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

We prove a two-term Weyl-type asymptotic law, with error term O(1n), for the eigenvalues of the operator ψ(−Δ)ψ(−Δ) in an interval, with zero exterior condition, for complete Bernstein functions ψ   such that ξψ′(ξ)ξψ′(ξ) converges to infinity as ξ→∞ξ→∞. This extends previous results obtained by the authors for the fractional Laplace operator (ψ(ξ)=ξα/2ψ(ξ)=ξα/2) and for the Klein–Gordon square root operator (ψ(ξ)=(1+ξ2)1/2−1ψ(ξ)=(1+ξ2)1/2−1). The formula for the eigenvalues in (−a,a)(−a,a) is of the form λn=ψ(μn2)+O(1n), where μnμn is the solution of μn=nπ2a−1aϑ(μn), and ϑ(μ)∈[0,π2) is given as an integral involving ψ.