We discuss the integral operator <i>A</i><sub><i>F</i></sub> which models the behavior of an unstable resonator having rectangular symmetry. Interest focuses on modes of the resonator that exhibit low loss for sizable values of the Fresnel number <i>F</i>; in the mathematical description, these correspond to eigenfunctions of <i>A</i><sub><i>F</i></sub> with eigenvalues of large modulus. A serious obstacle to a theoretical understanding of these entities is immediately presented by the non-Hermitian nature of the operator, which puts the very existence of eigenfunctions in doubt. Moreover, as the equation is hard to handle numerically, direct computational evidence has been sparse. In this paper we remove some of the conceptual difficulty by introducing a notion of approximate eigenvalues and eigenfunctions which, we argue, model the physical problem equally well. We then prove that each λ with |λ| = 1 becomes an approximate eigenvalue of <i>A</i><sub><i>F</i></sub> for <i>F</i> sufficiently large, and that the number of mutually orthogonal approximate eigenfunctions corresponding to λ grows at least linearly with √<i>F</i>. Interpreted in physical terms, this implies that for large Fresnel number, losses in the unstable resonator can be made arbitrarily low, but the resonator will not be capable of mode selection. These conclusions differ considerably from predictions contained in a recent paper on this subject.
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