Integral solutions to Schlesinger equations
The paper discusses the parameterized linear system of differential equations of Fuchsian class \[ \frac{dy}{dz}=\left(\sum_{i=1}^{n}\frac{{{B}_{i}}(a)}{z-{{a}_{i}}}\right)y,\tag{1} \] where \({{a}_{1}},\dots ,{{a}_{n}}\) are parameters, \({{a}_{i}}\neq {{a}_{j}}(i\neq j),\mathbf a=({{a}_{1}},\dots ,{{a}_{n}}),\mathbf B=({{B}_{1}}(a),\dots ,{{B}_{n}}(a))\) is a tuple of \(p\times p\) matrix-valued functions. One needs to find such a tuple of functions \(\mathbf B\) for which the monodromy (Galois) group of system (1) will not depend on the parameters. It is well known if a tuple of functions \(\mathbf B\) satisfies the system of Schlesinger differential equations \[ d{{B}_{i}}(a)=-\sum_{j=1,i\neq j}^{n}{\frac{[{{B}_{i}}(a),{{B}_{j}}(a)]}{{{a}_{i}}-{{a}_{j}}}d({{a}_{i}}-{{a}_{j}})},\,i=1,\dots ,n,\tag{2} \] then \(\mathbf B\) is a solution of the above problem. Unfortunately, if \(n> 3\) and \(p> 1\), it is extremely difficult to find a solution of system (2). The author shows that if all matrices of \(\mathbf B\) have eigenvalues which do not dependent on the parameters and if for each eigenvalue the inequality \(\text{Re}\beta >-1/n(p-1)\) holds, then the system (1) reduces to a triangular form, and the system (2) is reduced to a Pfaffian system. Moreover, for \(p=2,3\) the solution of the problem can be obtained in closed form.
scientific article; zbMATH DE number 6534011