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We investigate the spectral properties of the Steklov problem for the modified Helmholtz equation (p-) u = 0 in the exterior of a compact set, for which the positive parameter p ensures exponential decay of the Steklov eigenfunctions at infinity. We obtain the small-p asymptotic behavior of the eigenvalues and eigenfunctions and discuss their features for different space dimensions. These results find immediate applications to the theory of stochastic processes and unveil the long-time asymptotic behavior of probability densities of various first-passage times in exterior domains. Theoretical results are validated by solving the exterior Steklov problem by a finite-element method with a transparent boundary condition.
Grebenkov et al. (Sat,) studied this question.