explicitly computed for \(n=2\) and reduced to a single infinite sum for \(n=3\).NEWLINENEWLINEFor the entire collection see [Zbl 0967.00059].