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

It is proved that the integral equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∫₋₁¹G(x)F(xy)H(y)f(y)dy = λ f(x)</tex> 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.