Kac-Akhiezer formula for normal integral operators (Q1077669)

Let \(T_ tf=\int^{t}_{0}k(x-y)f(y)dy\) be an integral operator acting in \(L_ 2(0,t)\) and let \(\{\lambda_ n\}^{\infty}_{n=1}\) denote the set of all eigenvalues of \(T_ t\). The authors prove the Kac-Akhiezer formula for \(\{\lambda_ n\}\) with kernel K(x) satisfying the following conditions: 1) K(x) is such that the operator \(T_ t\) is normal for all \(0\leq t<\infty\); 2) \(K(x)\in L_ 2^{loc}(0,\infty)\); 3) \(K(x)\in L_ 1(0,\infty)\). Under these assumptions if \(\lambda\) satisfies condition \(\lambda \lambda_ n\neq 1\) for all n then \[ \prod^{\infty}_{n=1}(1- \lambda \lambda_ n)\exp \lambda \lambda_ n=\exp (- \int^{t}_{0}h(0,s,\lambda)ds\quad) \] where \(h(x,t,\lambda)=\alpha (x,t,\lambda)-K(x)\) and \(\alpha\) (x,t,\(\lambda)\) is the unique solution of the integral equation \[ \alpha (x,t,\lambda)=\lambda K(x)+\lambda \int^{t}_{0}K(x-y)\alpha (y,t,\lambda)dy. \]