On the eigenvalues of a second order elliptic operator in an unbounded domain

Refined asymptotics for eigenvalues on domains of infinite measure

“…Finally, as in the previous proof, This result improves the upper bounds obtained in [1,8,9], which only gives an upper bound for N(λ, Ω) whenever g(x) = x −1/d . It would be desirable to obtain a better knowledge of the asymptotic behavior, namely, N(λ, Ω) ∼ cλ 1/2 f (λ 1/2 ) (for certain constant c) as in [16], and even a second term as in [19].…”

“…We refer the interested reader to [19,1,8,9,16] where a special class of sets in R N is considered (horn-shaped domains, an (N − 1)-dimensional set scaled in the other dimension). In [1,8,9], an upper bound for the growth of N(λ) was derived by using a trace estimate in the class of Hilbert-Schmidt operators, obtained with the aid of some inequalities for the Green function of an elliptic operator.…”

“…In [1,8,9], an upper bound for the growth of N(λ) was derived by using a trace estimate in the class of Hilbert-Schmidt operators, obtained with the aid of some inequalities for the Green function of an elliptic operator. In [16] the asymptotic behavior of eigenvalues was refined by using the Trotter product formula in order to obtain another trace estimate by generalizing the Golden-Thompson inequality, and in [19] were obtained more terms in the asymptotic expansion by exploiting certain self-similarity of the horns.…”