Higher dimensional nonclassical eigenvalue asymptotics

In this article, we extend Simon's construction and results [B. Simon, J. Funct. Anal. 53(1), 84-98 (1983)] for leading order eigenvalue asymptotics to n-dimensional Schrodinger operators with non-confining potentials given by H-n(alpha) = -Delta + Pi(n)(i=1) vertical bar xi vertical bar(alpha i) on R-n (n > 2), alpha := (alpha(1), ..., alpha(n)) is an element of (R+*)(n). We apply the results to also derive the leading order spectral asymptotics in the case of the Dirichlet Laplacian -Delta(D) on domains Omega(alpha)(n) = {x is an element of R-n : Pi(n)(j=1) vertical bar x(j)vertical bar(alpha j/alpha n) < 1}. (C) 2015 AIP Publishing LLC.