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

Research paper by Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki

Indexed on: 14 Mar '16Published on: 09 Mar '16Published in: Journal of Mathematical Analysis and Applications



Abstract

We prove a two-term Weyl-type asymptotic law, with error term <img height="20" border="0" style="vertical-align:bottom" width="38" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16002286-si1.gif">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 <img height="20" border="0" style="vertical-align:bottom" width="139" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16002286-si8.gif">λn=ψ(μn2)+O(1n), where μnμn is the solution of <img height="20" border="0" style="vertical-align:bottom" width="132" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16002286-si10.gif">μn=nπ2a−1aϑ(μn), and <img height="18" border="0" style="vertical-align:bottom" width="90" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16002286-si11.gif">ϑ(μ)∈[0,π2) is given as an integral involving ψ.