Let P be a symmetric 2a-order classical strongly elliptic pseudodifferential operator with even symbol p(x, ξ) on Rⁿ (0 < a < 1), for example a perturbation of (−Δ)^a. Let Ω ⊂ Rⁿ be bounded, and let P_D be the Dirichlet realization in L₂(Ω) defined under the exterior condition u = 0 in Rⁿ \ Ω. When p(x, ξ) and Ω are C^∞, it is known that the eigenvalues λ_j (ordered in a nondecreasing sequence for j → ∞) satisfy a Weyl asymptotic formula λ_j (P_D) = C(P, Ω)j^2a/n + o(j^2a/n) for j → ∞, with C(P, Ω) determined from the principal symbol of P. We now show that this result is valid for more general operators with a possibly nonsmooth x-dependence, over Lipschitz domains, and that it extends to P^∼ = P + P^′ + P^′′, where P^′ is an operator of order < min{2a, a + ₂¹ } with certain mapping properties, and P^′′ is bounded in L₂(Ω) (e.g. P^′′ = V (x) ∈ L_∞(Ω)). Also the regularity of eigenfunctions of P_D is discussed.