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

作者：Kaleta Kamil; Kwasnicki Mateusz^*; Malecki Jacek

来源：Journal of Mathematical Analysis and Applications, 2016, 439(2): 896-924.

DOI：10.1016/j.jmaa.2016.03.010

• Eigenvalues
• Interval
• Pseudo-differential operator
• Levy process
• Weyl-type law

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 xi psi'(xi) converges to infinity as xi -> infinity. This extends previous results obtained by the authors for the fractional Laplace operator (psi(xi) = xi(alpha/2)) and for the Klein-Gordon square root operator (psi(xi) = (1 + xi(2))(1/2) - 1). The formula for the eigenvalues in (-a, a) is of the form lambda(n) = psi(mu(2)(n)) + O(1/n), where mu(n) is the solution of mu(n) = 1/a v(mu(n)) and v(mu) is an element of [0, pi/2) is given as an integral involving psi.

• 出版日期2016-7-15