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

It is proved that the integral equation -₁^1G (x) F (xy) H (y) f (y) dy = f (x) has at least one nonzero eigenvalue if F is any integral function of finite order, G and H are any bounded functions on − 1, 1, and the trace of the kernel G (x) F (xy) H (y) does not vanish. In particular, this theorem furnishes 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.