Existence of Eigenvalues of a Class of Integral Equations Arising in Laser Theory

It is 'proved that the integral equation f G(x)F(xy)H(y)f(y) dy = X/(*) has at least one nonzero eigenvalue if F is any integral function of finite order, G and II are any bounded functions on [ -- 1,1], and the trace of the kernel G(x)F(xy)Il(y) does not vanish. In particular, this theorem f urnishes the first rigorous proof that the kernel exp [ik(x -- y)2], which arises in the theory of the gas laser, has an eigenvalue for arbitrary complex k.