Asymptotics of the eigenvalues of operators with a Poisson algebra of symmetries in the highest symbol
Let on \({\mathfrak X}\), \({\mathcal R}^ n_ q\oplus {\mathcal R}^ n_ p\) (n\(\geq 2)\) the action of the group \(R_+\) be given such that dp\(\wedge dq\), \(H_ i\) are homogeneous of the degrees 1, L-i respectively, \(H_ i\) real valued. Let \({\mathcal H}=H(q,-i\partial_ q)\) have the symbol \(H=H_ 0+H_ 1+H_ 2\) and be essentially self-adjoint in \(L^ 2\) and on the range of \(H_ 0|_{{\mathfrak X}}\) the spectrum of H be discrete. Let \(H_ 0\) be an integrable symbol and satisfy a certain non-resonance condition. Then H has an eigenvalue series of degree m provided \(H_ 1\), \(H_ 2\) satisfy certain conditions; here \(m=1,2,4\) depends on these conditions. In particular, \({\mathcal H}=-\Delta +V(q)\) on \({\mathcal S}^ n\) with an odd potential V(q) has a fourth order eigenvalue series \(E(k)=k^ 2+k(n-1)+c_ 1k^{-2}+c_ 2k^{-3}.\)
scientific article; zbMATH DE number 3901703