Systems of Partial Differential Equations for a Class of Operator Determinants

Abstract

We consider one-dimensional integral operators with kernel of the form $[K(x,y) = {\mkern 1mu} \frac{1}{{x - y}}\sum\limits_{{\alpha ,\beta = 1}}^{n} {{{c}_{{\alpha ,\beta {\kern 1pt} \varphi \beta }}}(x)} \varphi \alpha (y)$ acting on functions on the union of intervals $J{\mkern 1mu} = {\mkern 1mu} \bigcup\limits_{{k = 1}}^{m} {[{{a}_{{2k - 1,}}}{{a}_{{2k}}}].}$.