Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval

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We prove a two-term Weyl-type asymptotic law, with error term O(1/n), for the eigenvalues of the operator psi(-Delta) in an interval, with zero exterior condition, for complete Bernstein functions psi such that x psi'(x) converges to infinity as x goes to infinity. This extends previous results obtained by the authors for the fractional Laplace operator (psi(x) = xalpha/2) and for the Klein-Gordon square root operator (psi(x) = (1+x²)¹/² - 1). The formula for the eigenvalues in (-a,a) is of the form lambdaₙ = psi(muₙ²) + O(1/n), where muₙ is the solution of muₙ = (n pi)/(2 a) - theta(muₙ)/a, and theta(mu) is given as an integral involving psi.