The notion of approximate eigenvalues applied to an integral equation of laser theory

The integral operator with kernel ${\left ( {i\eta /\pi } \right )^{1/2}}\exp \left [ { - i\eta {{\left ( {x - y} \right )}^2}} \right ]$ on the interval $\left | x \right |$, $\left | y \right | \le 1$ serves to model the behavior of a class of lasers. Although the kernel is simple, it is not Hermitian; this presents a major obstacle to a theoretical understanding of the equation—indeed, even the existence of eigenvalues is difficult to prove. We here introduce a definition of approximate eigenvalues and eigenfunctions, and argue that these will model the physical problem equally well. We then show that, for $\eta$ sufficiently large, each point $\lambda$ with $\left | \lambda \right | = 1$ is an approximate eigenvalue, and that the number of mutually orthogonal approximate eigenfunctions corresponding to $\lambda$ grows faster than any constant multiple of $\sqrt \eta$ This confirms a conjecture of J. A. Cochran and E. W. Hinds, supported by numerical evidence. In physical terms, it shows that for large Fresnel number the laser cannot be expected to settle to a single mode.