Quantum states of monodromy groups (Q1972785)

Consider systems in the form \[ dx/dt= A(t)X,\tag{1} \] with \(t\in\mathbb{C} P^1\), and \(A(t)= \sum^{p+ 1}_{j=1} A_j/(t- a_j)\), \(A_j\) being constant \(n\times n\)-matrices satisfying \(\sum_j A_j= 0\). Denote by \(\lambda_{\nu,j}\) the eigenvalues of \(A_j\); they satisfy \[ \sum_\nu \sum_j \lambda_{\nu,j}= 0.\tag{2} \] Let \(M_j\in \text{GL}(n, \mathbb{C})\) be the local monodromy around \(a_j\). Definitions. Fix \(\{M_1,\dots, M_p\}\) and denote by \(\sigma_{\nu,j}\) the eigenvalues of \(M_j\). Let \(\{\lambda_{\nu,j}\}\) be a set of complex numbers satisfying (2) and such that \(\exp(2\pi i\lambda_{\nu,j})= \sigma_{\nu, j}\). Call two such sets \(\{\lambda^s_{\nu, j}\}\), \(s= 1,2\), equivalent if there exist integers \(l_1,\dots, l_{p+ 1}\), \(l_1+\cdots+ l_{p+ 1}= 0\), such that \(\lambda^1_{\nu, j}= \lambda^2_{\nu, j}+ l_j\). A quantum state of the fixed monodromy group is a class of equivalent sets \(\{\lambda_{\nu, j}\}\). Any such set is called its representative. For a fixed irreducible monodromy group and a set of poles \(a_1,\dots, a_{p+ 1}\), a quantum state is called forbidden if there is no Fuchsian system with poles at \(a_j\) and only those whose monodromy group is the given one and whose eigenvalues of the matrices-residues \(A_j\) are equal to \(\lambda_{\nu, j}\). A non-forbidden quantum state is called admissible. For fixed eigenvalues of the matrices \(A_i\), \(i\neq j\), consider all quantum states obtained by changing the eigenvalues only of \(A_j\). They are called \(a_j\)-quantum states. Forbidden and admissible \(a_j\)-quantum states are defined in an obvious way. The main result reads: when the monodromy group of (1) is irreducible, the author gives a necessary and sufficient condition on (1) to have infinitely many forbidden \(a_j\)-quantum states.