We consider exterior elliptic problems with coefficients stabilizing at infinity and study the critical value β cr of the coupling constant (the coefficient at the potential) that separates operators with a discrete spectrum and those without it. The dependence of β cr on the boundary condition and on the distance between the boundary and the support of the potential is described. The discrete spectrum of a non-symmetric operator with the FKW boundary condition (that appears in diffusion processes with traps) is also investigated.

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